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Article| Volume 98, ISSUE 1, P138-146, January 06, 2010

# Water and Backbone Dynamics in a Hydrated Protein

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## Abstract

Rotational immobilization of proteins permits characterization of the internal peptide and water molecule dynamics by magnetic relaxation dispersion spectroscopy. Using different experimental approaches, we have extended measurements of the magnetic field dependence of the proton-spin-lattice-relaxation rate by one decade from 0.01 to 300 MHz for 1H and showed that the underlying dynamics driving the protein 1H spin-lattice relaxation is preserved over 4.5 decades in frequency. This extension is critical to understanding the role of 1H2O in the total proton-spin-relaxation process. The fact that the protein-proton-relaxation-dispersion profile is a power law in frequency with constant coefficient and exponent over nearly 5 decades indicates that the characteristics of the native protein structural fluctuations that cause proton nuclear spin-lattice relaxation are remarkably constant over this wide frequency and length-scale interval. Comparison of protein-proton-spin-lattice-relaxation rate constants in protein gels equilibrated with 2H2O rather than 1H2O shows that water protons make an important contribution to the total spin-lattice relaxation in the middle of this frequency range for hydrated proteins because of water molecule dynamics in the time range of tens of ns. This water contribution is with the motion of relatively rare, long-lived, and perhaps buried water molecules constrained by the confinement. The presence of water molecule reorientational dynamics in the tens of ns range that are sufficient to affect the spin-lattice relaxation driven by 1H dipole-dipole fluctuations should make the local dielectric properties in the protein frequency dependent in a regime relevant to catalytically important kinetic barriers to conformational rearrangements.

## Introduction

Water molecule dynamics in and around proteins may contribute to the molecular motions that are critical to catalytic and biological function. The water-proton-spin-lattice-relaxation-rate constant, 1/T1, is dependent on water dynamics and is a primary determinant of contrast in magnetic images. For tissues as well as isolated immobilized protein systems, the spin-lattice-relaxation-rate constant for the water protons is a function of magnetic field strength and described by a power law in the proton Larmor frequency (
• Bryant R.G.
• Mendelson D.A.
• Lester C.C.
The magnetic field dependence of proton spin relaxation in tissues.
,
• Koenig S.H.
• Brown 3rd, R.D.
The importance of the motion of water for magnetic resonance imaging.
,
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
,
• Nusser W.
• Kimmich R.
Protein backbone fluctuations and NMR field-cycling relaxation spectroscopy.
,
• Korb J.-P.
• Bryant R.G.
Magnetic field dependence of proton spin-lattice relaxation of confined proteins.
,
• Korb J.-P.
• Bryant R.G.
Noise and functional protein dynamics.
,
• Korb J.-P.
• Bryant R.G.
Magnetic field dependence of proton spin-lattice relaxation times.
). This magnetic field dependence derives from the dynamics of protein-bound water and magnetic coupling of the water protons to the protons of nonrotating macromolecular components of the tissue, particularly proteins (
• Lester C.C.
• Bryant R.G.
Water-proton nuclear magnetic relaxation in heterogeneous systems: hydrated lysozyme results.
). This intermolecular magnetic coupling transfers the magnetic field dependence of the immobilized protein protons to the bulk water protons. Depending on composition, the effects of finite magnetization-transfer rates may limit the water-proton-relaxation-rate constant at the lowest magnetic field strengths and cause a low field plateau for this contribution. Although spectral resolution for a rotationally immobilized protein is low, the dynamic information available is not limited by the protein rotational correlation time. Therefore, it is possible to study protein and water dynamics at low frequencies.
The immobilized protein is magnetically a solid in the sense that the dipole-dipole couplings among the protons are not averaged by rapid rotational motion; the resulting proton NMR line width is large, on the order of 35 kHz. The critical consequence of this unaveraged dipolar coupling is that spin-spin communication among the protein protons is rapid (
• Bryant R.G.
Nuclear magnetic cross-relaxation spectroscopy.
).
In the nonrotating protein, these strong dipolar couplings bring the protons into equilibrium with each other rapidly with time constants on the order of the transverse relaxation time, which is ∼10 μs (
• Goldman M.
Spin Temperature and Nuclear Magnetic Resonance in Solids.
). Thus, motion causing 1H spin-lattice relaxation anywhere in the protein causes relaxation in the whole strongly coupled protein-spin system. In this study, we focus on the dynamics in a pure protein system although there are important consequences for more complex systems such as tissues.
The underlying nuclear spin-lattice-relaxation mechanism for protein protons derives from structural fluctuations that modulate interproton dipolar couplings (
• Nusser W.
• Kimmich R.
Protein backbone fluctuations and NMR field-cycling relaxation spectroscopy.
,
• Kimmich R.
• Fatkullin N.
• Gille K.
• et al.
Chain dynamics in entangled polymers: power laws of the proton and deuteron spin-lattice relaxation dispersions.
). A “spin-fracton” theory for a direct spin-lattice-relaxation process has accounted successfully for the frequency and temperature dependence of 1H spin-lattice relaxation in immobilized protein systems (
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
,
• Korb J.-P.
• Bryant R.G.
Noise and functional protein dynamics.
,
• Korb J.-P.
• Van-Quynh A.
• Bryant R.
Low-frequency localized spin-dynamical coupling in proteins.
). The temperature dependence for the protein-proton relaxation is unusual in that the rate constant is a linear function of temperature in the low field regime as predicted by this theory and confirmed experimentally over a temperature range of 150 K (
• Goddard Y.
• Korb J.-P.
• Bryant R.G.
The magnetic field and temperature dependences of proton spin-lattice relaxation in proteins.
). The term “fracton” is appropriate because the system is not periodic and the excitations are localized on a length scale determined by the structural elements of the protein. The basic idea is that the proton relaxation derives from small changes in the relative positions and orientations of protons that are caused by structural fluctuations. The spin-fracton theory uses a vibrational model for the direct spin-phonon process, as opposed to a Raman process. Unlike a three-dimensional (3D) ionic solid, the dynamical connectivity within a protein is not uniform in three-space because the polypeptide folds with side-chains of differing volume and packing characteristics. The nonuniform connectivity reduces the number of propagation pathways the model for the vibrational density of states has a different frequency dependence than that associated with the 3D model of Debye (
• Orbach R.
Dynamics of fractal networks.
). The Gaussian network models for protein dynamics use the same class of ideas to construct a network based on the proximity between structural units in the folded structure; the dynamical connectivity is reduced and the vibrational density of states deduced from this approach is characterized by the spectra dimension ds that is substantially <3. Thus, ds − 1 is much less than the Debye value of d − 1 where d = 3 for a uniformly connected 3D solid (
• Granek R.
• Klafter J.
Fractons in proteins: can they lead to anomalously decaying time autocorrelations?.
,
• Reuveni S.
• Granek R.
• Klafter J.
Proteins: coexistence of stability and flexibility.
). This spectral dimension enters the spin-fracton theory through the vibrational density of states that determines, in part, the magnetic field dependence of the relaxation-rate constant. In the same spirit, the theory also includes the dimensionality df for the distribution of mass in space, which is fractal and close to, but not precisely, 3 (
• Enright M.B.
• Leitner D.M.
Mass fractal dimension and the compactness of proteins.
,
• Enright M.B.
• Yu X.
• Leitner D.M.
Hydration dependence of the mass fractal dimension and anomalous diffusion of vibrational energy in proteins.
). A final aspect of the spin-fracton model is the nature of the spatial localization of the structural disturbances. The spatial extent, $ℓα$, ranges from a minimal size on the order of the monomer unit to the length of the polypeptide, and the spatial extent is related to the frequency of the fluctuation ωα. For a region of size, $ℓα$, modes for which ω > ωα are strongly overdamped. Scaling arguments show that the product $ℓαdfωαdS$ is a constant, thus, there is an anomalous dispersion relation $qα=1/ℓα∝ωαdS/df$ between q-space and frequency, instead of the usual dispersion relation q ∝ 1/ω encountered in crystals. We note that these ideas are not new, nor are they unique to models for spin-lattice relaxation; the same parameters enter recent theoretical studies of proteins (
• Reuveni S.
• Granek R.
• Klafter J.
Proteins: coexistence of stability and flexibility.
).
Halle and co-workers have criticized this approach and built a model (EMOR) for the magnetic field dependence of spin-lattice relaxation that depends on the dynamics of the water-molecule exchange rates as the source of the critical fluctuations in hydrated protein systems (
• Chávez F.V.
• Halle B.
Molecular basis of water proton relaxation in gels and tissue.
,
• Halle B.
Molecular theory of field-dependent proton spin-lattice relaxation in tissue.
,
• Halle B.
Cell water dynamics on multiple time scales.
). The essence of the idea applied to 1H relaxation is that the dipolar couplings are interrupted by water molecule or 1H exchange processes that lead to a dispersion like that seen for pure water at very low frequencies because of exchange interruption of the rotationally invariant 1H-17O scalar coupling. An individual water binding site then contributes a Lorentzian dispersion, but the power-law shape is achieved by a broad distribution of exchange times. A critical explicit assumption for this model is that the cross-relaxation among the protein protons is negligible. As a result, the water spins sense only adjacent protein spins, not the whole protein-proton-spin population or its magnetic field dependence. In this case, the effective dynamics are only those that modulate the intramolecular water dipolar coupling or the water-proton coupling to nearest neighbor protein protons. This assumption effectively suppresses the coupling of the magnetic field dependence of the protein-proton population with the water-proton-field dependence. The water molecule exchange processes should be thermally activated so that an exponential dependence on temperature is expected; however, the apparent temperature dependence may be affected by the different activation energies associated with the distribution of exchange sites. These authors suggest that the dipolar coupling between the water and the protein is quenched by the exchange process, and therefore, unimportant. Although first order dipolar splittings of 1H2O are rarely observed and the mixing caused by chemical exchange would suppress them, the spin relaxation depends on the perturbation carried to second order, and the coupling is not attenuated by the exchange although the coupling time may be reduced. Thus, the chemical exchange need not make the intermolecular cross-relaxation negligible. The importance of exchange modulation and cross-relaxation depends on the efficiency of the relaxation coupling that is dependent on several factors including composition, frequency, and temperature.
We note that although there are good reasons to use the spin-fracton model in this analysis, the conclusions concerning local water molecule dynamics do not depend on it. One could equivalently proceed with a strictly numerical parameterization of the background protein-proton-spin-lattice-relaxation rates where $1/T1protein=Aω−b$.
Previous studies of proton-spin-lattice relaxation as a function of water content in protein systems indicated that the apparent power-law exponent was a function of the water content (
• Korb J.-P.
• Bryant R.G.
Magnetic field dependence of proton spin-lattice relaxation of confined proteins.
). Such a dependence is not unreasonable because resolution is generally lost with dehydration in scattering experiments and it is likely that the protein structure responds to increased electrostatic strain caused by removal of water electric dipoles and hydrogen bonds (
• Martin R.W.
• Zilm K.W.
Preparation of protein nanocrystals and their characterization by solid state NMR.
,
• Morcombe C.R.
• Gaponenko V.
• Zilm K.W.
• et al.
13C CPMAS spectroscopy of deuterated proteins: CP dynamics, line shapes, and T1 relaxation.
). Nevertheless, we show here that the power-law exponent describing the field dependence of 1H spin-lattice relaxation does not change with hydration over nearly 5 decades in Larmor frequency when a sufficiently wide range of frequencies is studied. Using deuterium for proton isotope substitution, we show that hydration of lyophilized protein with deuterated water even up to the level of 100 g water/15 g protein in a cross-linked gel, leaves the magnetic field dependence of the proton-spin-lattice relaxation unperturbed with a power law, 1/T1 = Aω-b, and b = 0.78 ± 0.06. However, when the water molecules bear protons, the local motion of the water protons in long-lived protein sites provides another spin-relaxation contribution in the range of several MHz that may be resolved from the protein backbone dynamics and distort the appearance of the underlying magnetic field dependence. For bovine serum albumin (BSA), this contribution from rare bound-water molecule motion has an effective correlation time of tens of ns.

