Slepian functions and their use in signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla

Bodily tides near spin-orbit resonances

Spin-orbit coupling can be described in two approaches. The method known as "the MacDonald torque" is often combined with an assumption that the quality factor Q is frequency-independent. This makes the method inconsistent, because the MacDonald theory tacitly fixes the rheology by making Q scale as the inverse tidal frequency. Spin-orbit coupling can be treated also in an approach called "the Darwin torque". While this theory is general enough to accommodate an arbitrary frequency-dependence of Q, this advantage has not yet been exploited in the literature, where Q is assumed constant or is set to scale as inverse tidal frequency, the latter assertion making the Darwin torque equivalent to a corrected version of the MacDonald torque. However neither a constant nor an inverse-frequency Q reflect the properties of realistic mantles and crusts, because the actual frequency-dependence is more complex. Hence the necessity to enrich the theory of spin-orbit interaction with the right frequency-dependence. We accomplish this programme for the Darwin-torque-based model near resonances. We derive the frequency-dependence of the tidal torque from the first principles, i.e., from the expression for the mantle's compliance in the time domain. We also explain that the tidal torque includes not only the secular part, but also an oscillating part. We demonstrate that the lmpq term of the Darwin-Kaula expansion for the tidal torque smoothly goes through zero, when the secondary traverses the lmpq resonance (e.g., the principal tidal torque smoothly goes through nil as the secondary crosses the synchronous orbit). We also offer a possible explanation for the unexpected frequency-dependence of the tidal dissipation rate in the Moon, discovered by LLR

Scalar and vector Slepian functions, spherical signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences, for scalar and vectorial signals defined on the surface of a unit sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag. This is a slightly modified but expanded version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the Handbook, when it was called: Slepian functions and their use in signal estimation and spectral analysi

Phenomenology of the Lense-Thirring effect in the Solar System

Recent years have seen increasing efforts to directly measure some aspects of the general relativistic gravitomagnetic interaction in several astronomical scenarios in the solar system. After briefly overviewing the concept of gravitomagnetism from a theoretical point of view, we review the performed or proposed attempts to detect the Lense-Thirring effect affecting the orbital motions of natural and artificial bodies in the gravitational fields of the Sun, Earth, Mars and Jupiter. In particular, we will focus on the evaluation of the impact of several sources of systematic uncertainties of dynamical origin to realistically elucidate the present and future perspectives in directly measuring such an elusive relativistic effect.Comment: LaTex, 51 pages, 14 figures, 22 tables. Invited review, to appear in Astrophysics and Space Science (ApSS). Some uncited references in the text now correctly quoted. One reference added. A footnote adde

A resonant-term-based model including a nascent disk, precession, and oblateness: application to GJ 876

Investigations of two resonant planets orbiting a star or two resonant satellites orbiting a planet often rely on a few resonant and secular terms in order to obtain a representative quantitative description of the system's dynamical evolution. We present a semianalytic model which traces the orbital evolution of any two resonant bodies in a first- through fourth-order eccentricity or inclination-based resonance dominated by the resonant and secular arguments of the user's choosing. By considering the variation of libration width with different orbital parameters, we identify regions of phase space which give rise to different resonant ''depths,'' and propose methods to model libration profiles. We apply the model to the GJ 876 extrasolar planetary system, quantify the relative importance of the relevant resonant and secular contributions, and thereby assess the goodness of the common approximation of representing the system by just the presumably dominant terms. We highlight the danger in using ''order'' as the metric for accuracy in the orbital solution by revealing the unnatural libration centers produced by the second-order, but not first-order, solution, and by demonstrating that the true orbital solution lies somewhere ''in-between'' the third- and fourth-order solutions. We also present formulas used to incorporate perturbations from central-body oblateness and precession, and a protoplanetary or protosatellite thin disk with gaps, into a resonant system. We quantify these contributions to the GJ 876 system, and thereby highlight the conditions which must exist for multi-planet exosystems to be significantly influenced by such factors. We find that massive enough disks may convert resonant libration into circulation; such disk-induced signatures may provide constraints for future studies of exoplanet systems.Comment: 39 pages of body text, 21 figures, 5 tables, 1 appendix, accepted for publication in Celestial Mechanics and Dynamical Astronom

