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Uncertainty Principle (uncertainty + principle)
Selected AbstractsNMR and the uncertainty principle: How to and how not to interpret homogeneous line broadening and pulse nonselectivity.CONCEPTS IN MAGNETIC RESONANCE, Issue 5 2008IV. (Un?)certainty Abstract Following the treatments presented in Parts I, II, and III, I herein address the popular notion that the frequency of a monochromatic RF pulse as well as that of a monochromatic FID is "in effect" uncertain due to the (Heisenberg) Uncertainty Principle, which also manifests itself in the fact that the FT-spectrum of these temporal entities is spread over a nonzero frequency band. I will show that the frequency spread should not be interpreted as "in effect" meaning a range of physical driving RF fields in the former, and "spin frequencies" in the latter case. The fact that a shorter pulse or a more quickly decaying FID has a wider FT-spectrum is in fact solely due to the Fourier Uncertainty Principle, which is a less well known and easily misunderstood concept. A proper understanding of the Fourier Uncertainty Principle tells us that the FT-spectrum of a monochromatic pulse is not "broad" because of any "uncertainty" in the RF frequency, but because the spectrum profile carries all of the pulse's features (frequency, phase, amplitude, length, temporal location) coded into the complex amplitudes of the FT-spectrum's constituent eternal basis harmonic waves. A monochromatic RF pulse's capability to excite nonresonant magnetizations is in fact a purely classical off-resonance effect that has nothing to do with "uncertainty". Analogously, "Lorentzian lineshape" means exactly the same thing physically as "exponential decay," and all inferences as to the physical reasons for that decay must be based on independent assumptions or observations. © 2008 Wiley Periodicals, Inc. Concepts Magn Reson Part A 32A: 373,404, 2008. [source] NMR and the uncertainty principle: How to and how not to interpret homogeneous line broadening and pulse nonselectivity.CONCEPTS IN MAGNETIC RESONANCE, Issue 4 2008Abstract Following the treatments presented in Parts I and II, I herein discuss in more detail the popular notion that the frequency of a monochromatic RF pulse as well as that of a monochromatic FID is "in effect" uncertain due to the (Heisenberg) Uncertainty Principle, which also manifests itself in the fact that the FT-spectrum of these temporal entities is spread over a nonzero frequency band. In Part III, I continue my preliminary review of some further fundamental concepts, such as the Heisenberg and Fourier Uncertainty Principles, that are needed to understand whether or not the NMR linewidth and the RF excitation bandwidth have anything to do with "uncertainty". The article then culminates in re-addressing our Two NMR Problems in a more conscientious frame of mind by using a more refined formalism. The correct interpretation of these problems will be discussed in Part IV. © 2008 Wiley Periodicals, Inc. Concepts Magn Reson Part A 32A: 302,325, 2008. [source] NMR and the uncertainty principle: How to and how not to interpret homogeneous line broadening and pulse nonselectivity.CONCEPTS IN MAGNETIC RESONANCE, Issue 5 2008IV. (Un?)certainty Abstract Following the treatments presented in Parts I, II, and III, I herein address the popular notion that the frequency of a monochromatic RF pulse as well as that of a monochromatic FID is "in effect" uncertain due to the (Heisenberg) Uncertainty Principle, which also manifests itself in the fact that the FT-spectrum of these temporal entities is spread over a nonzero frequency band. I will show that the frequency spread should not be interpreted as "in effect" meaning a range of physical driving RF fields in the former, and "spin frequencies" in the latter case. The fact that a shorter pulse or a more quickly decaying FID has a wider FT-spectrum is in fact solely due to the Fourier Uncertainty Principle, which is a less well known and easily misunderstood concept. A proper understanding of the Fourier Uncertainty Principle tells us that the FT-spectrum of a monochromatic pulse is not "broad" because of any "uncertainty" in the RF frequency, but because the spectrum profile carries all of the pulse's features (frequency, phase, amplitude, length, temporal location) coded into the complex amplitudes of the FT-spectrum's constituent eternal basis harmonic waves. A monochromatic RF pulse's capability to excite nonresonant magnetizations is in fact a purely classical off-resonance effect that has nothing to do with "uncertainty". Analogously, "Lorentzian lineshape" means exactly the same thing physically as "exponential decay," and all inferences as to the physical reasons for that decay must be based on independent assumptions or observations. © 2008 Wiley Periodicals, Inc. Concepts