Showing papers on "Biorthogonal system published in 2022"
TL;DR: In this paper , an efficient algorithm based on the Galerkin method using biorthogonal Hermite cubic spline multi-wavelets (BHCSMWs) was proposed.
34 citations
19 Sep 2022
TL;DR: A compact wavelet representation with a pair of coarse and detail coefficient volumes to implicitly represent 3D shapes via truncated signed distance functions and multi-scale biorthogonal wavelets is proposed, enabling direct generative modeling on a continuous implicit representation in wavelet domain.
Abstract: This paper presents a new approach for 3D shape generation, enabling direct generative modeling on a continuous implicit representation in wavelet domain. Specifically, we propose a compact wavelet representation with a pair of coarse and detail coefficient volumes to implicitly represent 3D shapes via truncated signed distance functions and multi-scale biorthogonal wavelets, and formulate a pair of neural networks: a generator based on the diffusion model to produce diverse shapes in the form of coarse coefficient volumes; and a detail predictor to further produce compatible detail coefficient volumes for enriching the generated shapes with fine structures and details. Both quantitative and qualitative experimental results manifest the superiority of our approach in generating diverse and high-quality shapes with complex topology and structures, clean surfaces, and fine details, exceeding the 3D generation capabilities of the state-of-the-art models.
23 citations
14 citations
TL;DR: In this paper , the authors demonstrate that the non-Hermitian topology can be realized in monitored quantum circuits, exemplified by the paradigmatic non-Schrieffer-Heeger model.
Abstract: We demonstrate that genuinely non-Hermitian topological phases and corresponding topological phase transitions can be naturally realized in monitored quantum circuits, exemplified by the paradigmatic non-Hermitian Su-Schrieffer-Heeger model. We emulate this model by a 1D chain of spinless electrons evolving under unitary dynamics and subject to periodic measurements that are stochastically invoked. The non-Hermitian topology is visible in topological invariants adapted to the context of monitored circuits. For instance, the topological phase diagram of the monitored realization of the non-Hermitian Su-Schrieffer-Heeger model is obtained from the biorthogonal polarization computed from an effective Hamiltonian of the monitored system. Importantly, our monitored circuit realization allows direct access to steady state biorthogonal expectation values of generic observables, and hence, to measure physical properties of a genuine non-Hermitian model. We expect our results to be applicable more generally to a wide range of models that host non-Hermitian topological phases.
13 citations
TL;DR: In this article , the bulk-boundary correspondence for a Chern insulator model with non-Hermitian skin effect was established by combining two approaches (non-Bloch approach and biorthogonal approach) based on different mathematical tools.
Abstract: The non-Hermitian skin effect can arise in materials that have asymmetric hoppings between atoms or resonating units, which makes the bulk eigenspectrum sensitive to boundary conditions. When skin effect emerges, eigenstates in the bulk continuum can become localized on the edges, making the distinction between edge and bulk states challenging. We establish the bulk-boundary correspondence for a Chern insulator model with non-Hermitian skin effect by combining two approaches ("non-Bloch" approach and "biorthogonal" approach). Both approaches can suppress the skin effect but they are based on different mathematical tools. A biorthogonal inverse participation ratio is used as a measure to distinguish between bulk states and edge states, and a non-Bloch Chern number is used to characterize the topology and predict the number of topological edge bands. In addition to tangential degeneracies, crossing degeneracies are found to occur between the bulk and edge bands. Their presence enriches the (de)localization behavior of the edge states but does not affect the Chern number. The phase diagram of the system has interesting features that are not found in Hermitian systems. For example, one topological transition and two non-Hermitian phase transitions can be induced by tuning a single parameter. The gapless phase is topologically protected due to the stable existence of the non-Hermitian band degeneracies guaranteed by nonzero discriminant numbers.
10 citations
TL;DR: In this article , the sensitivity of the spatial boundary conditions and the interplay of the non-Hermitian skin effect with many-body phenomena are investigated for a MBL system in the presence of non-reciprocal tunneling and random disorder potential.
