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Artykuły w czasopismach na temat „Orthogonality condition”

Autor: Grafiati

Data publikacji: 3 czerwca 2025

Data aktualizacji: 31 lipca 2025

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1

Kawahigashi, K., and M. Ichimura. "Non-Orthogonality Problem in Continuum RPA Studied by Orthogonality Condition." Progress of Theoretical Physics 85, no. 4 (1991): 829–38. http://dx.doi.org/10.1143/ptp/85.4.829.

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2

Li, Zheng, Tie Zhang, and Changjun Li. "Trapezoid Orthogonality in Complex Normed Linear Spaces." Mathematics 13, no. 9 (2025): 1494. https://doi.org/10.3390/math13091494.

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Let Gp(x,y,z)=∥x+y+z∥p+∥z∥p−∥x+z∥p−∥y+z∥p be defined on a normed space X. The special case G2(x,y,z)=0,∀z∈X, where X is a real normed linear space, coincides with the trapezoid orthogonality (T-orthogonality), which was originally proposed by Alsina et al. in 1999. In this paper, for the case where X is a complex inner product space endowed with the inner product ⟨·,·⟩ and induced norm ∥·∥, it is proved that Sgn(G2(x,y,z))=Sgn(Re⟨x,y⟩),∀z∈X, and a geometric explanation for condition Re⟨x,y⟩=0 is provided. Furthermore, a condition G2(x,iy,z)=0,∀z∈X is added to extend the T-orthogonality to the

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3

HERCZYNSKI, ANDRZEJ, and ROBERT T. FOLK. "ORTHOGONALITY CONDITION FOR THE POCHHAMMER-CHREE MODES." Quarterly Journal of Mechanics and Applied Mathematics 42, no. 4 (1989): 523–36. http://dx.doi.org/10.1093/qjmam/42.4.523.

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4

Yang, Q., X. Chen, and W. Y. Zhou. "Relationship between normality structure and orthogonality condition." Mechanics Research Communications 32, no. 5 (2005): 582–89. http://dx.doi.org/10.1016/j.mechrescom.2005.01.010.

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5

Shirshova, Elen E. "On groups with the almost orthogonality condition." Communications in Algebra 28, no. 10 (2000): 4803–18. http://dx.doi.org/10.1080/00927870008827121.

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6

Qian, Hailong. "REDUNDANCY OF MOMENT CONDITIONS AND THE EFFICIENCY OF OLS IN SUR MODELS." Econometric Theory 24, no. 5 (2008): 1456–60. http://dx.doi.org/10.1017/s0266466608080687.

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In this note, based on the generalized method of moments (GMM) interpretation of the usual ordinary least squares (OLS) and feasible generalized least squares (FGLS) estimators of seemingly unrelated regressions (SUR) models, we show that the OLS estimator is asymptotically as efficient as the FGLS estimator if and only if the cross-equation orthogonality condition is redundant given the within-equation orthogonality condition. Using the condition for redundancy of moment conditions of Breusch, Qian, Schmidt, and Wyhowski (1999, Journal of Econometrics 99, 89–111), we then derive the necessary

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7

Abdallah, Abdallah, Ahmed Atallah Alsaraireh, Malik Bataineh, Nabeela Abu Abu-Alkishik, Isra Al Al-Shbeil, and Abdallah Al Al-Husban. "On The Real Inner Products and Orthogonality for Plithogenic Vector Spaces of Orders 4 and 5." International Journal of Neutrosophic Science 24, no. 1 (2024): 94–103. http://dx.doi.org/10.54216/ijns.240109.