## Materials and Methods

BSA (Fraction V, Sigma-Aldrich, St. Louis, MO) was dialyzed against deionized water and lyophilized to constant weight using a mechanical vacuum at room temperature. To make BSA/H2O gels 0.1 mL of 25% aqueous glutaraldehyde (Sigma Chemical, St. Louis, MO) was added to 0.9 mL of 10 or 20% (g protein/100 g water) solutions of BSA in deionized H2O contained in 5 or 10 mm tubes; gels formed within 10 min. The gels in D2O were prepared by adding 0.1 mL of 25% aqueous glutaraldehyde to 0.9 mL of 15% BSA solution in D2O. The gel was soaked in D2O for 48 h with three changes of D2O per day then placed in a 10-mm tube for measurement. The hydrated protein samples were prepared to contain 0.32 g water per 1.0 g of protein by adding water to the lyophilized protein and equilibrating for at least 3 days at 310 K. The water content was measured using a Karl Fischer titrator (Aquatest 8; Photovolt Instruments, Indianapolis, IN).
The nuclear spin relaxation rates at Larmor frequencies between 0.01–30 MHz were recorded using a Stelar FFC-2000 spectrometer (Stelar, Mede, Italy). The field-switching time was 3 ms, the polarization field was 30 MHz and free induction decays were recorded after a single (5.5 μs) 90° excitation pulse applied at 15.8 MHz using a receiver-delay time of 11 μs. Temperature was controlled using a Stelar VTC90 variable temperature controller, which was calibrated using an external thermocouple inserted into a surrogate sample at the resonance position in the probe.
The NMR experiments for proton Larmor frequencies between 36–300 MHz were made using the fringe field of a 7.05 T magnet (Magnex Scientific, Oxford, UK) operating in conjunction with a Tecmag Apollo transceiver (Tecmag, Houston, TX), Miteq preamplifier (Miteq, Haupaug, NY), an AMT power amplifier (American Microwave Technologies, Brea, CA) and a probe constructed in this laboratory using a transmission-line design. The 90° pulse length was 8.5 μs. The sample was polarized in the resonance field of the superconducting magnet, after which the magnetization was inverted with a 180° pulse. The sample, then, was pneumatically shuttled to a calibrated position in the fringe field of the 7.05 T magnet for a variable relaxation delay and pneumatically returned to the resonance field, where the spectrum was promptly recorded using a 90° pulse. The integrated intensity of the NMR signal measured as a function of the relaxation delay provided the relaxation rate at the value of the calibrated fringe-field position. Within experimental error, all the decay/recovery curves of longitudinal magnetization were exponential. Fringe-field measurements were conducted at ambient laboratory temperature of ∼20°C.

## Results

Fig. 1 shows the protein-proton magnetic relaxation dispersion profile for dry BSA and a cross-linked gel at 15 g BSA/100 g of D2O. The water is strongly depleted in 1H and labile protein protons have been exchanged for deuterons by successive exchanges from D2O to minimize mobile 1H. The data sets are identical within experimental error except that deuteration of the amide protons decreases the effects of proton coupling to the amide 14N at magnetic field when the proton transitions match or cross the 14N transitions in the 0.5–3 MHz range (
• Kimmich R.
• Nusser W.
• Winter F.
In vivo NMR field-cycling relaxation spectroscopy reveals 14N1H relaxation sinks in the backbones of proteins.
,
• Winter F.
• Kimmich R.
14N1H and 2H1H cross-relaxation in hydrated proteins.
). The deuteration adds similar cross-relaxation peaks between the immobilized deuterons and the protons around 0.1–0.2 MHz. The environment in the cross-linked gel is highly aqueous and the protein structure is native. The protein is made rotationally immobile by the cross-linking reaction, and yet the magnetic relaxation dispersion (MRD) profile reproduces that for the dry solid. Therefore, by the measure of the magnetic-field dependence of the protein-proton-spin-lattice-relaxation-rate constant, the hydration effects on the protein-proton-spin-lattice-relaxation-dispersion profile are negligible if the water is 2H2O. The variety of water-protein interactions associated with the full hydration of the protein does not alter the underlying protein dynamics that cause the fluctuations in the proton-proton-dipole-dipole couplings that drive the spin-relaxation process. This observation does not depend on the details of the model for the nuclear spin-lattice-relaxation process and is somewhat different from our earlier conclusion based on MRD studies of proteins as a function of water content over a limited range of magnetic field strengths as discussed below.
Neglecting the 14N and 2H quadrupole relaxation peaks, the theory for the dry protein is summarized in Eq. 1 (
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
):
$1T1=4πkBTM2hds{341Ev∥2(1+12b)(hωoEv∥)−b+161Ev⊥2(72+12b)(hωoEv⊥)−b},$
(1)

where M2 is the proton second moment, ds the spectral dimension, $Ev∥$ and $Ev⊥$ are the vibrational frequencies for parallel and perpendicular motions of the polypeptide chain, ωo is the nuclear Larmor frequency, $b=3−2ds/df−ds$, and the other symbols have their usual meaning. The vibrational frequencies present a challenge; we have used the amide (I) frequency for the parallel mode of 1560 cm−1. The perpendicular mode appropriate to the whole chain is somewhat smaller; originally we assumed a value of order 200 cm−1 (
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
), but the more extensive data suggests that this value is too small. As the two vibrational frequencies become similar, the importance of separating the terms for parallel and perpendicular contributions is lost. The solid line in Fig. 1 was obtained with just the first term of Eq. 1 using the Levenburg-Marquardt algorithm and the fit parameters are listed in the Fig. 1 legend.
Fig. 2 summarizes relaxation rates for water and protein protons in rehydrated lyophilized BSA samples. The water signal is isolated from the protein signal by sampling the free induction decay after the solid component has decayed. Extrapolation of this component to zero time permits isolation of the solid component or protein proton relaxation rate constant. In general, the magnetic coupling of the two spin populations produces two decay components for the longitudinal magnetization. Because of finite magnetic field switching times, we are experimentally restricted to observe the slowly decaying longitudinal component of the total proton magnetization whether we observe water or protein protons. This slow longitudinal magnetization decay constant is
$1T1s=12{1T1W+1T1P+1TWP(1+1F)−[(1T1P−1T1W−1TWP(1−1F))2+4FTWP2]12},$
(2)

