Enhanced approximate cloaking by optimal change of variables

2014 ◽  
Vol 30 (3) ◽  
pp. 035014 ◽  
Author(s):  
Roland Griesmaier ◽  
Michael S Vogelius
Keyword(s):  
Change Of Variables ◽  
Optimal Change

On Krylov’s estimates for optional semimartingales

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
M. Abdelghani ◽  
A. Melnikov ◽  
A. Pak
Keyword(s):  
Sobolev Space ◽  
Measurable Function ◽  
Diffusion Processes ◽  
Stochastic Integrals ◽  
Controlled Diffusion ◽  
Convergence Of Solutions ◽  
Change Of Variables ◽  
General Class Of Functions ◽  
Class Of Functions ◽  
Controlled Diffusion Processes

Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.

Polynomial beta functions

1994 ◽  
Vol 14 (3) ◽  
pp. 453-474 ◽  
Author(s):  
Valerio De Angelis
Keyword(s):  
Spectral Radius ◽  
Algebraic Function ◽  
Beta Function ◽  
Complex Numbers ◽  
Algebraic Integers ◽  
Change Of Variables ◽  
Necessary And Sufficient ◽  
Positive Complex

AbstractThe pointwise spectral radii of irreducible matrices whose entries are polynomials with positive, integral coefficients are studied in this paper. Most results are derived in the case that the resulting algebraic function, the beta function of S. Tuncel, is in fact a polynomial. We show that the set of beta functions forms a semiring, and the spectral radius of a matrix of beta functions is again a beta function. We also show that the coefficients of a polynomial beta function p must be real algebraic integers, and p satisfies (after a change of variables if necessary) the inequality for non-zero (and not all positive) complex numbers z1,…,zd. If and the ordered sequence of exponents appearing in p is of the form (m,m+1,…,M−,1,M) for some integers m and M, the same inequality is necessary and sufficient for p to be a beta function.

Direct Fourier migration for vertical velocity variations

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1504-1514 ◽  
Author(s):  
Gary F. Margrave
Keyword(s):  
Phase Shift ◽  
Vertical Velocity ◽  
Constant Velocity ◽  
Delta Function ◽  
Fourier Domain ◽  
Input Frequency ◽  
Output Frequency ◽  
Dirac Delta ◽  
Change Of Variables ◽  
Velocity Variations

The Stolt f‐x migration algorithm is a direct (i.e. nonrecursive) Fourier‐domain technique based on a change of variables, or equivalently a mapping, that converts the unmigrated spectrum to the migrated spectrum. The algorithm is simple and efficient but limited to constant velocity. A v(z) f‐k migration method, capable of very high accuracy for vertical velocity variations, can be formulated as a nonstationary filter that avoids the change of variables. The result is a direct Fourier‐domain process that, for each wavenumber, applies a nonstationary migration filter to a vector of input frequency samples to create a vector of output frequency samples. The filter matrix is analytically specified in the mixed domain of input frequency and migrated time. It can be moved to the full‐Fourier domain of input frequency and output frequency by a fast Fourier transform. When applied for constant velocity, the v(z) f‐k algorithm is slower than the Stolt method but without the usual artifacts related to complex‐valued frequency‐domain interpolation. Vertical velocity variations, through an rms‐velocity (straight‐ray) assumption, are handled by the v(z) f‐k method with no additional cost. Greater accuracy at slight additional expense is obtained by extending the method to a WKBJ phase‐shift integral. This has the same accuracy as recursive phase shift and is similar in cost. For constant velocity, the full‐Fourier domain migration filter is a discrete approximation to a Dirac delta function whose singularity tracks along a hyperbola determined by the migration velocity. For variable velocity, the migration filter has significant energy between hyperbolic trajectories determined by the minimum and maximum instantaneous velocities. The full‐Fourier domain offers interesting conceptual parallels to Stolt’s algorithm. However, unless a more efficient method of calculating the Fourier filter matrix can be found, the mixed‐domain method will be faster. The mixed‐domain nonstationary filter moves the input data from the Fourier domain to the migrated time domain as it migrates. It is faster because the migration filter is known analytically in the mixed domain.