Abstract: The explorations of non-Hermiticity have been devoted to investigate the disorder-induced many-body localization (MBL). However, the sensitivity of the spatial boundary conditions and the interplay of the non-Hermitian skin effect with many-body phenomena are not yet clear. For a MBL system in the presence of non-reciprocal tunnelings and random disorder potential, we identify two different complex-real spectral transitions, one is present for both open and periodic boundaries while the other is present only for open boundaries of a coupled non-Hermitian chains. The later is driven due to the inter-chain coupling at weak disorder where the level statistics of the real eigenenergy phase follows Gaussian orthogonal ensemble. We further characterize wavefunctions through the (biorthogonal) inverse participation ratio and fractal dimension, which reveal the suppression of skin effect in the non-Hermitian MBL phase. Finally, we demonstrate that the quench dynamics of the local particle density, spin imbalance, and entanglement entropy also signify the hallmark of the boundary effects and non-ergodic character of many-body localization.
7 citations
TL;DR: In this paper , a second-order topological superconductor (SOTSC) model that hosts Majorana zero modes (MZMs) was proposed, which is protected by mirror rotation symmetry and remains robust under onsite random disorder.
Abstract: Being motivated by intriguing phenomena, such as the breakdown of conventional bulk boundary correspondence and the emergence of skin modes in the context of non-Hermitian (NH) topological insulators, we here propose a NH second-order topological superconductor (SOTSC) model that hosts Majorana zero modes (MZMs). Employing the non-Bloch form of the NH Hamiltonian, we topologically characterize the above modes by biorthogonal nested polarization and resolve the apparent breakdown of the bulk boundary correspondence. Unlike the Hermitian SOTSC, we note that the MZMs inhabit only one corner out of four in the two-dimensional NH SOTSCs. Such a localization profile of MZMs is protected by mirror rotation symmetry and remains robust under on-site random disorder. We extend the static MZMs into the realm of the Floquet drive. We find the anomalous $\ensuremath{\pi}$ mode following low-frequency mass kick in addition to the regular 0 mode that is usually engineered in a high-frequency regime. We further characterize the regular 0 mode with biorthogonal Floquet nested polarization. Our proposal is not limited to the $d$-wave superconductivity only and can be realized in the experiment with strongly correlated optical lattice platforms.
7 citations
TL;DR: In this article , an extension of the Resta's electronic polarization to non-Hermitian systems with periodic boundary conditions was discussed. But the results were restricted to the Su-Schrieffer-Heeger model, where the polarization is zero between two topologically distinguished regions, and there is no correspondence between the polarization and the winding number.
Abstract: We discuss an extension of the Resta's electronic polarization to non-Hermitian systems with periodic boundary conditions. We introduce the ``electronic polarization'' as an expectation value of the exponential of the position operator in terms of the biorthogonal basis. We found that there appears a finite region where the polarization is zero between two topologically distinguished regions, and there is one-to-one correspondence between the polarization and the winding number which takes half-odd integers as well as integers. We demonstrate this argument in the non-Hermitian Su-Schrieffer-Heeger model.
6 citations
TL;DR: In this paper , a derivation of real-time (RT) time-dependent orbital-optimized Møller-Plesset (TDOMP2) theory and its biorthogonal companion, timedependent non-orthogonal OMP2 theory, starting from the timedependent bivariational principle and a parametrization based on the exponential orbital-rotation operator formulation commonly used in the time independent molecular electronic structure theory was presented.