where F the ratio of the number of solid protons to the number of water protons, $1/T1P$ is the protein-proton-relaxation rate given by Eq. 1, $1/TWP$ is the pseudo first order rate constant for magnetization transfer between protein and water-proton populations, and $1/T1W$ is the bulk water-relaxation rate constant (
• Lester C.C.
• Bryant R.G.
Water-proton nuclear magnetic relaxation in heterogeneous systems: hydrated lysozyme results.
,
• Edzes H.T.
• Samulski E.T.
Cross relaxation and spin diffusion in the proton NMR or hydrated collagen.
). At all magnetic field strengths, the observable relaxation constants scale with the relative size of the two proton populations (
• Bryant R.G.
The dynamics of water-protein interactions.
).
At low temperature and low water content, the protein and water proton data are identical within experimental error and both spin populations report the same magnetic field dependence for the spin-lattice relaxation rate constant as shown in Fig. 2. At higher temperature or water contents, the water-proton profile falls below the protein proton profile at low frequency caused in part by data sampling errors for the protein spins, and because the magnetization transfer rate constant between the protein and water spins is concentration dependent. The Fig. 2 inset summarizes calculations of the MRD profile based on the assumption that the protein protons are described by a power law with an exponent of 0.78, the water proton relax with a field independent rate constant of 0.33, which neglects surface effects that are important at higher field strengths, and a protein-proton/water-proton ratio of 0.069 that is appropriate to a 10% BSA gel sample. The different profiles correspond to decreasing protein-to-water-proton-transfer-rate constants ranging from 105, which is in the strong coupling limit appropriate to a rigid solid, to 10, which clearly limits the low field relaxation rate constant to the magnetization transfer rate constant.
Fig. 3 summarizes relaxation dispersion profiles for a series of cross-linked BSA gels of different water content that correspond to values of F from 0.416 to 0.047. We observe the water protons because the signal is very large compared to the protein proton signal. The inset shows that the relaxation constant is a linear function of the quantity F/1 + F, which is the fraction of protons in the sample that belong to the solid pool. We note that this concentration dependence is not unique to this model. In particular, because the number of water-molecule-protein interactions is proportional to the protein concentration, a similar dependence is expected for relaxation in the absence of strong magnetic coupling between the water and protein protons.
The expanded relaxation-rate axis of Fig. 4 emphasizes the details of proton magnetic relaxation profiles of BSA gels in H2O, the inset in Fig. 4 compares the relaxation dispersion profiles for dry BSA and 10% BSA gel in H2O at 302 K, illustrating the differences between two dispersions on the same scale. Hydration of the protein with H2O changes the MRD in several ways:
• 1.
For the protein gel, the large pool of water protons is magnetically coupled to the pool of protein protons, which changes the observable relaxation-rate constants of both the protein and water protons. The effects of this magnetic coupling have been examined thoroughly and the consequences as a function of the magnetization transfer rate shown in Fig. 2 (
• Lester C.C.
• Bryant R.G.
Water-proton nuclear magnetic relaxation in heterogeneous systems: hydrated lysozyme results.
,
• Edzes H.T.
• Samulski E.T.
Cross relaxation and spin diffusion in the proton NMR or hydrated collagen.
,
• Bryant R.G.
The dynamics of water-protein interactions.
).
• 2.
The low-field-relaxation rates in H2O deviate from the power law, which is caused in part by the effects of finite magnetization transfer between the protein protons and the water protons as shown in Fig. 2. We note that at sufficiently low magnetic field strengths a second cause for a plateau in the field dependence is that for the solid-spin system, the effective magnetic field cannot drop below the local magnetic field created by the neighboring magnetic dipoles in the solid, i.e., the local dipolar field (
• Kimmich R.
• Schnur G.
• Scheuermann A.
Spin-lattice relaxation and lineshape parameters in nuclear magnetic resonance of lamellar lipid systems: fluctuation spectroscopy of disordering mechanisms.
); however, other experiments suggest that it is of minor importance in these cases.
• 3.
The shape of the magnetic field dependence in Fig. 4 is changed from that in Fig. 1. The 1H relaxation rates in H2O samples are larger than in D2O samples at Larmor frequencies >5 MHz. When 1H2O relaxation-rate constants were measured only to a proton Larmor frequency of 30 MHz, the high field region appeared to be a plateau deriving from a different class of motions such as surface dynamics of water. However, several different NMR relaxation approaches using 1H, 2H, and 17O show that the relaxation caused by motions of water at the protein surface are relatively unimportant in this frequency range because they are orders of magnitude faster than the reciprocal of these Larmor frequencies (
• Bryant R.G.
The dynamics of water-protein interactions.
,
• Denisov V.P.
• Halle B.
Protein hydration dynamics in aqueous solution.
). This conclusion is consistent with earlier NMR measurements using paramagnetic localization of diffusive contributions to relaxation (
• Polnaszek C.F.
• Bryant R.G.
Nitroxide radical induced solvent proton relaxation: measurement of localized translational diffusion.
), high resolution NMR measurements (
• Otting G.
• Liepinsh E.
• Wüthrich K.
Protein hydration in aqueous solution.
), neutron diffraction measurements (
• Gabel F.
• Bicout D.
• Zaccai G.
• et al.
Protein dynamics studied by neutron scattering.
), optical spectroscopy (
• Pal S.K.
• Peon J.
• Zewail A.H.
• et al.
Biological water: femtosecond dynamics of macromolecular hydration.
), and molecular dynamics simulations (
• Dastidar S.G.
Structure, dynamics, and energetics of water at the surface of a small globular protein: a molecular dynamics simulation.
,
• Rocchi C.
• Bizzarri A.R.
• Cannistraro S.
Water dynamical anomalies evidenced by molecular-dynamics simulations at the solvent-protein interface.
). When the proton-relaxation rates are measured to sufficiently high magnetic field strengths, the relaxation rates at high field return to an extrapolation of the low-field MRD profile and in the intermediate range of fields there is an additional contribution or a “bump” in the field dependence. This intermediate field bump is absent when the protein is hydrated with D2O even to the level of a gel as shown in Fig. 1. One might suspect that hydration could significantly affect side-chain motions and that the bump derives from such a dynamical consequence of hydration. However, this source would be fully active in the samples hydrated with D2O, which is not observed. Therefore, this effect is caused by the water protons.
Substitution of 2H for 1H exchanges deuterons for protons at labile nitrogen and oxygen positions on the protein. The dipolar coupling between the amide 1H at and 14N, which relaxes rapidly by a nuclear electric quadrupole mechanism, causes the peaks in the relaxation-rate profile when the nitrogen and proton frequencies match. These peaks occur close to the pure nuclear quadrupole resonance frequencies of the nitrogen, i.e., at ∼0.8, 2.4, and 2.8 MHz. One may suppose that the protonation of the nitrogen sites and the consequent additional relaxation pathway through the nitrogen spin system could cause the relaxation bump of Fig. 4. However, as shown by the inset of Fig. 1, these quadrupole features are much narrower than that of the broad relaxation contribution in Fig. 4. Therefore, we conclude that dynamics associated with the water molecule protons causes the broad relaxation contribution in the H2O hydrated protein cases.

## Discussion

The magnetic field dependence of the protein-proton 1/T1 in Fig. 1 is described by the same power-law exponent for the hydrated protein gel as the lyophilized solid provided that the water protons are replaced by deuterons, which are not magnetically well coupled to the protein protons. This lack of spin coupling occurs because the magnetogyric ratio of the deuteron is smaller than that for the proton by approximately a factor of 6 that enters the relaxation equation as the square (
• Abragam A.
The Principles of Nuclear Magnetism.
). Therefore, cross-relaxation between protons and deuterons is weak.
If we adopt the spin-fracton model for 1/T1p, the power law exponent, b, has been related to df, the fractal distribution of mass in the protein, and ds, the spectral dimension characterizing the distribution of vibrational states and the propagation of the structural disturbances that create relaxation (
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
). Values of df in proteins have been reported for a large number of proteins to range from 2.3–3.0 and depend somewhat on how df is computed (
• Enright M.B.
• Leitner D.M.
Mass fractal dimension and the compactness of proteins.
,
• Enright M.B.
• Yu X.
• Leitner D.M.
Hydration dependence of the mass fractal dimension and anomalous diffusion of vibrational energy in proteins.
). That the value of b is 0.78 regardless of the hydration in D2O implies a constraint on the spectral dimension, ds. It is shown elsewhere (
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
) that b = 3 − 2ds/dfds; setting b = 0.78 and 2.3 <df < 3.0 implies that the value of ds is constrained between 1.19 and 1.33. We note that for the gel shown in Fig. 1, the proteins are covalently connected by cross-link bridges that create an interconnected molecular network that fills the sample volume. Computational approaches based on network models for the protein dynamics have suggested that the value of ds depends on molecular mass (
• Reuveni S.
• Granek R.
• Klafter J.
Proteins: coexistence of stability and flexibility.
,
• Burioni R.
• Cassi D.
• Vulpiani A.
• et al.
Topological thermal instability and length of proteins.
). In these experiments this may imply that the value of b should depend on molecular size; however, within experimental error we see no difference between the uncross-linked dry protein and the cross-linked network. Of course, it is possible that there may be changes in df on hydration that compensate for changes in ds and maintain the value of b constant. These data do not test this possibility. In addition, the lyophilized condition provides intermolecular contacts that may alter the effective connectivity in the spirit of network models, but the dynamic consequences of these intermolecular contact interactions are unclear. It may seem surprising that the exponent b is not sensitive to the addition of solvent in light of recent discussions of solvent slaving or protein dynamics (
• Fenimore P.W.
• Frauenfelder H.
• Parak F.G.
• et al.
Slaving: solvent fluctuations dominate protein dynamics and functions.
); however, the spin relaxation is dominated by small scale displacements that modulate the dipolar coupling and is insensitive to rare large scale fluctuations. Therefore, these observations do not address the validity of the solvent slaving ideas.
Fig. 2 shows that at low water content and temperature, the strong magnetic coupling limit is satisfied and that the water protons and the protein protons report identical relaxation dispersion profiles. The rate constant, 1/Twp, is a pseudo first-order-rate constant describing an intermolecular cross-relaxation process. Therefore, 1/Twp is implicitly a function of the size the participating proton pools because of detailed balance. As the water population grows relative to the protein population, the effective transfer rate constant drops, and the consequence is that the cross-relaxation contribution to the proton relaxation profile develops a low-field plateau as shown in Fig. 2. The calculations shown in the Fig. 2 inset range from the very strong coupling limit where the power law of the protein spin system is transferred to the water protons at all field strengths, to transfer rate limited case with a transfer rate of 10; nevertheless, when the protein-proton-relaxation-rate constants drop with increasing field, the higher field portions of the profile satisfy the strong coupling constraints. Therefore, the importance of the water-protein-proton-cross-relaxation rate is both magnetic field and composition dependent. The cross-relaxation contributions are significant at all frequencies even when limited by the effective magnetization transfer rate constant. At Larmor frequencies >10 MHz, the intermolecular cross-relaxation effects will usually be in the strong coupling limit.
Fig. 3 summarizes relaxation dispersion data for a series of BSA gels that span a significant range of the composition variable F. The increase in relaxation rate with concentration at all Larmor frequencies is consistent with Eq. 2. However, as noted, this dependence does not distinguish between models that depend on the effective concentration of bound labile water molecules or protons.
In the strong magnetic coupling limit, the total observable water-proton- spin-lattice-relaxation-rate constant is generally a function of the rate constants for the protein protons, the water protons, and the cross-relaxation between these two populations as summarized in Eq. 2. In earlier discussions, the water relaxation-rate constant in hydrated protein systems was treated as independent of frequency because there was no evidence that there were water correlation times that were as slow as the Larmor frequencies studied except for the chemical exchange processes that contribute to the cross-relaxation rate. The magnetic field dependence of the observed 1H2O relaxation rate was ascribed to the protein-proton population (
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
), which is a power law as shown in Fig. 1. Comparison of Fig. 1 and Fig. 4 shows that there are field dependent water-proton contributions. For the relaxation of water protons we may write
$1T1w(ω)=1T1o+∑i1T1ibnd(ω)+∑j1T1jsur(ω),$
(3)

