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The transverse inclination is a probable scenario when inclined structures experience an inflow of altered attack angles. This work investigates the effects of transverse inclination on an aeroelastic prism through forced-vibration wind tunnel experiments. The aerodynamic characteristics are tri-parametrically evaluated under different wind speeds, inclination angles, and oscillation amplitudes. Results show that transverse inclination fundamentally changes the wake phenomenology by impinging the fix-end horseshoe vortex and breaking the separation symmetry. The aftermath is a bi-polar, once-for-all change in the aerodynamics near the prism base. The suppression of the horseshoe vortex unleashes the Karman vortex, which significantly increases the unsteady crosswind force. After the initial morphology switch, the aerodynamics become independent of inclination angle and oscillation amplitude and depend solely on wind speed. The structure's upper portion does not feel the effect, so this phenomenon is called Base Intensification. The phenomenon only projects notable impacts on the low-speed and VIV regime and is indifferent in the high-speed, quasi-steady Galloping regime. In practice, Base Intensification will disrupt the pedestrian-level wind environment from the unleashed Bernard-Karman vortex shedding, making it erratic and gusty. Moreover, it increases the aerodynamic load at a structure base by as much as 4.3 times. Since fix-end stiffness prevents elastic dissipation, the load translates to massive stress, making detection trickier and failures, if they are to occur, more sudden, extreme, and without any warnings. The 4.3-time amplification also surpasses the safety factor of many standard designs, so transverse inclination deserves engineering attention., Comment: 49 pages, 20 figures, under review at Journal of Wind Engineering and Industrial Aerodynamics

This serial work presents a Linear-Time-Invariance (LTI) notion to the Koopman analysis, finding consistent and physically meaningful Koopman modes and addressing a long-standing problem of fluid-structure interactions: deterministically relating the fluid and structure. Part 1 (Li et al., 2022) developed the Koopman-LTI architecture and applied it to a pedagogical prism wake. By the systematic procedure, the LTI generated a sampling-independent Koopman linearization that captured all the recurring dynamics, finding six corresponding, orthogonal, and in-synch fluid excitation-structure response mechanisms. This Part 2 analyzes the six modal duplets' to underpin their physical interpretations, providing a phenomenological revelation of the subcritical prism wake. By the dynamical mode shape, results show that two mechanisms at St1=0.1242 and St5=0.0497 describe shear layer dynamics, the associated B\'ernard-K\'arm\'an shedding, and turbulence production, which together overwhelm the upstream and crosswind walls by instigating a reattachment-type of response. The on-wind walls' dynamical similarity renders them a spectrally unified fluid-structure interface. Another four harmonic counterparts, namely the subharmonic at St7=0.0683, the second harmonic at St3=0.2422, and two ultra-harmonics at St7 =0.1739 and St13=0.1935, govern the downstream wall. The 2P wake mode is also observed as an embedded harmonic of the bluff-body wake. Finally, this work discovered the vortex breathing phenomenon, describing the constant energy exchange in wake's circulation-entrainment-deposition processes. With the Koopman-LTI, one may pinpoint the exact excitations responsible for a specific structural response, or vice versa., Comment: 24 figures, 60 pages. Video files at https://drive.google.com/drive/folders/1AHdhUdAfNwlC1XUh-74PgQWW6jUHXJ5j?usp=sharing

