67 research outputs found

    On the accuracy of solving confluent Prony systems

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    In this paper we consider several nonlinear systems of algebraic equations which can be called "Prony-type". These systems arise in various reconstruction problems in several branches of theoretical and applied mathematics, such as frequency estimation and nonlinear Fourier inversion. Consequently, the question of stability of solution with respect to errors in the right-hand side becomes critical for the success of any particular application. We investigate the question of "maximal possible accuracy" of solving Prony-type systems, putting stress on the "local" behavior which approximates situations with low absolute measurement error. The accuracy estimates are formulated in very simple geometric terms, shedding some light on the structure of the problem. Numerical tests suggest that "global" solution techniques such as Prony's algorithm and ESPRIT method are suboptimal when compared to this theoretical "best local" behavior

    Fractal Weyl law for chaotic microcavities: Fresnel's laws imply multifractal scattering

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    We demonstrate that the harmonic inversion technique is a powerful tool to analyze the spectral properties of optical microcavities. As an interesting example we study the statistical properties of complex frequencies of the fully chaotic microstadium. We show that the conjectured fractal Weyl law for open chaotic systems [W. T. Lu, S. Sridhar, and M. Zworski, Phys. Rev. Lett. 91, 154101 (2003)] is valid for dielectric microcavities only if the concept of the chaotic repeller is extended to a multifractal by incorporating Fresnel's laws.Comment: 8 pages, 12 figure

    Eigenvectors in the Superintegrable Model II: Ground State Sector

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    In 1993, Baxter gave 2mQ2^{m_Q} eigenvalues of the transfer matrix of the NN-state superintegrable chiral Potts model with spin-translation quantum number QQ, where mQ=(NLLQ)/Nm_Q=\lfloor(NL-L-Q)/N\rfloor. In our previous paper we studied the Q=0 ground state sector, when the size LL of the transfer matrix is chosen to be a multiple of NN. It was shown that the corresponding τ2\tau_2 matrix has a degenerate eigenspace generated by the generators of r=m0r=m_0 simple sl2sl_2 algebras. These results enable us to express the transfer matrix in the subspace in terms of these generators Em±E_m^{\pm} and HmH_m for m=1,...,rm=1,...,r. Moreover, the corresponding 2r2^r eigenvectors of the transfer matrix are expressed in terms of rotated eigenvectors of HmH_m.Comment: LaTeX 2E document, using iopart.cls with iopams packages. 17 pages, uses eufb10 and eurm10 fonts. Typeset twice! vs2: Many changes and additions, adding 7 pages. vs3: minor corrections. vs4 minor improvement

    Excited-State Effective Masses in Lattice QCD

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    We apply black-box methods, i.e. where the performance of the method does not depend upon initial guesses, to extract excited-state energies from Euclidean-time hadron correlation functions. In particular, we extend the widely used effective-mass method to incorporate multiple correlation functions and produce effective mass estimates for multiple excited states. In general, these excited-state effective masses will be determined by finding the roots of some polynomial. We demonstrate the method using sample lattice data to determine excited-state energies of the nucleon and compare the results to other energy-level finding techniques.Comment: 18 pages, 6 figure

    Development of the Bélanger Equation and Backwater Equation by Jean-Baptiste Bélanger (1828)

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    A hydraulic jump is the sudden transition from a high-velocity to a low-velocity open channel flow. The application of the momentum principle to the hydraulic jump is commonly called the Bélanger equation, but few know that Bélanger's (1828) treatise was focused on the study of gradually varied open channel flows. Further, although Bélanger understood the rapidly-varied nature of the jump flow, he applied incorrectly the Bernoulli principle in 1828, and corrected his approach 10 years later. In 1828, his true originality lay in the successful development of the backwater equation for steady, one-dimensional gradually-varied flows in an open channel, together with the introduction of the step method, distance calculated from depth, and the concept of critical flow conditions

    Time-Frequency Analysis Using Local Taylor Series Expansion

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    Projected Nonlinear Least Squares for Exponential Fitting

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