where T1o is the relaxation time for bulk water and is field independent in the frequency range of present magnetic field strengths, T1ibnd is the relaxation time for water bound to the protein in the ith site, and T1jsur is the relaxation time for the water molecule at the jth surface site and the sums include all sites. The surface site relaxation contributions are discussed in detail elsewhere but make contributions that are logarithmic in the Larmor frequency and important only at the highest field strengths (
• Grebenkov D.S.
• Goddard Y.A.
• Bryant R.G.
• et al.
Dimensionality of water diffusive exploration at the protein interface in solution.
).
There are two classes of motion for bound water molecules in a protein binding site: 1), in a site without hydrogen bonds between the water and a protein donor or acceptor, the entrapped water may reorient by stochastic rotational diffusion within the angles permitted by the confinement; and 2), in a site where the water may be constrained by hydrogen bonds to the protein, it may reorient stochastically as the bonding pattern permits but the motions of the water and the surrounding protein are dynamically coupled by the intermolecular connection. These two classes of motion produce qualitatively different dependence of the spin-lattice-relaxation-rate constant on the magnetic field strength. For the uncoupled case, the relaxation may be modeled as a Lorentzian contribution; the effect is that the high field portion of the MRD profile returns to an extrapolation of the low field relaxation profile. In the dynamically coupled case, where water molecules couple to the protein for time longer then characteristic correlation time of protein backbone fluctuations, the relaxation equation is not a simple Lorentzian and the high-field relaxation rates are displaced to higher values compared to an extrapolation of the low field rates (
• Goddard Y.
• Korb J.-P.
• Bryant R.G.
The magnetic field and temperature dependences of proton spin-lattice relaxation in proteins.
,
• Goddard Y.
• Korb J.-P.
• Bryant R.G.
Nuclear magnetic relaxation dispersion study of the dynamics in solid homopolypeptides.
). Either approach provides an acceptable fit to the data with a correlation time between 20–50 ns. The solid lines in Fig. 4 result from fits to Eq. 2, using the tethered local dynamic model for bound water contribution. Although there may be a distribution of effective local correlation times for the long-lived-water sites in the protein, no distribution function is necessary with either model to provide an adequate description of the data. It is, of course, possible that both classes of motion contribute.
A correlation time of 20–50 ns is within a factor of 2 of the global rotational correlation time for the protein in solution. This near degeneracy of the local motion with the global reorientation correlation time will make this motion difficult to observe in solution phase MRD experiments (
• Lipari G.
• Szabo A.
Model-free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules. 1. Theory and range of validity.
). The low-frequency amplitude of the MRD profile for a protein solution is proportional to the number of long-lived or bound water molecules (
• Kiihne S.
• Bryant R.G.
Protein-bound water molecule counting by resolution of (1)H spin-lattice relaxation mechanisms.
). For serum albumin, we and others have reported 25 ± 3 such water molecules if it is assumed that there is no local motion of the water in the bound sites; i.e., that all internal water is rigid and reorients with the global rotational correlation time of the protein (
• Denisov V.P.
• Halle B.
Protein hydration dynamics in aqueous solution.
,
• Denisov V.P.
• Jonsson B.H.
• Halle B.
Hydration of denatured and molten globule proteins.
,
• Denisov V.P.
• Peters J.
• Halle B.
• et al.
Using buried water molecules to explore the energy landscape of proteins.
). In fact, N ×S2 is measured where S is a generalized order parameter, N is the number of bound water molecules, and this product for serum albumin is 25. The local motion detected in this work as an important contribution to the total proton relaxation of the nonrotating protein spin system implies that at least some water molecules suffer a local motion with an average correlation time of 20–50 ns. Therefore, S2 may be <1, but the near degeneracy of the local and global reorientation implies that the separation of local and global dynamics may be problematic for at least some bound water molecules (
• Lipari G.
• Szabo A.
Model-free approach to the interpretation of nuclear magnetic resonance relaxation in macromolecules. 1. Theory and range of validity.
) and quantitative refinement of the number of bound water molecules based on the amplitude of the solution phase relaxation dispersion profile remains a challenge. Some have advanced the idea that the bound water molecules cannot move sufficiently to affect relaxation significantly (
• Halle B.
Nanosecond to microsecond protein dynamics probed by magnetic relaxation dispersion of buried water molecules.
). It is unclear that the protein is sufficiently rigid for this possibility to account for all bound water molecules. Even the polypeptide backbone retains sufficient motion for the 14N to provide a relaxation sink for the proton-spin population even when global rotation is stopped as shown in Fig. 1.
If the strong coupling approximation fails in the frequency range of the bump, the critical importance of the water-proton contribution remains. If one rejects local motions of bound water molecules as the cause of the bump, then a Lorentzian contribution to 1H relaxation may be obtained by a water-molecule-exchange process that modulates both the inter and intramolecular dipolar couplings as advanced by Halle and co-workers in the EMOR model (
• Chávez F.V.
• Halle B.
Molecular basis of water proton relaxation in gels and tissue.
,
• Halle B.
Molecular theory of field-dependent proton spin-lattice relaxation in tissue.
). Because it is difficult to dismiss relaxation coupling at all frequencies studied here, and the chemical exchange that interrupts the dipolar couplings is critical to carry the magnetization transfer between the protein and the water spins, it is reasonable in this case to presume that both an exchange process as described by the EMOR model and cross-relaxation contribute to the total relaxation. As before, the deviations from the power law shown in Fig. 4 require only a single Lorentzian where now the correlation time is interpreted as a water-molecule-exchange time. Although the character of the motions presumed in each approach is different, the correlation time deduced is the same.
The water-molecule motions with correlation times in the tens of ns range correspond to frequencies of order 107 s−1 and these motions affect the dielectric properties of the environment. To the extent that one may use continuum concepts in a localized context, the dielectric response in the vicinity of the dynamically inhibited water molecules must be frequency dependent with a dispersion associated with the characteristic dynamics similar to that observed magnetically here. Correspondingly, one expects the electrostatic contribution to the energetic cost of conformational changes is influenced by the perturbed local dielectric constant. In the context of transition-state theory, the free energy barrier for structural rearrangement may be tuned by the number and dynamical characteristics of semimobile water molecules in the critical regions of the protein structure (
• Prakash M.K.
• Marcus R.A.
An interpretation of fluctuations in enzyme catalysis rate, spectral diffusion, and radiative component of lifetimes in terms of electric field fluctuations.
,
• Prakash M.K.
• Marcus R.A.
Dielectric dispersion interpretation of single enzyme dynamic disorder, spectral diffusion, and radiative fluorescence lifetime.
). These results support conclusions drawn from dielectric dispersion measurements on hydrated proteins (
• Jansson H.
• Bergman R.
• Swenson J.
Relation between solvent and protein dynamics as studied by dielectric spectroscopy.
,
• Mijovic M.
• Bian Y.
• Chen B.
• et al.
Dynamics of proteins in hydrated state and in solution as studied by dielectric relaxation spectroscopy.
,
• Rizvi T.Z.
• Khan M.A.
Dielectric relaxation in singly hydrated bovine tendon collagen.
,
• Rizvi T.Z.
• Khan M.A.
Temperature dependent dielectric properties of slightly hydrated horn keratin.
).
The magnitude of the bound water contribution to the MRD profile for rotationally immobilized protein in the intermediate field range of tens of MHz for the proton Larmor frequency is significant and partly masks the power-law that derives from the protein structural fluctuations. The combination of this contribution and the low-field limitations in the magnetization-transfer-rate constant alters the magnetic field dependence. Data over a limited field range, 0.01 to tens of MHz may be fitted by a power law with a smaller exponent. It is now clear that with more complete measurement of the magnetic field dependence of the relaxation-rate constants, the underlying power-law parameters are practically the same as in the dry protein or protein hydrated with 2H2O as in Fig. 1. Thus, the apparent water-content dependence of the power-law exponent is not supported by the more extensive data now available over the wider range of magnetic fields. Structural changes attending hydration do not affect the power-law exponent significantly, although spectral changes are clearly detected in magic-angle spinning NMR measurements on crystalline and noncrystalline proteins (
• Morcombe C.R.
• Paulson E.K.
• Zilm K.W.
• et al.
1H-15N correlation spectroscopy of nanocrystalline proteins.
,
• Wylie B.J.
• Rienstra C.M.
Multidimensional solid state NMR of anisotropic interactions in peptides and proteins.
).
The power law in the Larmor frequency describes the 1H relaxation-dispersion profile over the nearly 5 decades in frequency shown in Fig. 1. Further, the same power law is maintained as the protein is hydrated with 2H2O that is known to facilitate side-chain motion at the surface (
• Goddard Y.
• Korb J.-P.
• Bryant R.G.
The magnetic field and temperature dependences of proton spin-lattice relaxation in proteins.
,
• Diakova G.
• Goddard Y.A.
• Bryant R.G.
• et al.
Changes in protein structure and dynamics as a function of hydration from (1)H second moments.
). Therefore, these side-chain motions are not critical in the underlying relaxation process in this range of frequencies. The anomalous dispersion relation included in the spin-fracton theory relates the frequency ω to a length scale l according to the anomalous dispersion law (
• Korb J.-P.
• Bryant R.G.
The physical basis for the magnetic field dependence of proton spin-lattice relaxation rates in proteins.
,
• Korb J.-P.
• Bryant R.G.
Noise and functional protein dynamics.
); $ω∝l−df/ds$ where ds is the spectral dimension characterizing the propagation of fluctuations in the structure. Therefore, the frequency axes in Fig. 1 and Fig. 4 may also correspond to a length axis, the larger the frequency the smaller the length scale. The persistence of the power law from 10 kHz to 300 MHz implies that the character of the fluctuation is scale invariant over the nearly 4 or 5 decades shown in Fig. 4, which is consistent with a system described by fractal characteristics in both the spatial and spectral properties. The imperfect connectivity in the protein causes confinement of fluctuations and a dramatic increase in the density of low frequency modes relative to a classical 3D network. This conclusion is supported by network model calculations as noted earlier (
• Reuveni S.
• Granek R.
• Klafter J.
Proteins: coexistence of stability and flexibility.
). As discussed previously (
• Korb J.-P.
• Bryant R.G.
Noise and functional protein dynamics.
), the relaxation-rate constant, 1/T1, is proportional to the frequency dependence of the mean-square displacement that varies as a power law ωds-2. With the MRD experiment, we interrogate frequencies that are lower than those sampled by most other methods. Nevertheless, these measurements demonstrate that the displacement amplitudes increase according to the same power law over the frequency range that extends well into that for catalytic rate constants. As noted by Doruker et al. (
• Doruker P.
• Jernigan R.L.
• Bahar I.
Dynamics of large proteins through hierarchical levels of coarse-grained structures.
), the contributions of the low frequency modes make the largest contributions to the mean-square displacements. Although based on different theoretical development, these results imply constraints on the value of ds that are similar to predictions based on Gaussian network models for the protein (
• Reuveni S.
• Granek R.
• Klafter J.
Proteins: coexistence of stability and flexibility.
).