This work proposes a Linear-Time-Invariance (LTI) notion to the Koopman analysis, finding an invariant subspace on which Koopman modes are consistent and physically meaningful. It also develops the Koopman-LTI architecture -- a systematic procedure to associate fluid excitation and structure surface pressure by matching Koopman eigen tuples, solving a longstanding problem for fluid-structure interactions. The architecture is data-driven and modular, accommodating all types of data and Koopman algorithms. Through a pedagogical demonstration on a prism wake and the rudimentary Dynamic Mode Decomposition algorithm, results show a near-exact linearization of nonlinear turbulence, with mean and rms errors of O-12 and O-9, respectively. The DMD also approximated the Koopman modes with O-8 error. The LTI reduced the subcritical prism wake during shear layer transition II into only six dominant excitation-response Koopman modal duplets. The upstream and crosswind walls constitute a dynamically unified interface dominated by only two mechanisms. The downstream wall remains a distinct interface and is dominated by four other mechanisms. The complete revelation of the prism wake essentially comes down to understanding the six mechanisms, which Part 2 (Li et al., 2022) will address by investigating the physical interpretations of the duplets' in-synch, phenomenological features. Finally, the current analysis also revealed w's trivial role in this convection-dominated free-shear flow, Reynolds stresses' spectral description of cascading eddies, vortices' sensitivity to dilation and indifference to distortion, and structure responses' origin in vortex activities., Comment: 15 figures, 63 pages

In this work, we introduce a novel data-driven formulation, the Koopman-Linearly-Time-Invariant (Koopman-LTI) analysis, for analyzing Fluid-Structure Interactions (FSI). An implementation of the Koopman-LTI on a subcritical free-shear flow over a prism at Re=22,000 corroborated a configuration-wise universal Koopman system, which approximated the configuration's nonlinear dynamics with stellar accuracy. The Koopman-LTI also successfully decomposed the entwined morphologies of raw measurement into a linear superposition of frequency-based constituents. Most importantly, with random and anisotropic turbulence, the Koopman-LTI yielded frequency-wise identical modes for structure response and fluid excitation, thus establishing direct constitutive relations between the phenomenology of fluid and structure., Comment: 12 pages, 6 figures, 2 multimedia files

This work continues the parametric investigation on the sampling nuances of the Dynamic Mode Decomposition (DMD) under the Koopman analysis. Through turbulent wakes, the investigation corroborated the generality of the universal convergence states for all DMD implementations. It discovered the implications of sampling range and resolution -- the determinants of the spectral discretisation by discrete frequency bins and the highest resolved frequency, respectively. The work reaffirmed the necessity of the Convergence state for sampling independence, too. Results also suggested that the observables derived from the same flow may contain dynamically distinct information, thus altering the DMD output. The static pressure and vortex identification criteria are optimal variables for characterising structural response and fluid excitation. The pressure, velocity magnitude, and turbulence kinetic energy fields also suffice for general applications, but the Reynolds stresses and velocity components shall be avoided. Mean-subtraction is recommended for best approximations of the Koopman eigen tuples. Furthermore, the parametric investigation on truncation discovered some low-energy states that dictate a system's temporal integrity. The best practice for order reduction is to avoid truncation and employ dominant mode selection on a full-state subspace, though large-degree truncation supports fair data reconstruction with low computational cost. Finally, this work demonstrated the synthetic noise resulting from pre-decomposition interpolation. In unavoidable interpolations to increase the spatial dimension n, high-order schemes are recommended for better retention of the original dynamics. Finally, the observations herein, derived from inhomogeneous anisotropic turbulence, offer constructive references for DMD on fluid systems, if not also others beyond fluid mechanics., Comment: 75 page, 21 figures, Under Review at Physics of Fluids

Scientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The Dynamic Mode Decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear systems into periodically distinct constituents on reduced-order subspaces. As a novel mathematical hatchling, the DMD bears vast potentials yet an equal degree of unknown. This serial effort investigates the nuances of DMD sampling with an engineering-oriented emphasis. This Part I aimed at elucidating how sampling range and resolution affect the convergence of DMD modes. We employed the most classical nonlinear system in fluid mechanics as the test subject--the turbulent free-shear flow over a prism--for optimal pertinency. We numerically simulated the flow by the dynamic-stress Large-Eddies Simulation with Near-Wall Resolution. With the large-quantity, high-fidelity data, we parametrized and identified four global convergence states: Initialization, Transition, Stabilization, and Divergence with increasing sampling range. Results showed that the Stabilization is the optimal state for modal convergence, in which DMD output becomes independent of the sampling range. The Initialization state also yields sufficient accuracy for most system reconstruction tasks. Moreover, defying popular beliefs, over-sampling causes algorithmic instability: as the temporal dimension, n, approaches and transcends the spatial dimension, m (i.e., m < n), the output diverges and becomes meaningless. Additionally, the convergence of the sampling resolution depends on the mode-specific dynamics, such that the resolution of 15 frames per cycle for target activities is suggested for most engineering implementations. Finally, a bi-parametric study revealed that the convergence of the sampling range and resolution are mutually independent., Comment: 57 pages, 19 figures, under review at Nonlinear Dynamics