Magn Reson Part A 32A: 373,404, 2008. [source] NMR and the uncertainty principle: How to and how not to interpret homogeneous line broadening and pulse nonselectivity.CONCEPTS IN MAGNETIC RESONANCE, Issue 4 2008Abstract Following the treatments presented in Parts I and II, I herein discuss in more detail the popular notion that the frequency of a monochromatic RF pulse as well as that of a monochromatic FID is "in effect" uncertain due to the (Heisenberg) Uncertainty Principle, which also manifests itself in the fact that the FT-spectrum of these temporal entities is spread over a nonzero frequency band. In Part III, I continue my preliminary review of some further fundamental concepts, such as the Heisenberg and Fourier Uncertainty Principles, that are needed to understand whether or not the NMR linewidth and the RF excitation bandwidth have anything to do with "uncertainty". The article then culminates in re-addressing our Two NMR Problems in a more conscientious frame of mind by using a more refined formalism. The correct interpretation of these problems will be discussed in Part IV. © 2008 Wiley Periodicals, Inc. Concepts Magn Reson Part A 32A: 302,325, 2008. [source] Herbivore and pathogen damage on grassland and woodland plants: a test of the herbivore uncertainty principleECOLOGY LETTERS, Issue 4 2002Stefan A. Schnitzer Researchers can alter the behaviour and ecology of their study organisms by conducting such seemingly benign activities as non-destructive measurements and observations. In plant communities, researcher visitation and measurement of plants may increase herbivore damage in some plant species while decreasing it in others. Simply measuring plants could change their competitive ability by altering the amount of herbivore damage that they suffer. Currently, however, there is only limited empirical evidence to support this `herbivore uncertainty principle' (HUP). We tested the HUP by quantifying the amount of herbivore and pathogen damage in 13 plant species (> 1400 individuals) at four different visitation intensities at Cedar Creek Natural History Area, Minnesota, USA. Altogether, we found very little evidence to support the HUP at any intensity of visitation. Researcher visitation did not alter overall plant herbivore damage or survival and we did not detect a significant visitation effect in any of the 13 species. Pathogen damage also did not significantly vary among visitation treatments, although there was some evidence that high visitation caused slightly higher pathogen damage. Based on our results, we question whether this phenomenon should be considered a `principle' of plant ecology. [source] Localized spectral analysis on the sphereGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 3 2005Mark A. Wieczorek SUMMARY It is often advantageous to investigate the relationship between two geophysical data sets in the spectral domain by calculating admittance and coherence functions. While there exist powerful Cartesian windowing techniques to estimate spatially localized (cross-)spectral properties, the inherent sphericity of planetary bodies sometimes necessitates an approach based in spherical coordinates. Direct localized spectral estimates on the sphere can be obtained by tapering, or multiplying the data by a suitable windowing function, and expanding the resultant field in spherical harmonics. The localization of a window in space and its spectral bandlimitation jointly determine the quality of the spatiospectral estimation. Two kinds of axisymmetric windows are here constructed that are ideally suited to this purpose: bandlimited functions that maximize their spatial energy within a cap of angular radius ,0, and spacelimited functions that maximize their spectral power within a spherical harmonic bandwidth L. Both concentration criteria yield an eigenvalue problem that is solved by an orthogonal family of data tapers, and the properties of these windows depend almost entirely upon the space,bandwidth product N0= (L+ 1) ,0/,. The first N0, 1 windows are near perfectly concentrated, and the best-concentrated window approaches a lower bound imposed by a spherical uncertainty principle. In order to make robust localized estimates of the admittance and coherence spectra between two fields on the sphere, we propose a method analogous to Cartesian multitaper spectral analysis that uses our optimally concentrated data tapers. We show that the expectation of localized (cross-)power spectra calculated using our data tapers is nearly unbiased for stochastic processes when the input spectrum is white and when averages are made over all possible realizations of the random variables. In physical situations, only one realization of such a process will be available, but in this case, a weighted average of the spectra obtained using multiple data tapers well approximates the expected