## Conclusions

Rotational immobilization of the protein provides the opportunity to examine the dynamics of both the protein and associated water in time regimes that are not directly accessible for proteins in solution because of rotational averaging of dipolar couplings by global rotation of the protein. The 1H magnetic relaxation dispersion profile over the Larmor frequency range from 0.01 to 300 MHz for rotationally immobilized proteins is described by a power law in the Larmor frequency that is independent of the hydration level of the protein provided that water 1H is replaced by 2H. Therefore, the character of the underlying protein dynamics that determine the nature of the fluctuations in the proton-proton dipolar couplings, which drive spin relaxation, is not substantially altered over this frequency range and is consistent with a spin-fracton relaxation theory where the values of the spectral dimension and fractal dimension for the distribution of mass are either constant or change in compensating ways as a function of hydration. In contrast, hydration with H2O provides an additional 1H-relaxation mechanism caused by motions of rare bound water molecules in the range of 20–50 ns at 302 K. By comparison, water motions at the protein surface are characterized by correlation times in the range of tens of picoseconds and make only small contributions to the total observed spin relaxation in cross-linked protein systems or hydrated solids over this range of Larmor frequencies.
Helpful discussions with P. Levitz, J. Klafter, D. Petit, and B. Sapoval are gratefully acknowledged.
This work was supported by the National Institutes of Health (RO1 NIBIB2805), the University of Virginia, and the CNRS, France.

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Hydration of denatured and molten globule proteins.
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text:
# Wave parameters¶

This page outlines the techniques and methods used to acquire and analyse data from radar satellite altimeters and how RADWave package uses this dataset to investigate wave parameters, enabling wave climate analysis of varying spatial and temporal resolution.

Important

RADWave uses post-processed altimeter dataset to analyse historical wave climate and trends but can also be used to determine cyclone-generated wave conditions.

## Satellite altimeter data¶

Altimeter observations of the ocean surface are been recorded since 1985, with a short break between 1989-1991 due to no operating satellites [Chelton2001] (fig. below from source). Thirteen altimeters, named GEOSAT, ERS-1, TOPEX, ERS-2, GFO, JASON-1, ENVISAT, JASON-2, HAI-YANG-2A, SARAL, JASON-3 and SENTINEL-3A, provides detailed and global coverage.

Note

Satellites were predominantly placed in sun-synchronous, near-polar orbits, covering the same ground track every 3-10 days. Observations are therefore not made every day, however, observation frequency has increased in recent years due to the addition of more altimeters.

To increase analysis ability between altimeter missions, [Ribal2019] combined and reprocessed data from Globwave, Radar Altimeter Data System and the National Satellite Ocean Application Service to provide a single dataset spanning all thirteen altimeter missions from 1985-2019.

Altimeters measure the ocean surface by emitting a radar pulse and determining the shape, power and time delay of the return pulse. The shape is converted into $$H_{s}$$. The power of the return pulse, also known as the backscatter coefficient, is used to determine ocean surface properties including surface roughness. By applying a relationship between uncalibrated wind speed and backscatter coefficient, a calibrated wind speed $$U_{10}$$ (10 m above the sea surface and averaged over 10 minutes) can be determined.

Important

Overall, [Ribal2019] suggests that preprocessed and calibrated dataset is valid for wind speed below 24 m/s and Hs below 9 m, however, notes that values above this limit are likely still valid.

## Wave periods¶

However, altimeters do not directly measure wave period, an essential component to adequately characterise wave climate.

For waves that have been locally generated by wind, the local wind speed and wave height, both of which are accurately measured by altimeters, can be used to determine $$T_{z}$$ [Quilfen2004]. More accurate results are achieved when the wave age is calculated first, which is the length of time wind has been acting on a wave [Remya2010]. In RADWave, an algorithm derived by [Remya2010] is implemented. It estimates $$T_{z}$$ in both wind and swell-generated seas (rmse = 0.76 s).

First, the wave age is calculated through:

$\epsilon = 3.25 \frac{H_{s} g}{U_{10}}$

where $$H_{s}$$ is the significant wave height, $$g$$ the acceleration by gravity and $$U_{10}$$ the wind speed.

Wave age $$\epsilon$$ is then inputted into a Genetic Algorithm [Remya2010] to estimate $$T_{z}$$:

$T_{z} = (((\epsilon - 5.78)/(\epsilon + (U_{10}/H_{s} \times (U_{10}/H_{s}) + H_{s}))))) +(H_{s} + 5.70))$

## Wave energy, group velocity & power¶

Mean wave energy density ($$E$$) (J/m2) is calculated by:

$E = \frac{1}{8} \rho g H_{s}^{2}$

with $$\rho$$ the density of seawater (set to 1027 kg/m3).

Wave group velocity ($$C_{g}$$) in deep water conditions is approximated with:

$C_{g} = \frac{g T_{z}}{2 \pi}$

And wave power $$P$$ can, therefore, be estimated through:

$P = E C_{g}$

which is the wave energy flux per metre of wave-crest (W/m). This is then converted into kW/m for ease of analysis.

 [Chelton2001] Chelton, D.B., Ries, J.C., Haines, B.J., Fu, L.L. & Callahan, P.S. - Satellite Altimetry, Satellite altimetry and Earth sciences in L.L. Fu and A. Cazenave Ed., Academic Press, 2001.
 [Quilfen2004] Quilfen, Y., Chapron, B., Collard, F. & Serre, M. - Calibration/validation of an altimeter wave period model and application to Topex/Poseidon and Jason-1 altimeters, Marine Geodesy, 27(3-4), 535–549, 2004.
 [Ribal2019] (1, 2) Ribal, A. & Young, I. R. - 33 years of globally calibrated wave height and wind speed data based on altimeter observations. Scientific Data 6(77), p.100, 2019.
 [Remya2010] (1, 2, 3) Remya G., Kumar, R., Basu, S. & Sarkar, A. - Altimeter-derived ocean wave period using genetic algorithm. IEEE Geoscience and Remote Sensing Letters, 8(2), 354–358, 2010.

format: plain


text:
# How to show ProgressBar in ReactNative?

React NativeJavascriptMobile Development

ProgressBar is a way to tell users that the content will be available in sometime. It can best be used when you submit something to the server and wait for the server to respond.

To work with the progress bar component install react-native-paper module using npm.

The command to install react-native-paper is −

npm install --save-dev react-native-paper

The basic component of progress bar is as follows−

<ProgressBar progress={progress_number} color="progresscolorbar" />

To work with Progress Bar you need to import it from react-native-paper as follows −

import { ProgressBar} from 'react-native-paper';

Following are some of the important properties available on ProgressBar −

Sr.NoProps & Description
1progress
It takes value from 0 to 10. The number value to be given to show inside the progress bar.
2color
The color of the progress bar.
3visible
The values are true/false. It helps to show/hide progressbar.
4style
The style to be applied for the progressbar.

## Example: Display of Progress Bar

It is very simple to display a progressbar. Just import it first from react-native-paper.

import { ProgressBar} from 'react-native-paper';

The show a progress bar the code is as follows −

<ProgressBar style={{ marginTop:200}} progress={0.5} color="#00BCD4" />

The default value is 0.5 and it will increment till 10.

import * as React from 'react';
import { ProgressBar} from 'react-native-paper';
const MyComponent = () => (
<ProgressBar style={{ marginTop:200}} progress={0.5} color="#00BCD4" />
);
export default MyComponent;

## Output

Published on 01-Jul-2021 08:26:09


text:
## anonymous 5 years ago Find all the solutions of the Differential Equation dy/dt - 2y^2 = ty^2

1. anonymous

2. anonymous

After that take an integral.

3. anonymous

that helps, thanks.