Abstract: We present a derivation of real-time (RT) time-dependent orbital-optimized Møller–Plesset (TDOMP2) theory and its biorthogonal companion, time-dependent non-orthogonal OMP2 theory, starting from the time-dependent bivariational principle and a parametrization based on the exponential orbital-rotation operator formulation commonly used in the time-independent molecular electronic structure theory. We apply the TDOMP2 method to extract absorption spectra and frequency-dependent polarizabilities and first hyperpolarizabilities from RT simulations, comparing the results with those obtained from conventional time-dependent coupled-cluster singles and doubles (TDCCSD) simulations and from its second-order approximation, TDCC2. We also compare our results with those from CCSD and CC2 linear and quadratic response theories. Our results indicate that while TDOMP2 absorption spectra are of the same quality as TDCC2 spectra, including core excitations where optimized orbitals might be particularly important, frequency-dependent polarizabilities and hyperpolarizabilities from TDOMP2 simulations are significantly closer to TDCCSD results than those from TDCC2 simulations.
4 citations
TL;DR: In this article , a unified error estimate for weak biorthogonal greedy algorithms with respect to dictionaries in Banach spaces was obtained by using some kind of [Formula: see text]-functional.
Abstract: In this paper, we obtain the unified error estimate for some weak biorthogonal greedy algorithms with respect to dictionaries in Banach spaces by using some kind of [Formula: see text]-functional. From this estimate, we derive the sufficient conditions for the convergence and the convergence rates on sparse classes induced by the [Formula: see text]-functional. The results on convergence and the convergence rates are sharp.
4 citations
TL;DR: In this article , the physics of wave propagation in one-dimensional waveguide lattices, with randomly distributed gain or loss, were investigated. But the role of eigenstates' nonorthogonality and biorthogonal projection is not discussed.
Abstract: In the context of non-Hermitian Anderson localization, we study the physics of wave propagation in one-dimensional waveguide lattices, with randomly distributed gain or loss. Despite the Anderson localization of all eigenstates, the system exhibits counterintuitive propagation by quantized jumps between states located around distant sites. Such a novel effect was recently experimentally demonstrated in optical fiber loop networks. We provide a systematic way of understanding the underlying physical mechanism of such an effect. The role of eigenstates' nonorthogonality and biorthogonal projection is of central importance to our work, and is systematically examined in the symmetric non-Hermitian model, as well as the nonsymmetric Hatano-Nelson Hamiltonian. Our methodology can be applied to any non-Hermitian disordered system that contains complex elements with loss and/or gain, and thus exploits the meaning of wave transport in complex open systems.
TL;DR: In this paper , a biorthogonal approach based on temporal coupled-mode theory is proposed to unravel the underlying physics of chiral metasurfaces, which inherits the intrinsic properties of open optical cavities, including time-reversal symmetry and non-Hermitian Hamiltonians, which are found to be in excellent agreement with numerical results.
Abstract: The physical origins of chiroptical responses from artificial optically active media are significant for developing high-performance circular dichroism (CD) spectroscopic techniques. Here, we present a biorthogonal approach based on temporal coupled-mode theory to unravel the underlying physics of chiral metasurfaces. Equipped with physically meaningful parameters, this approach inherits the intrinsic properties of open optical cavities, including time-reversal symmetry and non-Hermitian Hamiltonians, which are found to be in excellent agreement with numerical results. Remarkably, it identifies that the intrinsic chirality of coupled chiral nanocavities arises from (i) the asymmetric coupling between interlayer cross-polarized resonant modes and (ii) a coherent interference between doubly degenerate states. Based on this formalism, a critical coupling condition capable of achieving zero transmission for circularly polarized light is proposed.
TL;DR: In this paper , a triplet of q-difference operators X, Y, Z is shown to play a role analogous to the pair of bispectral operators of orthogonal polynomials.
17 Nov 2022
TL;DR: In this paper , a non-Hermitian two-coupled SYK model was shown to provide thermodynamic structure equivalent to a Hermitian 2-couple model.
Abstract: We show that a non-Hermitian two coupled Sachdev-Ye-Kitaev (SYK) model can provide thermodynamic structure equivalent to a Hermitian two coupled SYK model. The energy spectrum, the entanglement degree of the ground states and the low energy effective action of this model are not influenced by the non-Hermiticity. The novel biorthogonal ground states demonstrates that two SYK sites, one of which can be in the ground state and the other in the Schwarzian excited state by tuning the non-Hermiticity. We find evidence that the free energy is independent of the non-Hermiticity.
