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Previous Years JEE Advanced Questions

Q. Three vectors $\overrightarrow{\mathrm{P}}, \overrightarrow{\mathrm{Q}}$ and $\overrightarrow{\mathrm{R}}$ are shown in the figure. Let S be any point on the vector $\overrightarrow{\mathrm{R}}$. The distance between the points P and S is b $|\overrightarrow{\mathrm{R}}|$. The general relation among vectors $\overrightarrow{\mathrm{P}}, \overrightarrow{\mathrm{Q}}$ and $\overrightarrow{\mathrm{S}}$ is :

$(\mathrm{A}) \overrightarrow{\mathrm{S}}=(1-\mathrm{b}) \overrightarrow{\mathrm{P}}+\mathrm{b}^{2} \overrightarrow{\mathrm{Q}}$

(B) $\overrightarrow{\mathrm{S}}=(b-1) \overrightarrow{\mathrm{P}}+b \overrightarrow{\mathrm{Q}}$

(C) $\overrightarrow{\mathrm{S}}=(1-\mathrm{b}) \overrightarrow{\mathrm{P}}+\mathrm{b} \overrightarrow{\mathrm{Q}}$

$(\mathrm{D}) \overrightarrow{\mathrm{S}}=\left(1-\mathrm{b}^{2}\right) \overrightarrow{\mathrm{P}}+\mathrm{b} \overrightarrow{\mathrm{Q}}$

[JEE Advanced – 2017]

Sol. (C)

Let vector from point P to point S be $\overrightarrow{\mathrm{c}}$

$\Rightarrow \overrightarrow{\mathrm{c}}=\mathrm{b}|\overrightarrow{\mathrm{R}}| \hat{\mathrm{R}}=\mathrm{b}|\overrightarrow{\mathrm{R}}|\left(\frac{\overrightarrow{\mathrm{R}}}{|\overrightarrow{\mathrm{R}}|}\right)=\mathrm{b} \overrightarrow{\mathrm{R}}=\mathrm{b}(\overrightarrow{\mathrm{Q}}-\overrightarrow{\mathrm{P}})$

from triangle rule of vector addition

$\overrightarrow{\mathrm{P}}+\overrightarrow{\mathrm{C}}=\overrightarrow{\mathrm{S}}$

$\overrightarrow{\mathrm{P}}+\mathrm{b}(\overrightarrow{\mathrm{Q}}-\overrightarrow{\mathrm{P}})=\overrightarrow{\mathrm{S}}$

$\Rightarrow \overrightarrow{\mathrm{S}}=(1-\mathrm{b}) \overrightarrow{\mathrm{P}}+\mathrm{b} \overrightarrow{\mathrm{Q}}$

Q. Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density  remains uniform throughout the volume. The rate of fractional change in density $\left(\frac{1}{\rho} \frac{\mathrm{d} \rho}{\mathrm{dt}}\right)$ is constant. The velocity v of any point on the surface of the expanding sphere is proportional to :

(A) $\mathrm{R}^{3}$

(B) $\frac{1}{\mathrm{R}}$

(C) R

(D) $\mathrm{R}^{2 / 3}$

[JEE Advanced – 2017]

Sol. (C)

Density of sphere is $\rho=\frac{\mathrm{m}}{\mathrm{v}}=\frac{3 \mathrm{m}}{4 \pi \mathrm{R}^{3}}$

$\Rightarrow \frac{1}{\rho} \frac{\mathrm{d} \rho}{\mathrm{dt}}=-\frac{3}{\mathrm{R}} \frac{\mathrm{d} \mathrm{R}}{\mathrm{dt}}$

since $\Rightarrow \frac{1}{\rho} \frac{\mathrm{d} \rho}{\mathrm{dt}}$ is constant

$\therefore \frac{\mathrm{d} \mathrm{R}}{\mathrm{dt}} \propto \mathrm{R}$

Velocity of any point on the circumfrence V is equal to $\frac{\mathrm{d} \mathrm{R}}{\mathrm{dt}}$ (rate of change of radius of outer layer)