4. anonymous

$y'[t]-2 y^2=t y^2\to y'[t]=t y^2+2 y^2$ $\int\limits y'[t] \, dt=\int\limits \left(-2 y^2+t y^2\right) \, dt$ $y[t]=-2 t y^2+\frac{t^2 y^2}{2}$


text:
## Bulk viscosity and conformal symmetry breaking in the dilute Fermi gas near unitarity    [PDF]

Thomas Schaefer, Kevin Dusling
The dilute Fermi gas at unitarity is scale invariant and its bulk viscosity vanishes. We compute the leading contribution to the bulk viscosity when the scattering length is not infinite. A measure of scale breaking is provided by the ratio $(P-\frac{2}{3}{\cal E})/P$, where $P$ is the pressure and ${\cal E}$ is the energy density. In the high temperature limit this ratio scales as $\frac{z\lambda}{a}$, where $z$ is the fugacity, $\lambda$ is the thermal wave length, and $a$ is the scattering length. We show that the bulk viscosity $\zeta$ scales as the second power of this parameter, $\zeta \sim (\frac{z\lambda}{a})^2 \lambda^{-3}$.
View original: http://arxiv.org/abs/1305.4688


text:
# Remove the white space between the item label and the equations [duplicate]

Here is a question. I want to know how to remove the white space between the item label and the equations? For example, I want to remove the blank between the number 1 and the equation f(X)=...

The code and picture are as follows

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
\begin{enumerate}
\item
\begin{eqnarray*}
f(X)&=& AD+SF SF SDF+DFDF FA \\
\end{eqnarray*}

\item
\begin{equation*}
\end{equation*}

\end{enumerate}

## marked as duplicate by campa, Phelype Oleinik, Raaja, siracusa, MarijnJun 28 at 12:41

Welcome to tex.SE... Hope the below codes may helps you:

\documentclass[10pt]{book}
\usepackage{amsmath}
\newcommand\mathitem{\item\leavevmode\vspace*{-\dimexpr\baselineskip+\abovedisplayskip\relax}}

\begin{document}

\begin{enumerate}
\mathitem
\begin{align*}
f(X)&= AD+SF SF SDF+DFDF FA \\
\end{align*}

\mathitem
\begin{equation*}
\end{equation*}

\item This is for test
\end{enumerate}

\end{document}

Note that please try to avoid eqnarray which is little bit older, use amsmath environments like align, gather, etc. Also hope your operator alignment was wrong, + sign should be quad indent with =

PS: Big thanks to Bernard who already gave this suggestion, refer Remove white line after \item in enumerate and before display math

• Ingenious! (+1) – Bernard Jun 28 at 12:04
• @Bernard Thanks... :-D – MadyYuvi Jun 28 at 12:28
• I try this, it looks excellent. Many thanks. Also, thanks a lot for the suggestions. – Trick1234 Jun 28 at 12:37

Here is a solution based on \makebox[\linewidth] and aligned:

\documentclass{article}
\usepackage{mathtools}
\usepackage[showframe]{geometry}

\begin{document}

XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX
\begin{enumerate}
\item \makebox[\linewidth]{\displaystyle\begin{aligned}[t] f(X)={} &AD+SF SF SDF+DFDF FA \\ &+SADSADDDDDDDDDDDA D+D DADS AD \end{aligned}}
\item \makebox[\linewidth]{$\displaystyle G(X)=ADSA DASD SAD ASD ASD$}
\end{enumerate}

\end{document}


text:
## FANDOM

17 Pages

A technique to solve first order differential equations of the form

${{dy \over dx} + p(x)y = q(x)}$.

• Calculate the integration factor ${\rho (x) = e^{\int p(x) dx}}$.
• Multiply each side of the equation by ${\rho (x)}$
• Integrate both sides of the equation to get ${\rho(x)y(x) = \int \rho(x)q(x) dx}$.
• Sove for y(x).

## Example

${{dy \over dx} - y = {11 \over 8}e^{-x/3}}$

${\Rightarrow \rho(x) = e^{\int (-1) dx} = e^{-x}}$

${\Rightarrow e^{-x}{dy \over dx} - e^{-x}y = {11 \over 8}e^{-4x/3}}$

${\Rightarrow e^{-x}y = \int {11 \over 8}e^{-4x/3} dx = {-33 \over 32} e^{(-4x/3)} + C}$

${\Rightarrow y(x) = {-33 \over 32} e^{(-x/3)} + Ce^{x}}$


text:
# What is the Schwarzschild metric with proper radial distance?

Reading the marvellous book "The Membrane Paradigm" I stumbled upon a suggested change of variable that I'm not able to deal with. Starting with the usual Schwarzschild metric for the spatial 3-geometry

$$ds^2 = \frac{1}{f(r)} dr^2 + r^2 \left( d\theta^2 + \sin ^2 \theta \ d\phi\right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)$$

where $f(r) = 1-\frac{2M}{r}$ (that is simply Schwarzschild metric without time element $-f(r)\ dt^2$) they suggest a new radial coordinate $R$ with the "property that $R-2M$ measures the proper radial distance outward from the horizon". This new variable is

$$R= 2M + \sqrt{r(r-2M)}+\ln \left[ \sqrt{\frac{r}{2M} -1} + \sqrt{\frac{r}{2M}} \right].\ \ \ \ \ \ \ \ \ \ (2)$$

I'd like to see the new metric but I'm not able to invert (2) (namely find $r(R)$) to perform the coordinate change $$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=g_{\mu\nu}\frac{\partial x^{\mu}}{\partial x'^{\rho}}\frac{\partial x^{\nu}}{\partial x'^{\sigma}}dx'^{\rho}dx'^{\sigma}=g'_{\rho\sigma}dx'^{\rho}dx'^{\sigma}$$

I tried also to use the inverse of the Jacobian as suggested in http://physics.stackexchange.com/a/43084 but at the end I always need at least to change the variable $r^2$ in the angular part of (1).

Do you have any idea on how to deal with it or how to invert (2) ?

-
You have certainly an error in $(2)$ : there should be a length term $[L]$ multiplying the $ln$ term. –  Trimok Aug 13 '14 at 13:29
@Trimok I see your point. Actually it is correctly copied from the book, so the error should be in the book. BTW here $c=1$ everywhere. –  miskio1646 Aug 13 '14 at 14:25

$$R= 2M + \sqrt{r(r-2M)}+2M\ln \left[ \sqrt{\frac{r}{2M} -1} + \sqrt{\frac{r}{2M}} \right] \tag{1}$$
You have (if no error) : $dR = \dfrac{dr}{\sqrt{ 1 - \dfrac{2M}{r}}}$, so $dR^2 = \dfrac{dr^2}{f(r)}$, and this simplifies your metrics.
However, you cannot invert the formula $(1)$ to get $r$ as an explicit function of $R$. You will simply write the metrics :
$$ds^2 = dR^2 + (r(R))^2 \left( d\theta^2 + \sin ^2 \theta \ d\phi\right) \tag{2}$$ where $r(R)$ is implicitely defined by the equation $(1)$


text:
# How far out can a pre-telescope human society (naked eye observations only) detect planets?

There are many questions on WBSE that cover most aspects of building a planetary system. (some examples below):

Creating a realistic world(s) map - planetary systems

How many planets should I have in my planetary system?

What distances would be involved in this planetary system?

But it occurred to me that while a sci-fi story setting might need the complete details of the system, a story set in a medieval or any similar society with pre-telescope or primitive telescope technology would only need to build the parts of the system that would be observable by the inhabitants of participants of the story, or the well known and widespread knowledge of the society that those participants are members of.

So ... How far away (maximum distance) can humans detect planets with the naked eye?

Assume best observable conditions for naturally occurring planets. As far as I can tell from my limited research, this should limit the albedo of the observed planet to about .8 (unless you can give a reasonable explanation of why it should be more or less than that), and the radius of the observed planet should be no more than about that of Jupiter (again, unless a reasonable reason for an exception is given). Assume ideal observing conditions, such as no light pollution, good (perfect specimen human) eyesight, ideal alignment of star and observing planet and observed planet for best lighting of observed planet, etc. Also assume a sun-like star, and earth-like planet as far as atmosphere and other observation characteristics, though human habitability is not a requirement except as it directly relates to human-like observation (a fictional atmosphere is allowed, as long as there is an explanation of how it improves observation, while not entirely preventing complex life in general). Earth-like orbit is NOT required for either planet.

This question has applications not only for general world-building, but can also be used as a basis for calculating orbit times which then apply to things like creating mythologoes, calendars, religious influences, cultural iconography, and much more.

• Are multiple start systems allowed? Any size of planet up to the brown dwarf limit (i.e. about 13 Jupiter masses?) – Rafael Dec 18 '18 at 21:50
• @Rafael Yes, but in order to show that it would be better observation than what's possible in a system with a single star, the observation characteristics of the single-star system would also be needed, for comparison. – Dalila Dec 18 '18 at 21:52
• Do they have to know that the bright point in the sky is not a star, but a planet? – DarthDonut Dec 19 '18 at 8:23
• @DarthDonut It should be possible to tell that it moves differently across the sky, over time, yes. Though I place no restrictions on how long it takes to compile the observations required to reach that conclusion (as this could be applied to fantasy settings, immortal or near immortal races could observe for hundreds or thousands of years to accomplish it, or even multiple generations of actual humans could do it as well) – Dalila Dec 19 '18 at 16:22
• I tweaked the title of your question. Feel free to roll back or to Edit further if you disagree with the edit. – user Dec 19 '18 at 18:24

With the naked eye, humans can see approximately 6th-magnitude objects. We can compute the apparent magnitude of a planet at a given distance, and find the distance corresponding to an apparent magnitude of +6. The formula is $$m_p=M_p+5\log\left(\frac{d}{10\text{ pc}}\right)$$ where $$m_p$$ and $$M_p$$ are the apparent and absolute magnitudes and $$d$$ is the distance from Earth to the planet. $$M_p$$ can be calculated if we know $$M_S$$, the absolute magnitude of the Sun: $$M_p=M_S-2.5\log\left(a\frac{R_p^2}{4d_s^2}\right)$$ where $$d_s$$ is the distance from the planet to the Sun and $$r_p$$ is its radius. Let's say that $$d_s\approx d$$, by assuming that the planet is much further from the Sun than Earth is. Finally, we get $$m_p=M_S-2.5\log\left(a\frac{R_p^2}{4d^2}\right)+5\log\left(\frac{d}{10\text{ pc}}\right)$$ This has the solution $$d=\sqrt{\frac{a^{1/2}R_J\cdot10\text{ pc}}{2}}10^{\cfrac{m_p-M_S}{10}}$$ Let's say $$R_p\approx R_J$$, the radius of Jupiter, and $$a=0.5$$ - also like Jupiter. Then, given that $$M_S=4.83$$, I get 24 astronomical units - about halfway between Uranus and Neptune. We can't see Neptune with our naked eyes, but we can see Uranus under good conditions, which matches our calculations (although note that the ice giants have different albedos). Let's say we choose an even higher albedo - say, $$a=0.8$$, as you suggested. This gets me up to 27, even closer to Neptune.