TL;DR: In this article , a methodology for analyzing chemical bonds embedded in the electronic wave function of molecules, especially in terms of spin correlations or so-called "local spin," is presented.
Abstract: We present a methodology for analyzing chemical bonds embedded in the electronic wave function of molecules, especially in terms of spin correlations or so-called "local spin." In this paper, based on biorthogonal second quantization, the spin correlation functions of molecules are naturally introduced, which enables us to extract local singlet and local triplet elements from the wave function. We also clarify the relationship between these spin correlations and traditional chemical concepts, i.e., resonance structures. Several chemical reactions, including the intramolecular radical cyclization and the formation of preoxetane, are demonstrated to verify the analysis method numerically.
01 Feb 2022
TL;DR: In this paper , the authors show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo.
Abstract: We show that non-Hermitian biorthogonal many-body phase transitions can be characterized by the enhanced decay of Loschmidt echo. The quantum criticality is numerically investigated in a non-Hermitian transverse field Ising model by performing the finite-size dynamical scaling of Loschmidt echo. We determine the equilibrium correlation length critical exponents that are consistent with previous results from the exact diagonalization. More importantly, we introduce a simple method to detect quantum phase transitions with the short-time average of rate function motivated by the critically enhanced decay behavior of Loschmidt echo. Our studies show how to detect equilibrium many-body phase transitions with biorthogonal Loschmidt echo that can be observed in future experiments via quantum dynamics after a quench.
TL;DR: In this paper , the wavelet Galerkin method was used to solve the fractional Riccati equation and the convergence rate was shown to be O(2−s).
Abstract: This paper is devoted to the wavelet Galerkin method to solve the Fractional Riccati equation. To this end, biorthogonal Hermite cubic Spline scaling bases and their properties are introduced, and the fractional integral is represented based on these bases as an operational matrix. Firstly, we obtain the Volterra integral equation with a weakly singular kernel corresponding to the desired equation. Then, using the operational matrix of fractional integration and the Galerkin method, the corresponding integral equation is reduced to a system of algebraic equations. Solving this system via Newton’s iterative method gives the unknown solution. The convergence analysis is investigated and shows that the convergence rate is O(2−s). To demonstrate the efficiency and accuracy of the method, some numerical simulations are provided.
TL;DR: In this article , the authors considered the Muttalib-Borodin ensemble of Laguerre type, a determinantal point process on [0,∞] which depends on the varying weights xαe−nV(x), α>−1, and a parameter θ.
TL;DR: A survey of polynomials P n arising from convergence acceleration and numerical integration that satisfy "biorthogonality" conditions such as ∫ a b P n x ϕ j x w x d x = 0, for appropriate functions ϕ x and weights w is given in this article.
TL;DR: In this paper , a new multivariate Toda hierarchy of nonlinear partial differential equations adapted to multivariate biorthogonal polynomials is discussed, which is associated with non-standard multivariate BORthogonality.
TL;DR: In this paper , it was shown that a Banach space with a fundamental biorthogonal system admits a polyhedral norm that locally depends on finitely many coordinates (LFC).