The Dynamic Mode Decomposition (DMD) is a Koopman-based algorithm that straightforwardly isolates individual mechanisms from the compound morphology of direct measurement. However, many may be perplexed by the messages the DMD structures carry. This work investigates the modal implications of the DMD/Koopman modes through the prototypical subcritical free-shear flow over a square prism. It selected and analysed the fluid mechanics and phenomenology of the ten most dominant modes. The results showed that the reduced-order description is morphologically accurate and physically insightful. Mode 1 renders the mean-field. Modes 2 depicts the roll-up of the Strouhal vortex. Mode 3 delineates the Bloor-Gerrard vortex resulting from the Kelvin-Helmholtz instability inside shear layers, its superposition onto the Strouhal vortex, and the concurrent flow entrainment. Modes 4, 5, 7, 8, and 9 portray the harmonic excitation. Modes 6 and 10 describe the low-frequency shedding of turbulent separation bubbles (TSBs) and turbulence production, respectively, which contribute to the beating phenomenon in the lift time history and the flapping motion of shear layers. Finally, this work demonstrates the capability of the DMD in providing insights into similar fluid problems. It also serves as an excellent reference for an array of other nonlinear systems., Comment: 39 pages, 12 figures, Under review at Journal of Wind Engineering and Industrial Aerodynamics

The construction of exactly solvable refractive indices allowing guided TE modes in optical waveguides is investigated within the formalism of Darboux-Crum transformations. We apply the finite-difference algorithm for higher-order supersymmetric quantum mechanics to obtain complex-valued refractive indices admitting all-real eigenvalues in their point spectrum. The new refractive indices are such that their imaginary part gives zero if it is integrated over the entire domain of definition. This property, called condition of zero total area, ensures the conservation of optical power so the refractive index shows balanced gain and loss. Consequently, the complex-valued refractive indices reported in this work include but are not limited to the parity-time invariant case., Comment: 21 pages, 7 figures

We report a new set of Laguerre-Gaussian wave-packets that propagate with periodical self-focusing and finite beam width in weakly guiding inhomogeneous media. These wave-packets are solutions to the paraxial form of the wave equation for a medium with parabolic refractive index. The beam width is defined as a solution of the Ermakov equation associated to the harmonic oscillator, so its amplitude is modulated by the strength of the medium inhomogeneity. The conventional Laguerre-Gaussian modes, available for homogenous media, are recovered as a particular case., Comment: 11 pages, 5 figures

The energy spectra of two different quantum systems are paired through supersymmetric algorithms. One of the systems is Hermitian and the other is characterized by a complex-valued potential, both of them with only real eigenvalues in their spectrum. The superpotential that links these systems is complex-valued, parameterized by the solutions of the Ermakov equation, and may be expressed either in nonlinear form or as the logarithmic derivative of a properly chosen complex-valued function. The non-Hermitian systems can be constructed to be either parity-time-symmetric or non-parity-time-symmetric., Comment: 9 pages, 2 figures (affiliation institution corrected)