Cassini Radio Science

Cassini radio science investigations will be conducted both during the cruise (gravitational wave and conjunction experiments) and the Saturnian tour of the mission (atmospheric and ionospheric occultations, ring occultations, determinations of masses and gravity fields). New technologies in the construction of the instrument, which consists of a portion on-board the spacecraft and another portion on the ground, including the use of the Ka-band signal in addition to that of the S- and X-bands, open opportunities for important discoveries in each of the above scientific areas, due to increased accuracy, resolution, sensitivity, and dynamic range.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/43765/1/11214_2004_Article_1436.pd

Climatic oscillations during the Precambrian era

A remarkably regular cyclicity with a fundamental period of ~11–12 cycles is preserved in the 680 million year old Elatina formation of South Australia. All but one of the many periods present can be interpreted as resulting from the combined influences of the sunspot cycle and the lunar nodal tide - in particular, beating between these two cycles gives rise to a long period phase alternation. Available paleontological evidence is used to constrain the lunar distance 680 Ma ago, thereby constraining the length of the lunar nodal tide. We then infer from the beat period that the sunspot cycle was 10.8 ± 0.2 years, which is in agreement with independent astronomical evidence suggesting that the sunspot cycle would then have been some 3–10% shorter than it is at present. Although this interpretation is not consistent with the 12.0 year sunspot cycle counted by Williams and Sonett (1985), we demonstrate that unavoidable random errors made in discriminating unusually indistinct varves gives rise to a systematic overcount of varves of this magnitude. The clarity of this evidence for solar and lunar signals in the climate 680 Ma ago lends support to reports of their importance today.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42576/1/10584_2004_Article_BF00143906.pd

Zen and the Art of Living Mindfully: The Health-Enhancing Potential of Zen Aesthetics

Amidst the burgeoning enthusiasm for mindfulness in the West, there is a concern that the largely secular ‘de-contextualized’ way in which it is being harnessed is denuding it of its potential to improve health and well-being. As such, efforts are underway to ‘re-contextualize’ mindfulness, explicitly drawing on the wider framework of Buddhist ideas and practices in which it was initially developed. This paper aims to contribute to this, doing so by focusing on Zen Buddhism, and in particular on Zen aesthetic principles. The article concentrates on the seven principles identified by Hisamatsu (1971) in his classic text Zen and the Fine Arts: kanso (simplicity); fukinsei (asymmetry); koko (austere sublimity); shizen (naturalness); daisuzoku (freedom from routine); sei-jaku (tranquillity); and yūgen (profound grace). The presence of these principles in works of art is seen as reflecting and communicating insights that are central to Buddhism, such as non-attachment. Moreover, these principles do not only apply to the creation and appreciation of art, but have clear applications for treating health-related issues, and improving quality of life more generally. This paper makes the case that embodying these principles in their lives can help people enhance their psychosomatic well-being, and come to a truer understanding of the essence of mindful living

Genesis, growth and agency of National bibliography (National bibliography 1)

92-96<span style="font-size:12.0pt;line-height: 115%;font-family:" times="" new="" roman";mso-fareast-font-family:"times="" roman";="" mso-ansi-language:en-us;mso-fareast-language:en-us;mso-bidi-language:ar-sa"="" lang="EN-US">Gives the several definitions of National bibliography according to various authors and evaluates them. Enumerates the different types. Explains the purpose of selective bibliography. Brings out the need for National bibliography. Describes the purposes of National bibliography on the basis of utility and users. Mentions the agencies bringing out National bibliography.</span