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This paper will study the necessary conditions and sufficient conditions for the problem of orthogonality in 4-plithogenic and 5-plithogenic vector spaces, where we provided a definition of the real scalar products defined above these spaces, and we used the new concept of scalar product that was defined in determining the necessary and sufficient condition for the orthogonality of symbolic 4-plithogenic and 5-plithogenic vectors, in addition to calculating various norms and their metric properties. We have also provided some examples that aim to explain the scientific contribution of this wor

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8

Kukulin, Vladimir, V. Pomerantsev, and Amand Faessler. "Generalized orthogonality-condition model for the NN interaction." Physical Review C 59, no. 6 (1999): 3021–34. http://dx.doi.org/10.1103/physrevc.59.3021.

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9

Zhukov, A. A., G. A. Maslennikov, and M. V. Chekhova. "Operationalistic orthogonality condition for single-mode biphotons (qutrits)." Journal of Experimental and Theoretical Physics Letters 76, no. 10 (2002): 596–99. http://dx.doi.org/10.1134/1.1541042.

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10

Yakymiv, R. Ya. "Representations of canonical anticommutation relations with orthogonality condition." Ukrainian Mathematical Journal 64, no. 9 (2013): 1440–47. http://dx.doi.org/10.1007/s11253-013-0726-5.

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11

ETEDADIALIABADI, MAHMOOD. "Generic behavior of a measure-preserving transformation." Ergodic Theory and Dynamical Systems 40, no. 4 (2018): 904–22. http://dx.doi.org/10.1017/etds.2018.62.

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Del Junco–Lemańczyk [Generic spectral properties of measure-preserving maps and applications. Proc. Amer. Math. Soc., 115 (3) (1992)] showed that a generic measure-preserving transformation satisfies certain orthogonality conditions. More precisely, there is a dense $G_{\unicode[STIX]{x1D6FF}}$ subset of measure preserving transformations such that, for every $T\in G$ and $k(1),k(2),\ldots ,k(l)\in \mathbb{Z}^{+}$, $k^{\prime }(1),k^{\prime }(2),\ldots ,k^{\prime }(l^{\prime })\in \mathbb{Z}^{+}$, the convolutions $$\begin{eqnarray}\unicode[STIX]{x1D70E}_{T^{k(1)}}\ast \cdots \ast \unicode[STI

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12

Bagchi, B., та C. Quesne. "Pseudo-Hermiticity, weak pseudo-Hermiticity and η-orthogonality condition". Physics Letters A 301, № 3-4 (2002): 173–76. http://dx.doi.org/10.1016/s0375-9601(02)00929-5.

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13

Yuan, King, and Chen-Meng Hu. "Nonlinear Modeling and Partial Linearizing Control of a Slewing Timoshenko-Beam." Journal of Dynamic Systems, Measurement, and Control 118, no. 1 (1996): 75–83. http://dx.doi.org/10.1115/1.2801154.

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The modeling and control of a horizontally slewing inextensible Timoshenko beam, taking into account the centrifugal stiffening effect and a tip payload, are considered. Partial differential equations of motion and orthogonality conditions for the constrained modes are derived. A finite dimensional dynamic model simplified by using the orthogonality conditions is obtained. To achieve the joint angle trajectory tracking with simultaneous suppression of elastic vibrations, a nonlinear controller is designed using input-output linearization and elastic-mode stabilization. A sufficient condition f

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14

Dehghani, Mahdi. "Characterization of inner product spaces by unitary Carlsson type orthogonalities." Filomat 36, no. 14 (2022): 4655–63. http://dx.doi.org/10.2298/fil2214655d.

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In this study, we consider the Hermite-Hadamard type of unitary Carlsson?s orthogonality (UHH-C-orthogonality) to characterize real inner product spaces. We give a necessary and sufficient condition weaker than the homogeneity of symmetric HH-C-orthogonalities which characterizes inner product spaces among normed linear spaces of dimension at least three. In conclusion, some more characterizations of real inner product spaces are provided.

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15

DECKERS, KARL, and DORON S. LUBINSKY. "CHRISTOFFEL FUNCTIONS AND UNIVERSALITY LIMITS FOR ORTHOGONAL RATIONAL FUNCTIONS." Analysis and Applications 10, no. 03 (2012): 271–94. http://dx.doi.org/10.1142/s0219530512500133.