Abstract:
Let $\mathcal {X}$ be a Banach space with a fundamental biorthogonal system, and let $\mathcal {Y}$ be the dense subspace spanned by the vectors of the system. We prove that $\mathcal {Y}$ admits a $C^\infty $-smooth norm that locally depends on finitely many coordinates (LFC, for short), as well as a polyhedral norm that locally depends on finitely many coordinates. As a consequence, we also prove that $\mathcal {Y}$ admits locally finite, $\sigma $-uniformly discrete $C^\infty $-smooth and LFC partitions of unity and a $C^1$-smooth locally uniformly rotund norm. This theorem substantially generalises several results present in the literature and gives a complete picture concerning smoothness in such dense subspaces. Our result covers, for instance, every weakly Lindelöf determined Banach space (hence, all reflexive ones), $L_1(\mu )$ for every measure $\mu $, $\ell _\infty (\Gamma )$ spaces for every set $\Gamma $, $C(K)$ spaces where $K$ is a Valdivia compactum or a compact Abelian group, duals of Asplund spaces, or preduals of Von Neumann algebras. Additionally, under Martin Maximum MM, all Banach spaces of density $\omega _1$ are covered by our result.
TL;DR: In this paper , the authors compared the influence of selecting a mother wavelet and decomposition level on the forecast accuracy of time series compiled from the daily remaining balances in the TSA.
Abstract: Improving the accuracy of cash flow forecasting in the TSA is key to fulfilling government payment obligations, minimizing the cost of maintaining the cash reserve, providing the absence of outstanding debt accumulation and ensuring investment in financial instruments to obtain additional income. This study aims to improve the accuracy of traditional methods of forecasting the time series compiled from the daily remaining balances in the TSAbased on prior decomposition using a discrete wavelet transform. The paper compares the influence of selecting a mother wavelet out of 570 mother wavelet functions belonging to 10 wavelet families (Haar;Dabeshies; Symlet; Coiflet; Biorthogonal Spline; Reverse Biorthogonal Spline; Meyer; Shannon; Battle-Lemarie; and Cohen–Daubechies–Feauveau) and the decomposition level (from 1 to 8) on the forecast accuracy of time series compiled from the daily remaining balances in the TSA in comparison with the traditional forecasting method without prior timeseries decomposition. The model with prior time series decomposition based on the Reverse Biorthogonal Spline Wavelet [5.5] mother wavelet function, upon the eighth iteration, features the highest accuracy, significantly higher than that of the traditional forecasting models. The choice of the mother wavelet and the decomposition level play an important role in increasing the accuracy of forecasting the daily remaining balances in the TSA.
TL;DR: In this paper , a generalization of Laurent biorthogonal polynomials is proposed and its recurrence relation and Christoffel transformation are derived, which yields an extension of the fully discrete relativistic Toda lattice, one of which can reduce to the Narita-Itoh-Bogoyavlensky lattice.
Abstract: This paper is concerned about certain generalization of Laurent biorthogonal polynomials together with the corresponding related integrable lattices. On one hand, a generalization for Laurent biorthogonal polynomials is proposed and its recurrence relation and Christoffel transformation are derived. On the other hand, it turns out the compatibility condition between the recurrence relation and the Christoffel transformation for the generalized Laurent biorthogonal polynomials yields an extension of the fully discrete relativistic Toda lattice. And also, it is shown that isospectral deformations of the generalized Laurent biorthogonal polynomials lead to two different generalizations of the continuous-time relativistic Toda lattice, one of which can reduce to the Narita–Itoh–Bogoyavlensky lattice.
TL;DR: In this article , the boundary controllability of a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation is studied, and an estimate on the null-control cost is provided.
Abstract: <p style='text-indent:20px;'>In this paper we study the boundary controllability for a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation. We analyze both approximate and null boundary controllability properties. Besides, we provide an estimate on the null-control cost. The proofs are based on a detailed spectral analysis and the use of the moment method by Fattorini and Russell together with some results on biorthogonal families.</p>
TL;DR: In this paper , a wavelet-based adaptive version of the proper orthogonal decomposition (the wPOD) is proposed to reduce the amount of data to be analyzed by compressing them using biorthogonal wavelets, yielding a sparse representation.