• I think this answer defines every variable used in the formulae, except actually for $d$ (so you don't really tell us what you're calculating). We can see that from the question, but it'd be a small change to make the answer stand more on its own. Also, you might consider \cfrac for the exponential term in the last formula, but that may require a little bit of rearranging to make it read nicely. – user Dec 19 '18 at 18:21
• @aCVn Thanks for the suggestions; I've implemented them. The final equation's still a bit bulky, but that's the best I can do for now. – HDE 226868 Dec 19 '18 at 18:36
• We can see Uranus with our naked eyes, under good conditions. It was seen in the pre-telescope age, but nobody realized it was a planet. To be classified as a planet, it has to attract enough attention so someone looks for it more than once, with enough elapsed time to get noticeable motion. There's lots and lots of sixth-magnitude stars, and nobody's going to track all of them. The minimum brightness for a visible planet as detected is 1.86 for Mars, and the lowest mean brightness is 0.71 for Mars. There's very few stars that bright, so it would be easy to note one moving. – David Thornley Dec 19 '18 at 22:58
• @DavidThornley Thank you for the correction. – HDE 226868 Dec 19 '18 at 23:17

Uranus was detected by naked eye in the classical age but because it is too slow it wasn't recoginzed as a planet. In the "discovery" section. So the classical astrological manuals that are aware of its existence classify the planet as a fixed star.

So it's not only visibility that matters but also the speed because if it is too slow it will be classified as fixed star.

• Very good points, but not actually an answer to the question ... A Jupiter sized planet (significantly larger than Uranus) that is more reflective than any in our current solar system should (I suspect) be visible at a significantly greater distance, and while speed would make it hard to differentiate from fixed stars, it should still be possible with enough long term observations. – Dalila Dec 18 '18 at 21:55
• An addendun - since the question deals with ""primitive"" societies it also deals with astrology, specifically something like the classical, pagan, astrology. In this system what matters is what you see, the human perception of the world. So, invisible objects are irrelevant, even if they exist. Also an object that is too slow won't be classified as a planet but as a fixed star. Also the ancients were aware that the fixed stars werent that fixed in a long enough time, like the crab nebula supernova shown. – Geronimo Dec 20 '18 at 21:40
• Yes, astrology in stories is one purpose I intended for this question. And the fact that invisible objects are irrelevant in that sense is also a basis for the question. Determining a full planetary system isn't useful in that type of setting, because anything outside the range of observation couldn't possibly have an effect on the society in that way. However, even if said society classified it as a star, a distant planet could still "behave" noticeably different ("special/important star") from actual stars, and could be given preferential status in their astrology/mythology as a result. – Dalila Dec 21 '18 at 22:41


text:
# Descartes’ Rule of Signs

The purpose of the Descartes’ Rule of Signs is to provide an insight on how many real roots a polynomial P\left( x \right) may have. We are interested in two kinds of real roots, namely positive and negative real roots. The rule is actually simple.

Here is the Descartes’ Rule of Signs in a nutshell.

## Breakdown or Explanation of the Descartes’ Rule of Signs

Suppose P\left( x \right) is a polynomial where the exponents are arranged from highest to lowest, with real coefficients excluding zero, and contains a nonzero constant term.

The number of positive real roots is either

• equal to the number of sign changes in P\left( x \right)
• or, less than the number of sign changes in P\left( x \right) by some multiple of 2.

The number of negative real roots is either

• equal to the number of sign changes in P\left( { - x} \right)
• or, less than the number of sign changes in P\left( { - x} \right) by some multiple of 2.

In summary, if n is the number of sign changes in either P\left( x \right) or P\left( { - x} \right), then the number of positive or negative roots may equal to n, n-2, n-4, n-6, etc.

Note that we start with the number of sign changes, “n“, then we keep subtracting it by some multiple of 2 (positive even integers) such as 2, 4, 6, etc.

We stop subtracting until such time when the difference becomes 0 or 1. That is it!

Quick examples for both cases:

• Let P\left( x \right) has n=7 number of sign changes, the possible number of positive real roots will be

7, 5, 3 or 1

• Let P\left( x \right) has n = 6 number of sign changes, the possible number of negative real roots will be

6, 4, 2 or 0

### Examples of Descartes’ Rule of Signs

Let’s take a look at a few examples to see this rule in action!

Example 1: Find the number of real roots of the polynomial below using Descartes’ Rule of Signs.

Start by clearly marking off the sign of each term in the polynomial. I will use the color red for positive symbol (+), and black for negative symbol (). This would allow us to easily keep track of the change in sign.

It is considered a sign change if the two signs of adjacent coefficients switch (or alternate). For instance, it can go from positive to negative, or negative to positive.

For the positive real roots:

Use the given function itself because the “x” inside the parenthesis, P\left( x \right), is positive.

There are two sign changes as shown by the arrows. Since n = 2, therefore there are 2 or 0 positive real roots.

For the negative real roots:

Use the modified version of the function, P\left( { - x} \right), where the “x” inside the parenthesis is negative.

Before we count the sign change, we will need some side calculation. Substitute “-x” in P\left( x \right) to get P\left( { - x} \right).  Here we go…

Now, let’s do the counting…

There are three sign changes as pointed out by the arrows. Since n = 3, therefore there are 3 or 1 negative real roots.

For our final answer, we say, there are 2 or 0 positive real roots, and 3 or 1 negative real roots.

Here is the graph of the polynomial showing that indeed our “guess” is spot on! In fact, it has two (2) positive roots, and three (3) negative roots.

Example 2: Find the number of real roots of the polynomial below using Descartes’ Rule of Signs.

Before we start on this problem, I must caution you not to treat this like a synthetic division problem where we place zeroes on the missing powers of x‘s. As long as the polynomial is arranged with a decreasing number of exponents, that is good enough.

For the positive real roots:

Use the given polynomial and count the number of sign change.

There are three sign changes for P\left( x \right), that means there can be 3 or 1 positive real roots.

For the negative real roots:

Evaluate -x into P\left( x \right) to get P\left( { - x} \right) then count the sign change.

There is only one sign change for P\left( { - x} \right), that means there is exactly 1 negative real root.

For our final answer, there are 3 or 1 positive real roots, and exactly 1 negative real root.

Example 3: Find the number of real roots (positive and/or negative) of the polynomial below.

To find the positive roots:

Observe that all terms in the polynomial are all positive.

Since there is no sign change in P\left( x \right), this implies that the polynomial has NO positive real roots.

To find the negative roots:

Solve for P\left( { - x} \right) then count the variation in signs.

Because we have seven (7) sign changes in P\left( { - x} \right), there are 7, 5, 3, or 1 negative real roots.

For our final answer, there are no positive real roots, and there are 7, 5, 3, or 1 negative real roots.

Example 4: Find the number of real roots of the polynomial (positive and/or negative) below.

To find the positive roots:

Count the number of alternating signs in P\left( x \right).

We have six sign changes here which imply that there are 6, 4, 2, or 0 positive real roots.

To find the negative roots:

Solve for P\left( { - x} \right) first then count the variation in signs.

No sign change in P\left( { - x} \right) that means there are no negative real solutions.

There are 6, 4, 2, or 0 positive real roots, and there are no negative real roots.


text:
Вернуться к Modern Robotics, Course 1: Foundations of Robot Motion

4.7
звезд
Оценки: 853

## О курсе

Do you want to know how robots work? Are you interested in robotics as a career? Are you willing to invest the effort to learn fundamental mathematical modeling techniques that are used in all subfields of robotics? If so, then the "Modern Robotics: Mechanics, Planning, and Control" specialization may be for you. This specialization, consisting of six short courses, is serious preparation for serious students who hope to work in the field of robotics or to undertake advanced study. It is not a sampler. In Course 1 of the specialization, Foundations of Robot Motion, you will learn fundamental material regarding robot configurations, for both serial robot mechanisms and robots with closed chains. You will learn about configuration space (C-space), degrees of freedom, C-space topology, implicit and explicit representations of configurations, and holonomic and nonholonomic constraints. You will also learn how to represent spatial velocities and forces as twists and wrenches. This material is at the core of the study of anything that moves (e.g., robots). This course follows the textbook "Modern Robotics: Mechanics, Planning, and Control" (Lynch and Park, Cambridge University Press 2017). You can purchase the book or use the free preprint pdf. You will build on a library of robotics software in the language of your choice (among Python, Mathematica, and MATLAB) and use the free cross-platform robot simulator V-REP, which allows you to work with state-of-the-art robots in the comfort of your own home and with zero financial investment....

## Лучшие рецензии

OO

29 дек. 2021 г.

Very Engineering driven course and you better be ready for the calculations... Made for Engineers.

Coming from an Electrical/Electronic background it was really intuitive. I plan on continuing...

AC

2 янв. 2019 г.

This is the place to start your journey into robotics. But stick to the entire 6 courses. This course is about laying out the basics so don't expect making robots straightaway. All the best!

Фильтр по:

## 1–25 из 200 отзывов о курсе Modern Robotics, Course 1: Foundations of Robot Motion

автор: Ashish R

3 февр. 2019 г.

Good course, very heavy on theory with little to no example problems. Its good to teach theory but should be followed up with multiple real world problem to solve by hand or code.

автор: Marcus G

9 февр. 2020 г.

Explanations are not clear. I had to get back to my old mechanic books to fully understand the content of the course. As it doesn't show how to apply what you learn it is boring and useless to me. Not worth the money at all.