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We establish limits for Christoffel functions associated with orthogonal rational functions, whose poles remain a fixed distance away from the interval of orthogonality [-1, 1], and admit a suitable asymptotic distribution. The measure of orthogonality μ is assumed to be regular on [-1, 1], and to satisfy a local condition such as continuity of μ′. As a consequence, we deduce universality limits in the bulk for reproducing kernels associated with orthogonal rational functions.

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16

Gao, Hong Wei, and Lan Ran Fang. "The Nice Feature of a Sort of Orthogonal Finitely Supported Five-Variant Wavelet Packages." Advanced Materials Research 459 (January 2012): 289–92. http://dx.doi.org/10.4028/www.scientific.net/amr.459.289.

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In this article, the notion of orthogonal nonseparable five-variant wavelet packages, whi- ch is the generalization of orthogonal univariate wavelet packages, is introduced. A new approach for constructing the wavelet packages is presented by iteration method as well wavelet frames. The orthogonality properties for five-dimensional wavelet packages are discussed. Three orthogonality formulas concerning these wavelet packages are estabished. A necessary and sufficient condition for the existence of the pyramid decomposition scheme of space is presented.

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17

Tan, Weiwei, Heran Zhang, Lan Wang, Shengyang Nie, Jin Jiao, and Yingtao Zuo. "Effects of the Uncertainty of Wall Distance on the Simulation of Turbulence/Transition Phenomena." Aerospace 11, no. 11 (2024): 898. http://dx.doi.org/10.3390/aerospace11110898.

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The uncertainty of the turbulence/transition model is a problem with relatively high attention in the CFD area. Wall distance is an important physical parameter in turbulence/transition modeling, and its accuracy has a large effect on numerical simulation results. As most CFD solvers use the solving strategy to calculate the nearest distance to the wall based on mesh topology, this makes wall distance one important source of the uncertainty of the simulation results. To investigate the role of wall distance in turbulence/transition simulations, we have conducted simulations for various aerodyn

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18

Fadeev, S. N., та K. A. Gridnev. "Low energy α+16O scattering in the orthogonality condition model". Bulletin of the Russian Academy of Sciences: Physics 80, № 8 (2016): 867–70. http://dx.doi.org/10.3103/s1062873816080128.

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19

Fukatsu, K., K. Kato, and H. Tanaka. "Three-Alpha Potential in 3 and 4 Orthogonality Condition Models." Progress of Theoretical Physics 81, no. 4 (1989): 738–42. http://dx.doi.org/10.1143/ptp.81.738.

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20

Kato, K., and H. Tanaka. "A New Truncation Method for Multi-Cluster Orthogonality Condition Model." Progress of Theoretical Physics 81, no. 4 (1989): 841–57. http://dx.doi.org/10.1143/ptp.81.841.

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21

Clarke, T., S. J. Griffin, and J. Ensor. "Output feedback eigenstructure assignment using a new reduced orthogonality condition." International Journal of Control 76, no. 4 (2003): 390–402. http://dx.doi.org/10.1080/0020717031000082568.

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22

Folk, Robert, and Andrzej Herczynski. "Solutions of elastodynamic slab problems using a new orthogonality condition." Journal of the Acoustical Society of America 80, no. 4 (1986): 1103–10. http://dx.doi.org/10.1121/1.393853.

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23

Andrejić, Vladica. "Equivalence between Jacobi-orthogonality and Osserman condition in dimension four." Journal of Geometry and Physics 216 (October 2025): 105599. https://doi.org/10.1016/j.geomphys.2025.105599.

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24

Falset, J. García. "The fixed point property in banach spaces whose characteristic of uniform convexity is less than 2." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 54, no. 2 (1993): 169–73. http://dx.doi.org/10.1017/s1446788700037095.