Change of Variables in Factorization Method for Second-order Functional Equations

2004 ◽  
Vol 54 (11) ◽  
pp. 1257-1263 ◽  
Author(s):  
Alina Dobrogowska ◽  
Tomasz Goliński ◽  
Anatol Odzijewicz
Keyword(s):  
Functional Equations ◽  
Factorization Method ◽  
Second Order ◽  
Change Of Variables

scholarly journals Quantum square-well with logarithmic central spike

2018 ◽  
Vol 33 (02) ◽  
pp. 1850009 ◽  
Author(s):  
Miloslav Znojil ◽  
Iveta Semorádová
Keyword(s):  
Perturbation Theory ◽  
Boundary Conditions ◽  
Closed Form ◽  
Wave Functions ◽  
Change Of Variables ◽  
State Dependent ◽  
Square Well ◽  
Strength Formula ◽  
Schrödinger Perturbation ◽  
Dependent Interaction

Singular repulsive barrier [Formula: see text] inside a square-well is interpreted and studied as a linear analog of the state-dependent interaction [Formula: see text] in nonlinear Schrödinger equation. In the linearized case, Rayleigh–Schrödinger perturbation theory is shown to provide a closed-form spectrum at sufficiently small [Formula: see text] or after an amendment of the unperturbed Hamiltonian. At any spike strength [Formula: see text], the model remains solvable numerically, by the matching of wave functions. Analytically, the singularity is shown regularized via the change of variables [Formula: see text] which interchanges the roles of the asymptotic and central boundary conditions.

scholarly journals Meromorphic integrability of the Hamiltonian systems with homogeneous potentials of degree -4

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jaume Llibre ◽  
Yuzhou Tian
Keyword(s):  
Hamiltonian Systems ◽  
Homogeneous Polynomial ◽  
First Integral ◽  
Liouville Integrability ◽  
Linear Change ◽  
Change Of Variables ◽  
Polynomial First Integral ◽  
Homogeneous Potentials

<p style='text-indent:20px;'>We characterize the meromorphic Liouville integrability of the Hamiltonian systems with Hamiltonian <inline-formula><tex-math id="M2">\begin{document}$ H = \left(p_1^2+p_2^2\right)/2+1/P(q_1, q_2) $\end{document}</tex-math></inline-formula>, being <inline-formula><tex-math id="M3">\begin{document}$ P(q_1, q_2) $\end{document}</tex-math></inline-formula> a homogeneous polynomial of degree <inline-formula><tex-math id="M4">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> of one of the following forms <inline-formula><tex-math id="M5">\begin{document}$ \pm q_1^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ 4q_1^3q_2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \pm 6q_1^2q_2^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \pm \left(q_1^2+q_2^2\right)^2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ \pm q_2^2\left(6q_1^2-q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \pm q_2^2\left(6q_1^2+q_2^2\right) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ q_1^4+6\mu q_1^2q_2^2-q_2^4 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ -q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \mu&gt;-1/3 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ \mu\neq 1/3 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M15">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M16">\begin{document}$ \mu \neq \pm 1/3 $\end{document}</tex-math></inline-formula>. We note that any homogeneous polynomial of degree <inline-formula><tex-math id="M17">\begin{document}$ 4 $\end{document}</tex-math></inline-formula> after a linear change of variables and a rescaling can be written as one of the previous polynomials. We remark that for the polynomial <inline-formula><tex-math id="M18">\begin{document}$ q_1^4+6\mu q_1^2q_2^2+q_2^4 $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M19">\begin{document}$ \mu\in\left\{-5/3, -2/3\right\} $\end{document}</tex-math></inline-formula> we only can prove that it has no a polynomial first integral.</p>

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