автор: Lucas R

23 июня 2020 г.

More example exercises. Chapter 3 was very poor at having enough example exercises, and the discussion went deep too fast before showing more examples. The videos should be longer with more explanation of the theory. There are several exercises at the end of the chapter, but before solving those, more examples would be good.

On Youtube, you can find videos that explain a lot more than what is available on the course. The difference is that Youtube is free, and this Course is paid to get the certificate. 2-5min videos clearly are not enough. Any lecture should be is at least an 30-60min long. Relying just on the theory of the book to learn can be very difficult!

The video lectures and the book should complement each other, but this course puts all of the onus of passing the knowledge solely on the book.

автор: Arnab C

3 янв. 2019 г.

This is the place to start your journey into robotics. But stick to the entire 6 courses. This course is about laying out the basics so don't expect making robots straightaway. All the best!

автор: Andrew M

25 окт. 2018 г.

I wish there was a bit more explanation of some of the equations and derivations. Sometimes the textbook just skips to the final equation, and does not show how it got there. It would be nice if the videos showed the derivation, as opposed to just the equations.

автор: Ajmeet S D

10 окт. 2020 г.

One of the best courses I have ever taken. The textbook really helps with understanding concepts.

Can't wait to finish the remaining courses and get my specialization

автор: Saj M

6 нояб. 2018 г.

Could add more material to help with certain mathematical constructs.

11 дек. 2019 г.

The overall quality of the course is good, but there were a couple of issues which were a bit unsatisfactory.

For example, I didn't find the videos very engaging and helpful, but rather bland. Professor Lynch simply reads through a prepared text (which is in the background) and gives you an overview of the concepts which I found quite hard to follow. So you need to read the supplied book which, on the other hand, is very clear and rigorous. Thus you may completely ignore the videos, or just refer to them as a wrap-up.

автор: Pranesh G

10 июня 2019 г.

it was good intro to the field of robotics. this course has intrigued my interest to learn further in the area of robotics. I would recommend this course for aspiring roboticist

автор: John G

20 дек. 2020 г.

Plus 5 points for course content: nice to see a course that isn't afraid of using some mathematics. Very well presented. Minus 2 points for use of peer review for grading. While that might be defensible for a completely free MOOC, it is really unacceptable for a paid course. Take the \$ you are receiving and pay a TA to develop some better automated quizzes or code review bots, and get rid of the ridiculous and tedious peer reviews.

автор: Sichun X

17 мая 2020 г.

Really well structured and organized. The material can be slightly overwhelming at times because of the concepts, but once you get a hang of it, it becomes much better!

автор: Mahmoud A M M

29 июня 2018 г.

It's a great course but it need to make more detalied video

автор: Priyank S

23 июля 2021 г.

The course is not well structured at all. The videos are crisp, sure, but it takes several hours of studying from the book (and other sources too!) just to meet the expectations.

The videos convey not much, and so, the whole course is about reading the books and trying to get through the quiz.

And I don't know about the others, but I found the quiz to be extremely tough (though exhaustive).

Not to mention the several prerequisites that need to be met before even starting the course, which saps away bit of motivation.

All in all, not a typical Coursera course. I have never had to leave a course mid-way before, but I just can't be spending so much time (and money) anymore.

автор: SAMARTH

15 сент. 2018 г.

I thank both NORTHWESTERN UNIVERSITY and COURSERA for providing such an wonderfull course, please do this if you want to pursue your career in the field of robotics, and also the professors answer your doubts regularly, so do not think you wont have support, they are really sweet and YES 5 starts are less for this course : )

автор: Youssef D

4 апр. 2020 г.

very fundamental course for robotics & it covers all the basics you need to start in robotics field.

thank you Dr Kevin Lynch & Dr Frank Park.

автор: Juan C C Z

4 мая 2018 г.

I like this course and I like the textbook. The reason I gave it 4 stars instead of 5 is because I think it could be a bit more rigorous, e.g., it is explained that postmultiplication is associated to body frame transformations (which is true), but it's exposed dogmatically instead of derived mathematically. I think that approaching Robotics using screw theory and differential equations instead of a more geometric and traditional way is a very good idea, but it calls for a better formalism in the presentation of the themes.

автор: Jeff S

29 июня 2018 г.

A solid course, which could be improved by exploring from top down instead of bottom up. The course starts with fundamentals spaces, then of rotation matrices, translations then moves to screwstwists, wrenches. I think it would help to have this first course be an overview of robotics: Robots (types) of, the use of forward and inverse kinematics, dynamics, constrained motion, path planning, sensory systems (vision, touch), SLAM, etc.

автор: Haoran C

8 авг. 2018 г.

Rigorous mathematical introduction to rigid body motion combined with practical exercises in MATLAB and V-REP. I hope that the coursera platform can provide more exercises that test deep understanding and more difficult exercises.

автор: Rishipal S

29 апр. 2019 г.

This provides good understanding on the motion of robot. Although some notations were confusing but overall it is a very good course.

автор: abdelrahman s

3 сент. 2019 г.

The information in the videos are too little and lacks good example

the course need more solved examples

2 нояб. 2022 г.

I went up to course 5 out of 6 and I was at the end of course 5 and GOT NO HELP FROM NOBODY, no tutor, no professor, no student to help, give a hint. So, I abandoned the Specialisation. It's outrageous.

автор: Elena O

26 июня 2018 г.

Awful. It is just a glimpse of a course.

автор: Paul G

6 янв. 2021 г.

beyond me.

автор: Zhang K

31 мар. 2020 г.

I have to say it's a really great class with their textbook which can download, but I think it quite difficult for those people who didn't learn linear algebra and calculous. But I think this courses is excately great for me. I can learn a lot knowledges about Rotation matrix ect.. (Textbook is really great, I think people should download and read it during taking whole classes. (bcs you will find the class is too short it need textbook to let you truely understand those knowledge. ( and the discussion forum is really great place to find some hint from others. Really.)

And thanks for teachers and students who did these series classes for our! Respect!

автор: Roshan Z

17 мая 2020 г.

A perfect course for undergraduate engineering students who want to pursue robotics as a career. A perfect balance of difficulty level is maintained, meaning you need to be good at linear algebra (Matrices) and geometrical transformations if you want to excel in this course. And a very basic elementary knowledge of any programming language like Python or MATLAB is desirable. The textbook offered in this course has helped me understand the course at a deeper level whereas the lectures provided an overview of the important points. Awesome course, has been challenging and very interesting!


text:
Extracting countable chains from linear orders - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T05:01:25Z http://mathoverflow.net/feeds/question/85699 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85699/extracting-countable-chains-from-linear-orders Extracting countable chains from linear orders Tomek Kania 2012-01-15T00:11:06Z 2013-05-10T08:30:11Z
There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{*}$

(by $\omega^{*}$ I mean set of natural number with reversed order). It seems to be a non-trivial result - for example, one can derive it from the Baumgartner-Hajnal theorem but this is, in my taste, too heavy machinery.

Do you know who iss responsible for this result? Are there any cheaper ways (than the BH-theorem) to obtain it?
 http://mathoverflow.net/questions/85699/extracting-countable-chains-from-linear-orders/85701#85701 Answer by François G. Dorais for Extracting countable chains from linear orders François G. Dorais 2012-01-15T00:27:08Z 2012-01-15T00:27:08Z
You can derive it straight from Ramsey's Theorem. We may assume the linear order is countably infinite, say $x_0,x_1,x_2,\ldots$ enumerates it. Define the coloring $c:[\omega]^2\to2$ by $c(i,j) = 0$ iff $i \lt j$ and $x_i \lt_P x_j$ (and color 1 if the two orderings disagree). If $H$ is a homogeneous set for $c$, then the subsequence $\langle x_i \rangle_{i \in H}$ is either increasing or decreasing with respect to ${\lt_P}$.

This result is actually not quite as strong as Ramsey's Theorem (for pairs and two colors). See Hirschfeldt & Shore, Combinatorial principles weaker than Ramsey's theorem for pairs, JSL 72 (2007), 171-206.
 http://mathoverflow.net/questions/85699/extracting-countable-chains-from-linear-orders/85702#85702 Answer by Goldstern for Extracting countable chains from linear orders Goldstern 2012-01-15T00:27:22Z 2012-01-15T00:27:22Z
It seems to me that a natural solution is to use Ramsey's theorem $\aleph_0 \to (\aleph_0)^2_2$: enumerate a countable subset, and color two points depending on whether the enumeration agrees with the given order.

This proof seems "cheaper" to me: Wlog the linear order $P$ is a subset of the rationals. Find a limit point $r$. Wlog $r$ is a limit point of the points from $P$ below $r$ -- so you can find an increasing sequence converging to $r$.
 http://mathoverflow.net/questions/85699/extracting-countable-chains-from-linear-orders/85704#85704 Answer by Andreas Blass for Extracting countable chains from linear orders Andreas Blass 2012-01-15T02:03:56Z 2012-01-15T02:03:56Z
Here's another "cheap" proof. If your linear ordering doesn't have a decreasing $\omega$-sequence, then it's well-ordered, and therefore order-isomorphic to an ordinal. Since it's infinite, that ordinal is at least $\omega$, and so your ordering not only has a subset of order-type $\omega$ but has an initial segment of order-type $\omega$. (I've used the axiom of choice, or at least dependent choice, to infer "well-ordered" from the non-existence of a decreasing $\omega$-sequence, but some choice is needed in any proof. Without choice, there can be infinite linearly ordered sets with no countably infinite subsets.)
 http://mathoverflow.net/questions/85699/extracting-countable-chains-from-linear-orders/130237#130237 Answer by Butch Malahide for Extracting countable chains from linear orders Butch Malahide 2013-05-10T08:30:11Z 2013-05-10T08:30:11Z
Every infinite sequence has a monotone subsequence. (In introductory real analysis texts this is proved for sequences of real numbers, but the same proof works for sequences in a linearly ordered set.) So take an infinite sequence of distinct elements, etc.