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25

Potapov, Alexander N. "ABOUT THE FREE-VIBRATION MODE SHAPES OF ELASTOPLASTIC DISSIPATIVE SYSTEMS." International Journal for Computational Civil and Structural Engineering 14, no. 3 (2018): 114–25. http://dx.doi.org/10.22337/2587-9618-2018-14-3-114-125.

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The author presents the conditions of the generalized orthogonality of the free-vibration mode shapes of an elastic dissipative system, for which traditional classical orthogonality conditions are a private case. As opposed to these conditions, the above ratios contain the mass matrix, the damping matrix, and the diagonal form of the spectral characteristics (damping coefficients and mode-shape frequencies). Within the theory of time analysis, free-vibration mode shapes of an elastoplastic system are built on the basis of using a schematized diagram of strain with hardening. The author propose

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26

Sevilla, Francisco J., Andrea Valdés-Hernández, and Alan J. Barrios. "The Underlying Order Induced by Orthogonality and the Quantum Speed Limit." Quantum Reports 3, no. 3 (2021): 376–88. http://dx.doi.org/10.3390/quantum3030024.

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We perform a comprehensive analysis of the set of parameters {ri} that provide the energy distribution of pure qutrits that evolve towards a distinguishable state at a finite time τ, when evolving under an arbitrary and time-independent Hamiltonian. The orthogonality condition is exactly solved, revealing a non-trivial interrelation between τ and the energy spectrum and allowing the classification of {ri} into families organized in a 2-simplex, δ2. Furthermore, the states determined by {ri} are likewise analyzed according to their quantum-speed limit. Namely, we construct a map that distinguis

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27

Guo, Ming Pu. "The Optimality Characters of Vector-Valued Binary Small-Wave Wraps with a Constant Dilation Factor." Applied Mechanics and Materials 58-60 (June 2011): 1460–65. http://dx.doi.org/10.4028/www.scientific.net/amm.58-60.1460.

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Frame theory has been the focus of active research for twenty years, both in theory and applications. In this work, the notion of orthogonal vector-valued binary small-wave wraps, which is a generalization of uni-wavelet packets, is introduced. A procedure for constructing them is presented. Their orthogonality traits are investigated by using time-frequency analysis method, matrix theory and finite group theory. Orthogonality formulas concerning these binary small-wave wraps are established. Orthonormal wavelet packet bases are obtained. The pyramid decomposition scheme is obtained based on s

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28

Bilalov, Bilal, Aysel Guliyeva, and Sabina Sadigova. "On Riemann problem in weighted Smirnov classes with general weight." Acta et Commentationes Universitatis Tartuensis de Mathematica 25, no. 1 (2021): 33–56. http://dx.doi.org/10.12697/acutm.2021.25.03.

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Weighted Smirnov classes in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problems with a measurable coefficient whose argument is a piecewise continuous function are considered in these classes. A Muckenhoupt type condition is imposed on the weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered,and conditions on degeneration order are found.

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29

Kruppa, A. T., and K. Kato. "Resonances in Complex-Scaled Orthogonality Condition Model of Nuclear Cluster System." Progress of Theoretical Physics 84, no. 6 (1990): 1145–59. http://dx.doi.org/10.1143/ptp/84.6.1145.

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30

Szolgay, D., and T. Szirányi. "Optimal stopping condition for iterative image deconvolution by new orthogonality criterion." Electronics Letters 47, no. 7 (2011): 442. http://dx.doi.org/10.1049/el.2010.3692.

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31

Saad, S. M., V. B. Subbotin, K. A. Gridnev, E. F. Hefter та V. M. Semjonov. "The orthogonality condition model applied to (α, α) scattering on12C and16O". Il Nuovo Cimento A 98, № 5 (1987): 529–37. http://dx.doi.org/10.1007/bf02902010.

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Zhang, Qing Ge. "The Traits of Nontensor Five-Variant Wavelet Packs and Applications in Management Science and Materials Science." Advanced Materials Research 461 (February 2012): 868–71. http://dx.doi.org/10.4028/www.scientific.net/amr.461.868.