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# SQUID-based detection of ultra-low-field multinuclear NMR of substances hyperpolarized using signal amplification by reversible exchange

## Abstract

Ultra-low-field (ULF) nuclear magnetic resonance (NMR) is a promising spectroscopy method allowing for, e.g., the simultaneous detection of multiple nuclei. To overcome the low signal-to-noise ratio that usually hampers a wider application, we present here an alternative approach to ULF NMR, which makes use of the hyperpolarizing technique signal amplification by reversible exchange (SABRE). In contrast to standard parahydrogen hyperpolarization, SABRE can continuously hyperpolarize 1 H as well as other MR-active nuclei. For simultaneous measurements of 1 H and 19 F under SABRE conditions a superconducting quantum interference device (SQUID)-based NMR detection unit was adapted. We successfully hyperpolarized fluorinated pyridine derivatives with an up to 2000-fold signal enhancement in 19 F. The detected signals may be explained by two alternative reaction mechanisms. SABRE combined with simultaneous SQUID-based broadband multinuclear detection may enable the quantitative analysis of multinuclear processes.

## Introduction

Signal Amplification By Reversible Exchange (SABRE) is a relatively new technique to produce continuous hyperpolarization to boost nuclear magnetic resonance (NMR) signals and is even applied for magnetic resonance imaging (MRI)1,2. SABRE is based on a symmetry-breaking mechanism that converts the parahydrogen (para-H2) spin order into a non-Boltzmann polarization1,3,4. High-field NMR measurements show signal enhancements of up to a factor of 105 for 1H. The underlying mechanism is based on the interaction of para-H2 with a substrate of interest via an Ir-based catalyst system1.

Gong et al. first reported the use of SABRE at low-field in 20105. Further low-field (10–500 mT) examinations for the standard para-H2 induced polarization (PHIP) approach, where hydrogen atoms were added to a double or triple bond6,7, were done by Hamans et al.8 and Theis et al.9,10. In general, the advantage of SABRE against standard-PHIP is the possibility that the enhanced substrates can be continuously re-hyperpolarized by supplying a steady flow of parahydrogen11,12. It is important to note that most heteronuclei require special pulse sequences, or one has to lower the field strength to perform an efficient polarization transfer. Thus, for high-field MR, dedicated solutions have been developed such as SABRE-SHEATH13 or mechanically challenging shuttle mechanisms14. Most of the experiments have focused on the detection of hyperpolarized molecules in high magnetic fields, on increasing the number of hyperpolarized substrates15,16, or on measuring extracts of biofluids where spin densities down to sub-μM concentrations could have been detected17. The mechanism relies on transforming the spin order of the para-H2 singlet state into nuclear spin polarization of Ir hydride protons when an exchange reaction is performed in a low magnetic field1,18,19. The required field strength (2–10 mT) of the polarization transfer reaction to further spin-½ nuclei of the substrate such as pyridine is strongly dependent on the coupling constants1,20,21. So far the theoretical background of SABRE is based on level anti-crossings between magnetic field independent J-coupling energy levels and the magnetic field dependent chemical shift11,22,23.

Low-field and ultra-low-field (ULF, <10 mT) NMR provide an optimal and in comparison to high-field NMR less costly tool for investigating the polarization transfer mechanism. Even MR imaging can be realized at ultra-low fields24,25,26,27,28,29,30,31,32. More important, certain experiments that require complex and expensive technical efforts at high fields can be easily realized at ultra-low fields, such as the simultaneous detection of several nuclei33, the separation of detection field and polarizing field, or the variation of the polarizing field. However, a wider use of ULF NMR is still hampered by the inherent low signal-to-noise ratio (SNR), as the thermal equilibrium magnetization is increasing with the polarizing field, which is typically in the range of several mT as compared to high-field NMR with up to 23.5 T. Additionally, ULF NMR signals are in the range of some kHz (about 2 kHz at the earth’s magnetic field), which renders the signal in detection coils too low.

