Resolution theorem

Manne s resolution theorems clearly stated how the distribution of the concentration and spectral windows of the different components in a data set could affect the quality of the pure profiles recovered after data analysis [22], The correct knowledge of these windows is the cornerstone of some resolution methods, and in others where it is not essential, information derived from this knowledge can be introduced to generally improve the results obtained. 

Kedar-Cabelli, S. and McCarthy, L. Explanation-based generalization as resolution theorem proving Proceedings of the Fourth Intematiorud Workshop on Machine Learning, University of California, Irvine, pp. 383-389.1987. 

It is apparent that the projeetion matriees are orthogonal, and using the spectral resolution theorem we may write the representation matrix of H as... 

Finally, Chapter 5 by Hiroshi Watari and Yuhei Shimoyama describes density-matrix formalism for treating angular momenta in multi-quantum systems. The spectrum resolution theorem is used to obtain a linear combination representation of the spin Hamiltonian and greatly simplifies the manipulation of angular momenta with high quantum numbers. 

For differential equations with periodic coefficients, the theorems are the same but the calculation of the characteristic exponents meets with difficulty. Whereas in the preceding case (constant coefficients), the coefficients of the characteristic equation are known, in the present case the characteristic equation contains the unknown solutions. Thus, one finds oneself in a vicious circle to be able to determine the characteristic exponents, one must know the solutions, and in order to know the latter, one must know first these exponents. The only resolution of this difficulty is to proceed by the method of successive approximations.11... 

Keywords radar radar imaging tomography high resolution synthetic aperture radar interferometry polarimetry Radon transform projection slice theorem backprojection. 

The second term on the right hand side of Equation 3.16 introduces complications because it couples x[ and xj. The first term, on the other hand is easily solved because it involves no coupling. The resolution of the difficulty introduced by the second term is to take advantage of the symmetry of the fy matrix. Note that each f is an element of a symmetric matrix and the second derivatives fy are independent of the order of differentiation. There is a well known mathematical theorem on the diagonalization of symmetric matrices which states (as applied to Equation 3.16) that when we introduce a new coordinate Q ... 

Theorem 4.1 (Ginzburg and Kapranov [25], Y. Ito and I. Nakamura [42]). The re striction of the Hilbert-Chow morphism to X is the minimal resolution of singularities... 

Conversely, starting from this description, Kronheimer constructed resolutions of the simple singularities [49] as follows. In order to explain the hyper-Kahler structures, we use the description given in Theorem 2.1 rather than that in Theorem 1.14. Let M = Hom(l/, Q 0 F) Hom(VF, V) Hom(F, W). Then its F-hxed point is... 

It is of importance that expression (5.12) holds even when /(x) is known only in part of space, as is the case in a crystallography experiment at finite resolution determined by Hmax. Using the Fourier convolution theorem, we may write (Dunitz and Seiler 1973)... 

The convolution theorem plays a valuable role in both exact and approximate descriptions of functions useful for analyzing resolution distortion and in helping us understand the effects of these functions in Fourier space. Functions of interest and their transforms can be constructed from our directory in Fig. 2 by forming their sums, products, and convolutions. This technique adds immeasurably to our intuitive grasp of resolution limitations imposed by instrumentation. 

A proof of this relation may be found in Bracewell (1978). Note that the spectral variable used in this and the next chapter is the same as that defined in Eqs. (7) and (8). Now consider a spatial distribution /(x) and its Fourier spectrum F(w) that come close to satisfying the equality in Eq. (4). We may take Ax and Aw as measures of the width, and hence the resolution, of the respective functions. To see how this relates to more realistic data, such as infrared spectral lines, consider shifting the peak function /(x) by various amounts and then superimposing all these shifted functions. This will give a reasonable approximation to a set of infrared lines. To discuss quantitatively what is occurring in the frequency domain, note that the Fourier spectrum of each shifted function by the shift theorem is given simply by the spectrum of the unshifted function multiplied by a constant phase factor. The superimposed spectrum would then be... 

Then we can apply the approximation theorem of Stone,SS) to 22/ and 22/. The relation (7.10) then follows immediately if we note that the resolution of the identity corresponding to 22/ is obtained from that of 22 by a simple transformation, and further argument is the same as in the text. 

Complete intersections are particular cases of a. CM. schemes they are purely dimensional by Macaulay s theorem and the Koszul complex is a resolution of S. 

Because of property (a), the a. C M. schemes of positive dimension satisfy the hypothesis of theorem (9.1). Property (a) is satisfied more generally by every scheme X c Vr whose homogeneous coordinate ring has a graded free resolution of lenght i r-1. Hence any such scheme satisfies the hypothesis of theorem (9.1). 

The correct performance of any curve-resolution (CR) method depends strongly on the complexity of the multicomponent system. In particular, the ability to correctly recover dyads of pure profiles and spectra for each of the components in the system depends on the degree of overlap among the pure profiles of the different components and the specific way in which the regions of existence of these profiles (the so-called concentration or spectral windows) are distributed along the row and column directions of the data set. Manne stated the necessary conditions for correct resolution of the concentration profile and spectrum of a component in the 2 following theorems [22] ... 

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