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Materials science is an interdisciplinary field applying the properties of matter to various areas of science and engineering. In this article, the notion of orthogonal nonseparable five-variant wavelet packages is presented. A novel approach for constructing them is presented by iteration method and functional analysis method. A feasible approach for constructing two-directional orthogonal wavelet packs is developed. The orthogonality property of five-variant wavelet packs is discussed. Three orthogonality formulas concerning these wavelet packs are estabished. A constructive method for affin

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33

Liu, Chein-Shan, Chih-Wen Chang, and Chia-Cheng Tsai. "Numerical Simulations of Complex Helmholtz Equations Using Two-Block Splitting Iterative Schemes with Optimal Values of Parameters." AppliedMath 4, no. 4 (2024): 1256–77. http://dx.doi.org/10.3390/appliedmath4040068.

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For a two-block splitting iterative scheme to solve the complex linear equations system resulting from the complex Helmholtz equation, the iterative form using descent vector and residual vector is formulated. We propose splitting iterative schemes by considering the perpendicular property of consecutive residual vector. The two-block splitting iterative schemes are proven to have absolute convergence, and the residual is minimized at each iteration step. Single and double parameters in the two-block splitting iterative schemes are derived explicitly utilizing the orthogonality condition or th

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34

Sabitov, T. I., and A. V. Kiselev. "Simulation of echo signals of two-position radar station using matrix of five emitters." Issues of radio electronics 49, no. 5 (2020): 6–9. http://dx.doi.org/10.21778/2218-5453-2020-5-6-9.

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The problem of modeling the echo signals of radar systems which use two receiving antennas with spaced phase centers and overlapping beam patterns is considered. To solve it, it is proposed to use a matrix of five emitters. Its configuration is based on the results of the previous work of the authors and meets the conditions of compensation and orthogonality of signals. Independent simulation of targets for each antenna is carried out by two pairs of emitters, while the fifth one is introduced for phase compensation of signals. The orthogonality condition is met by configuring the radiation po

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35

Kotova, E. V., R. M. Klebleev, and V. A. Kudinov. "Method for determining eigen numbers in heat conduction problems for a cylinder." Power engineering: research, equipment, technology 25, no. 4 (2023): 71–82. http://dx.doi.org/10.30724/1998-9903-2023-25-4-71-82.

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RELEVANCE. Due to the difficulties of finding eigenvalues and proper functions for bodies with axial (cylinder) and central (ball) symmetries defined in classical methods from the edge Sturm-Liouville problems, including Bessel equations whose exact analytical solutions are not obtained (known only numerical solutions, described by approximation formulas), there is a need to develop analytical methods of their solution.THE PURPOSE. Using orthogonal methods of weighted residuals, an approximate analytical method for determining eigenfunctions and eigenvalues in boundary value problems with axia

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36

Lahaye, Noé, and Stefan G. Llewellyn Smith. "Modal Analysis of Internal Wave Propagation and Scattering over Large-Amplitude Topography." Journal of Physical Oceanography 50, no. 2 (2020): 305–21. http://dx.doi.org/10.1175/jpo-d-19-0005.1.

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AbstractCoupled-mode equations describing the propagation and scattering of internal waves over large-amplitude arbitrary topography in a two-dimensional stratified fluid are derived. They consist of a simple set of ordinary differential equations describing the evolution of modal amplitudes, based on an orthogonality condition that allows one to distinguish leftward- and rightward-propagating modes. The coupling terms expressing exchange of energy between modes are given in an analytical form using perturbation theory. This allows the derivation of a weak-topography approximate solution, gene

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37

Magomed-Kasumov, Magomedrasul Grozbekovich, Tadgidin Nurmagomedovich Shakh-Emirov, and Ramis Makhmudovich Gadzhimirzaev. "Basis property of the Legendre polynomials in variable exponent Lebesgue spaces." Sbornik: Mathematics 215, no. 2 (2024): 234–49. http://dx.doi.org/10.4213/sm9891e.