spectrum. While developed primarily to solve problems in planetary science, our method has applications in all areas of science that investigate spatiospectral relationships between data fields defined on a sphere. [source] Sure independence screening for ultrahigh dimensional feature spaceJOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 5 2008Jianqing Fan Summary., Variable selection plays an important role in high dimensional statistical modelling which nowadays appears in many areas and is key to various scientific discoveries. For problems of large scale or dimensionality p, accuracy of estimation and computational cost are two top concerns. Recently, Candes and Tao have proposed the Dantzig selector using L1 -regularization and showed that it achieves the ideal risk up to a logarithmic factor log (p). Their innovative procedure and remarkable result are challenged when the dimensionality is ultrahigh as the factor log (p) can be large and their uniform uncertainty principle can fail. Motivated by these concerns, we introduce the concept of sure screening and propose a sure screening method that is based on correlation learning, called sure independence screening, to reduce dimensionality from high to a moderate scale that is below the sample size. In a fairly general asymptotic framework, correlation learning is shown to have the sure screening property for even exponentially growing dimensionality. As a methodological extension, iterative sure independence screening is also proposed to enhance its finite sample performance. With dimension reduced accurately from high to below sample size, variable selection can be improved on both speed and accuracy, and can then be accomplished by a well-developed method such as smoothly clipped absolute deviation, the Dantzig selector, lasso or adaptive lasso. The connections between these penalized least squares methods are also elucidated. [source] Stable signal recovery from incomplete and inaccurate measurementsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 8 2006Emmanuel J. Candès Suppose we wish to recover a vector x0 , ,,, (e.g., a digital signal or image) from incomplete and contaminated observations y = A x0 + e; A is an ,, × ,, matrix with far fewer rows than columns (,, , ,,) and e is an error term. Is it possible to recover x0 accurately based on the data y? To recover x0, we consider the solution x# to the ,,1 -regularization problem where , is the size of the error term e. We show that if A obeys a uniform uncertainty principle (with unit-normed columns) and if the vector x0 is sufficiently sparse, then the solution is within the noise level As a first example, suppose that A is a Gaussian random matrix; then stable recovery occurs for almost all such A's provided that the number of nonzeros of x0 is of about the same order as the number of observations. As a second instance, suppose one observes few Fourier samples of x0; then stable recovery occurs for almost any set of ,, coefficients provided that the number of nonzeros is of the order of ,,/(log ,,)6. In the case where the error term vanishes, the recovery is of course exact, and this work actually provides novel insights into the exact recovery phenomenon discussed in earlier papers. The methodology also explains why one can also very nearly recover approximately sparse signals. © 2006 Wiley Periodicals, Inc. [source] Herbivore and pathogen damage on grassland and woodland plants: a test of the herbivore uncertainty principleECOLOGY LETTERS, Issue 4 2002Stefan A. Schnitzer Researchers can alter the behaviour and ecology of their study organisms by conducting such seemingly benign activities as non-destructive measurements and observations. In plant communities, researcher visitation and measurement of plants may increase herbivore damage in some plant species while decreasing it in others. Simply measuring plants could change their competitive ability by altering the amount of herbivore damage that they suffer. Currently, however, there is only limited empirical evidence to support this `herbivore uncertainty principle' (HUP). We tested the HUP by quantifying the amount of herbivore and pathogen damage in 13 plant species (> 1400 individuals) at four different visitation intensities at Cedar Creek Natural History Area, Minnesota, USA. Altogether, we found very little evidence to support the HUP at any intensity of visitation. Researcher visitation did not alter overall plant herbivore damage or survival and we did not detect a significant visitation effect in any of the 13 species. Pathogen damage also did not significantly vary among visitation treatments, although there was some evidence that high visitation caused slightly higher pathogen damage. Based on our results, we question whether this phenomenon should be considered a `principle' of plant ecology. [source] | ||||||||||