A combined strategy provides a solution to increase the SNR dramatically: first, detection is realized by a superconducting interference device (SQUID) that is very sensitive and can detect magnetic field down to some fT; second, the signal is enhanced by using the hyperpolarization technique SABRE.

ULF NMR provides the advantage that the polarizing field B p  > B 0, where B 0 is the detection field of the NMR signal, can be easily and reproducibly adapted to the individual field strengths required for the optimum polarization transfer of specific substances34. The B p field is usually realized by electromagnetic coils which allow field changes within milliseconds. Also imaging in the vicinity of metals is possible without distortion artefacts35. The sensitivity of different magnetic field detectors used for ULF MRI36,37, such as for example atomic magnetometers, is comparable to the already widely used SQUIDs. A SQUID-based system has not only a higher signal to noise ratio (SNR) compared to a system using a Faraday coil at ULF38, but also can detect the NMR signal of multiple nuclei simultaneously. SQUIDs are broadband detectors, which are able to detect the magnetic flux directly, rather than the change of the magnetic field, making them sensitive from DC up to the GHz range39.

Another advantage is that the NMR signal can be detected via a second order superconducting gradiometer, which acts as a surface coil39. The open geometry and the small field strengths needed for detecting the NMR signal makes a SQUID based system combinable with imaging techniques such as magnetoencephalography (MEG)40,41, where SQUIDs can be used to detect the MEG and NMR signals.

Since SQUIDs are sensitive to signals from DC up to the GHz range, the whole system is placed inside a cylindrical mu-metal shield with a diameter of 780 mm for magnetic shielding and inside a cubic steel-shielding chamber with an edge length of 2.8 m for shielding RF noise.

To allow for continuous hyperpolarization, a cylindrical/spherical PEEK vessel with 2 ml volume serving as reaction chamber was adapted. The para-H2 was continuously delivered at the bottom of the vessel and bubbled continuously through the sample. At the top part, a small hole permitted the para-H2 outflow. Since the evaporation rate of the substrate dramatically increases with the flow rate of the para-H2 and capillary action in combination with overpressure due to the para-H2 removes liquid from the vessel, a reservoir behind the hole in the top part was installed together with a backflow tube. The backflow tube feeds the liquid from the reservoir back to the bottom of the reaction chamber. A typical measurement session required about 2 to 3 hours. At a para-H2 flow rate of approx. 1.5 l/h, roughly 3–4 ml of the substrate evaporated. By filling the reservoir with additional 5 ml we were able to measure during the whole session without refilling. The temperature of the sample stayed constantly at room temperature during the whole measurement. The para-H2 was produced on-site during the experiment with a home-made para-H2 generator operating at about 21 K–30 K. The design of the generator is based on ref. 46. The concentration of para-H2 (in comparison to orthohydrogen) was determined before the experiments by using a home-made temperature-stabilized thermal-conductivity cell based on a heated tungsten filament47,48,49,50. With this method a quantitative analysis of the para-H2 concentration is difficult, however qualitatively we could see that the concentration was much higher than the equilibrium concentration at liquid nitrogen temperature (77 K).

A simple free-induction-decay (FID) pulse sequence served for acquiring the NMR signal and is shown in Fig. 1(d). A prepolarizing pulse with variable amplitude B p and variable length t Bp was used to increase the B 0 field strength. After the adiabatic switch-off of the B p field the sample magnetization is aligned with B 0 and a 90° B 1 pulse followed. A double resonant pulse, for example, a B 1 pulse exhibiting in its Fourier transform two peaks at the Larmor frequencies of nucleus X and Y, can be used to excite the magnetization of multiple nuclei simultaneously. In this study we used single resonant pulses with one exception in the SI (see spectrum in Figure S1). Subsequently, the FID was read out.

As model compounds 3,5-bis(trifluoromethyl)pyridine, ethyl-5-fluoronicotinic acid and 3-fluoropyridine were chosen to compare the hyperpolarizability of 1H and 19F quantitatively under ULF conditions. All measurements were performed in presence of an Ir catalyst [Ir(COD)(IMes)(Cl)]51. Substrates as well as the catalyst were dissolved in methanol and injected at room temperature into the reaction chamber before starting the first measurement. The exact mixture of the three substances was 7 mg of [Ir(COD)(IMes)(Cl)] and 0.23 mmol of the fluorinated pyridine derivative, dissolved in 10 ml methanol. The samples were not degassed.