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Sharapudinov proved that the Legendre polynomials form a basis of the Lebesgue space with variable exponent $p(x)$ if $p(x) > 1$ satisfies the Dini-Lipschitz condition and is constant near the endpoints of the orthogonality interval. We prove that the system of Legendre polynomials forms a basis of these spaces without the condition that the variable exponent be constant near the endpoints. Bibliography: 9 titles.

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38

Fukatsu, K., and K. Kato. "The 4 Orthogonality Condition Model for Low-Lying 0+ States of 16O." Progress of Theoretical Physics 87, no. 1 (1992): 151–67. http://dx.doi.org/10.1143/ptp/87.1.151.

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39

Ouyang, Xing, Octavia A. Dobre, Yong Liang Guan, and Jian Zhao. "Chirp Spread Spectrum Toward the Nyquist Signaling Rate—Orthogonality Condition and Applications." IEEE Signal Processing Letters 24, no. 10 (2017): 1488–92. http://dx.doi.org/10.1109/lsp.2017.2737596.

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Li, JianLin. "A necessary and sufficient condition for the finite µ M,D -orthogonality." Science China Mathematics 58, no. 12 (2015): 2541–48. http://dx.doi.org/10.1007/s11425-015-5014-4.

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Boerner, David E., Ron D. Kurtz, and Alan G. Jones. "Orthogonality in CSAMT and MT measurements." GEOPHYSICS 58, no. 7 (1993): 924–34. http://dx.doi.org/10.1190/1.1443483.

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The electric and magnetic fields from a single plane‐wave source on a one dimensional (1-D) earth, or a plane‐wave source polarized parallel or perpendicular to strike on a two-dimensional (2-D) earth, are orthogonal. On a layered earth and in the far‐field of a controlled source, the electric and magnetic fields are also orthogonal. Therefore, orthogonality of E and H data is a necessary condition to justify the application of 1-D or 2-D modeling algorithms having a plane wave source. A strict criterion to prove orthogonality, and thus provide a rationale for the choice of interpretation meth

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42

YAMADA, T., and Y. FUNAKI. "CLUSTER STRUCTURES AND SINGLE-CLUSTER MOTIONS IN 11B AND 13C." International Journal of Modern Physics E 20, no. 04 (2011): 910–14. http://dx.doi.org/10.1142/s0218301311018952.

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The structures of 11 B and 13 C are studied up to Ex ~ 15 MeV with the semi-microscopic three-body and four-body cluster models, 2α+t and 3α+n orthogonality condition models (OCM). We discuss the cluster states and Hoyle-analogue states in 11 B and 13 C .

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Wang, Guo Xin, and De Lin Hua. "The Excellent Traits of a Class of Orthogonal Quarternary Wavelet Wraps with Short Support." Advanced Materials Research 219-220 (March 2011): 496–99. http://dx.doi.org/10.4028/www.scientific.net/amr.219-220.496.

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The frame theory has been one of powerful tools for researching into wavelets. In this article, the notion of orthogonal nonseparable quarternary wavelet wraps, which is the generalizati- -on of orthogonal univariate wavelet wraps, is presented. A novel approach for constructing them is presented by iteration method and functional analysis method. A liable approach for constructing two-directional orthogonal wavelet wraps is developed. The orthogonality property of quarternary wavelet wraps is discussed. Three orthogonality formulas concerning these wavelet wraps are estabished. A constructive

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44

Salem, Ahmed, Faris Alzahrani, and Moustafa El-Shahed. "Orthogonality Property of the Discrete q -Hermite Matrix Polynomials." Mathematical Problems in Engineering 2022 (September 15, 2022): 1–8. http://dx.doi.org/10.1155/2022/3448290.