The Darboux method is commonly used in the coordinate variable to produce new exactly solvable (stationary) potentials in quantum mechanics. In this work we follow a variation introduced by Bagrov, Samsonov, and Shekoyan (BSS) to include the time-variable as a parameter of the transformation. The new potentials are nonstationary and define Hamiltonians which are not integrals of motion for the system under study. We take the stationary oscillator of constant frequency to produce nonstationary oscillators, and also provide an invariant that serves to define uniquely the state of the system. In this sense our approach completes the program of the BSS method since the eigenfunctions of the invariant are an orthonormal basis for the space of solutions of the related Schrodinger equation. The orthonormality holds when the involved functions are evaluated at the same time. The dynamical algebra of the nonstationary oscillators is generated by properly chosen ladder operators and coincides with the Heisenberg algebra. We also construct the related coherent states and show that they form an overcomplete set that minimizes the quadratures defined by the ladder operators. These states are not invariant under time-evolution since their time-dependence relies on the basis of states and not on the complex eigenvalue that labels them. Some concrete examples are provided., Comment: 25 pages, t figures

Bright and dark solitons of the cubic nonlinear Schrodinger equation are used to construct complex-valued potentials with all-real spectrum. The real part of these potentials is equal to the intensity of a bright soliton while their imaginary part is defined by the product of such soliton with its concomitant, a dark soliton. Considering light propagation in Kerr media, the real part of the potential refers to the self-focusing of the signal and the imaginary one provides the system with balanced gain-and-loss rates., Comment: 6 figures, 17 pages, LaTeX file. The manuscript has been re-organized (abstract, introduction and conclusions rewritten), and it now includes an appendix with detailed calculations of some relevant results reported in the paper. New references were added

The purposes of this work are (1) to show that the appropriate generalizations of the oscillator algebra permit the construction of a wide set of nonlinear coherent states in unified form; and (2) to clarify the likely contradiction between the nonclassical properties of such nonlinear coherent states and the possibility of finding a classical analog for them since they are P-represented by a delta function. In (1) we prove that a class of nonlinear coherent states can be constructed to satisfy a closure relation that is expressed uniquely in terms of the Meijer G-function. This property automatically defines the delta distribution as the P-representation of such states. Then, in principle, there must be a classical analog for them. Among other examples, we construct a family of nonlinear coherent states for a representation of the su(1,1) Lie algebra that is realized as a deformation of the oscillator algebra. In (2), we use a beam splitter to show that the nonlinear coherent states exhibit properties like anti-bunching that prohibit a classical description for them. We also show that these states lack second order coherence. That is, although the P-representation of the nonlinear coherent states is a delta function, they are not full coherent. Therefore, the systems associated with the generalized oscillator algebras cannot be considered `classical' in the context of the quantum theory of optical coherence., Comment: 26 pages, 10 figures, minor changes, misprints corrected

Using the Hubbard representation for $SU(2)$ we write the time-evolution operator of a two-level system in the disentangled form. This allows us to map the corresponding dynamical law into a set of non-linear coupled equations. In order to find exact solutions, we use an inverse approach and find families of time-dependent Hamiltonians whose off-diagonal elements are connected with the Ermakov equation. The physical meaning of the so-obtained Hamiltonians is discussed in the context of the nuclear magnetic resonance phenomenon

It is shown that each one of the Lie algebras su(1,1) and su(2) determine the spectrum of the radial oscillator. States that share the same orbital angular momentum are used to construct the representation spaces of the non-compact Lie group SU(1,1). In addition, three different forms of obtaining the representation spaces of the compact Lie group SU(2) are introduced, they are based on the accidental degeneracies associated with the spherical symmetry of the system as well as on the selection rules that govern the transitions between different energy levels. In all cases the corresponding generalized coherent states are constructed and the conditions to squeeze the involved quadratures are analyzed., Comment: Abridged version, includes new references

The metric dimension is quite a well-studied graph parameter. Recently, the adjacency dimension and the local metric dimension have been introduced and studied. In this paper, we give a general formula for the local metric dimension of the lexicographic product $G \circ \mathcal{H}$ of a connected graph $G$ of order $n$ and a family $\mathcal{H}$ composed by $n$ graphs. We show that the local metric dimension of $G \circ \mathcal{H}$ can be expressed in terms of the true twin equivalence classes of $G$ and the local adjacency dimension of the graphs in $\mathcal{H}$.