Abstract: Abstract The proper orthogonal decomposition (POD) is a powerful classical tool in fluid mechanics used, for instance, for model reduction and extraction of coherent flow features. However, its applicability to high-resolution data, as produced by three-dimensional direct numerical simulations, is limited owing to its computational complexity. Here, we propose a wavelet-based adaptive version of the POD (the wPOD), in order to overcome this limitation. The amount of data to be analyzed is reduced by compressing them using biorthogonal wavelets, yielding a sparse representation while conveniently providing control of the compression error. Numerical analysis shows how the distinct error contributions of wavelet compression and POD truncation can be balanced under certain assumptions, allowing us to efficiently process high-resolution data from three-dimensional simulations of flow problems. Using a synthetic academic test case, we compare our algorithm with the randomized singular value decomposition. Furthermore, we demonstrate the ability of our method analyzing data of a two-dimensional wake flow and a three-dimensional flow generated by a flapping insect computed with direct numerical simulation.
01 Jan 2022
TL;DR: In this article , the dual system of the Malmquist-takenaka complex orthonormal system on the upper half-plane was introduced and proved biorthogonality.
Abstract: Starting from recent result of Fridli and Schipp, we introduce the dual system of the Malmquist–Takenaka complex orthonormal system on the upper half-plane, and prove the biorthogonality of the systems. We show that the nodal points of the biorthogonal systems on the unite disc and on the upper half-plane satisfy the similar equilibrium condition as it was proved previously for the discrete orthogonal system on the unite circle and on the real line. In this way we generalize the results obtained in [1, 2] by Pap and Schipp.
TL;DR: A survey of polynomials P n arising from convergence acceleration and numerical integration that satisfy "biorthogonality" conditions such as ∫ a b P n x ϕ j x w x d x = 0 , for appropriate functions ϕ x and weights w is given in this article .
26 May 2022
TL;DR: In this paper , a multiscale LMMSE estimation method using un-decimated wavelet transforms (UWT) has been proposed to generate the wavelet transformed images, Daubechies and Biorthogonal filters are used.
Abstract: Visual information exchange in the form of a digital picture has become ubiquitous in the world of communication. During the transmission of images, the information is distorted due to noise. This noise component distorts the quality of images. De-noising methods are used to expand the grade of picture pixels or to restore the original form of incoming data. In this article, we will be introducing a de-noising method for upgrading the excellence of the image. A multiscale LMMSE estimation method using un-decimated wavelet transforms (UWT) has been proposed. To generate the wavelet transformed images, Daubechies & Biorthogonal filters are used. In this proposed method, a Hybrid filter is generated using these two filters. Image produced using this method seems to be agreeable as compared to individual filters. The result is observed by the PSNR & MSE value for the quantitative measure.
21 Jun 2022
TL;DR: In this article , a MATLAB wavelet toolbox with a soft thresholding method was used to denoise the desired signal, and the results showed that the highest SNR value was 63.0172 dB, indicating that the filter had a high ability to remove the noises in EEG signals.
Abstract: Denoising is crucial in electroencephalography (EEG) processing to remove undesired components contaminated in a signal. Wavelet filters are a powerful and robust denoising approach to eliminate the noises in EEG. However, a broad number of wavelet families and decomposition levels confused the selection of the optimal and most appropriate wavelet filter. Therefore, this study aims to determine the optimal wavelet filter based on the signal-to-noise ratio (SNR) for EEG denoising. This work used the semi-simulated EEG signal contaminated with ocular noise as the observed signal. The wavelet filter with various wavelet families that is Haar, Daubechies (db), Symlets (sym), coiflets (coif), Discrete Meyer (dmey), Fejer-Korovkin (fk), biorthogonal (bior), and Reverse Biorthogonal (rbior) from decomposition level 1 to 8 were applied. A MATLAB wavelet toolbox with a soft thresholding method was used to denoise the desired signal. The result showed that the highest SNR value was 63.0172 dB. The highest SNR indicated that the filter had a high ability to remove the noises in EEG signals. Therefore, this work suggested that the haar, db1, bior1.1, and rbior1.1 of the mother wavelet at decomposition level 8 were the most efficient for removing the ocular noise.