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In this paper, we prove that the solution of the autonomous q -difference system D q Y x = A Y q x with the initial condition Y 0 = Y 0 where A is a constant square complex matrix, D q is the Jackson q -derivative and 0 < q < 1 , is asymptotically stable if and only if ℜ λ < 0 for all λ ∈ σ A where σ A is the set of all eigenvalues of A (the spectrum of A ). This results are exploited to provide the orthogonality property of the discrete q -Hermite matrix polynomials.

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Hassanpour, P. A., E. Esmailzadeh, W. L. Cleghorn, and J. K. Mills. "Generalized Orthogonality Condition for Beams with Intermediate Lumped Masses Subjected to Axial Force." Journal of Vibration and Control 16, no. 5 (2010): 665–83. http://dx.doi.org/10.1177/1077546309106526.

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Magnanti, Thomas L., and Georgia Perakis. "The Orthogonality Theorem and the Strong-f-Monotonicity Condition for Variational Inequality Algorithms." SIAM Journal on Optimization 7, no. 1 (1997): 248–73. http://dx.doi.org/10.1137/s1052623493259227.

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Kerrian, Peter, and Teresa Kinney. "Doomed from the start, understanding the importance of boundary conditions in modal testing." Journal of the Acoustical Society of America 155, no. 3_Supplement (2024): A76. http://dx.doi.org/10.1121/10.0026859.

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Modal survey tests are commonly performed in the aerospace industry to validate the structural dynamics behavior of the finite element model used in the flight loads predictions for both hardware structural capability and expected crew loads. During the planning phase, great effort is spent on the selection of accelerometer locations to properly capture the modes and satisfy orthogonality conditions. The same due diligence is not always spent on the design of the test’s setup and boundary conditions. Logistical and programmatic concerns may influence the technical approach taken, which, unfort

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Chen, Y., DM McFarland, BF Spencer, and LA Bergman. "A beam with arbitrarily placed lateral dampers: Evolution of complex modes with damping." Journal of Vibration and Control 24, no. 2 (2016): 379–92. http://dx.doi.org/10.1177/1077546316641592.

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Free vibration of a beam with multiple arbitrarily placed lateral viscous dampers is investigated to gain insight into the intrinsic dynamic features of non-proportional damped systems. In terms of virtual boundary condition method, complex modes of a damper-beam system are achieved, and the solution is also suitable for the beams that have different boundary conditions. The features of the wave numbers satisfying the frequency equation were discussed in theory. The orthogonality analysis conducted in this paper provides two orthogonality conditions for complex modes. Pseudoundamped natural fr

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Sun, Yanmei, and Yakui Huang. "An alternate gradient method for optimization problems with orthogonality constraints." Numerical Algebra, Control & Optimization 11, no. 4 (2021): 665. http://dx.doi.org/10.3934/naco.2021003.

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<p style='text-indent:20px;'>In this paper, we propose a new alternate gradient (AG) method to solve a class of optimization problems with orthogonal constraints. In particular, our AG method alternately takes several gradient reflection steps followed by one gradient projection step. It is proved that any accumulation point of the iterations generated by the AG method satisfies the first-order optimal condition. Numerical experiments show that our method is efficient.</p>

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Howard, C. Douglas. "Orthogonality of Measures Induced by Random Walks with Scenery." Combinatorics, Probability and Computing 5, no. 3 (1996): 247–56. http://dx.doi.org/10.1017/s0963548300002017.

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Thinking of a deterministic function s: ℤ → ℕ as ‘scenery’ on the integers, a simple random walk on ℤ generates a random record of scenery ‘observed’ along the walk. We address this question: If t:ℤ → ℕ is another scenery on the integers and we are handed a random scenery record obtained from either s or t, under what circumstances can the source be distinguished? We allow ourselves to use information about s and t together with information contained in the scenery record. It has been conjectured that it is sufficient for t to be neither a translate of s nor a translate of the reflection of s.

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