It is often assumed that G is a Lie group, because then one can exploit the formulas of § 1.5 to lift transformations to the higher jets, without usage of the inverse function theorem. A function I is called a differential invariant of order k with respect to the action of G, if it...
The operator Ly=-y '' +q(t)y on the interval [0,∞) with a boundary condition sin α y(0)+cosαy ' (0)=0, under the assumption q(t+π)=q(t) is investigated. It is shown that the specification of only a finite number of eigenvalues leads to a unique determination of q(t...
By comparing the Laplace transform with the differential operator , we obtain a formula for the inverse Laplace transform = (1/)cos(), where is a unitary transformation operator. This helps us obtain an explicit spectral representation of . Some applications of the above relation are discussed.关...
The object image can be reconstructed according to the Fourier spectrum coefficients through inverse Fourier transform with the assistance of the Fourier algorithm. Realizing a 2D image with a single-pixel device in a limited space is a good starting point for intelligent photodetectors, which require...
of the laplacianand Sturm-Liouville operators. In addition we need restriction on the poten-tial. We investigate the stability of the solution of inverse problem for a self-adjoint discrete operator. Some of these results were published in (Sadovnichiiand Dubrovskii, 1979; Dubrovskii and Nagorny...
The normalized differential cross section for top-quark pair production in association with at least one jet is studied as a function of the inverse of the... R Poole,A Gamper,A Porter,... - 《Family Practice》 被引量: 29发表: 2011年 Measurement of the jet mass distribution of boosted ...
Inverse Fourier transform Now to find the inverse Fourier transform, in order to return of the linearized FSDDEs to time domain, for this we need to apply the inverse Fourier transform operator \(\mathcal {F}^{-1}\) to the Eq. (9) in the frequency domain given in form: $$\begin{...
The Plancherel formula for the latticeℏreads ℏ (2.2) To prove the inverse Fourier transform we perform the following computations: Letℓℏ. If we setℏ,, andℏℏ, thenℏis a function fromtoand using (2.1) we have: ℏℏℏ...
The Kepler problem, which describes the interaction of two point particles with an inverse square law of attraction, can be given by a system of three second order ordinary differential equations. The Lie algebra of its symmetry group contains the direct sum of the 2-dimensional non-abelian Lie...
The general idea of PINNs is to learn (or solve) the PDE solutions with DNNs, where the loss functions are formulated as a combination of the data mismatch and residuals of known PDEs, unifying forward and inverse problems within the same DNN optimization framework. The merits of PINNs have ...