The detection field B 0 was ~150 µT (about three times stronger than the earth’s magnetic field). Accordingly, the Larmor frequencies of the hyperpolarized nuclei were ~6140 Hz for 1H and ~5770 Hz for 19F (see peak positions in SI Figure S1). The field strength of ~150 µT was chosen, because the noise spectrum around 6 kHz did not show any additional noise due to external noise sources. Test experiments on solutions of the compounds and catalyst in methanol but without para-H2 show small MR signals of the methanol, which increases linearly with B p . Since this signal is much smaller than the hyperpolarized signal of the substrate, no deuterated methanol was used.

To determine the influence of the polarizing field B p on the achievable signal enhancements experiments were performed using the sequence shown in Fig. 1d. The B p field strength was varied between 0.144 mT (the B 0 field) and 10.3 mT. The acquired signal was Fourier transformed, and the absolute value of the signal was integrated over the range of the multiple peak structure (≈20 Hz around the excitation frequency). The results show (Fig. 2) that the signal enhancement is not only influenced by the polarizing field, but also by the substance to be hyperpolarized and the corresponding nuclei. Interestingly, the 19F signals of the monofluorinated compounds exhibit only a weak dependence on the polarizing field when compared to the according 1H signals. The field strengths that yield maximum signal enhancement for the 19F and the 1H signal of the substances were also different. The B p field for the maximum 19F signal enhancement is lower for the two monofluorinated pyridine derivatives lower than for 1H. 3,5-Bis(trifluoromethyl)pyridine seems to exhibit a different behavior. Here, the B p field dependency of the 19F is even less pronounced than for all the other substrates. The small increase of the area-under-peak for stronger B p field strengths is due to an increase in noise, which becomes dominant here, because of the small SNR.

A more detailed investigation can be made by plotting the spectra (real part of the MR signal) as a function of the B p field. Due to J-coupling, multiple peaks can be observed. For 3-fluoropyridine and 3,5-bis(trifluormethyl)pyridine the signal amplitudes of all resonances are correlated. The same could be observed for the 1H signal of ethyl-5-fluoronicotinic acid. An exception is observed for the 19F signal of ethyl-5-fluoronicotinic acid (see Fig. 3): the peak intensity below 5771 Hz and between 5773 Hz and 5776 Hz seems to be correlated. However, the peak at 5772 Hz (between dashed lines) and at 5778 Hz (between dotted lines) is changing its phase for increasing B p . The sequence parameters are listed in Table 1, measurement number 1 to 6.

The measurements support previous results that the polarization transfer from 1H to further spin-1/2 nuclei is field-dependent even at ultra-low fields52,53. In the next set of experiments we investigated the signal enhancement of 1H and 19F and the magnetization transfer between both nuclei when applying the SABRE technique. It was observed that a strong magnetization transfer reaction to 1H and 19F nuclei takes place in the range of 0–10 mT. However, while the transfer to 1H nuclei of pyridine derivatives was strongly field-dependent, the field dependence for 19F hyperpolarization is much less pronounced (see Fig. 2). To the best of our knowledge, this is the first time that such behavior could be detected.

As a next characterization step, the signal enhancement as a function of the hyperpolarization time t Bp was determined in order to gain the maximum SNR for multiple averages. The experimental setup was identical to the previous measurements; however, instead of sweeping B p , now t Bp was varied between 100 ms and 10.1 s. The polarizing field B p was fixed to the value where the maximum enhancement was observed (Fig. 4). The corresponding experimental parameters are listed in Table 1, rows 7 to 12.

An exponential saturation function

$$M({t}_{Bp})={M}_{HP}(1-\exp (-\frac{{t}_{Bp}-{t}_{0}}{{T}_{HP}}))$$
(1)

was fitted to the data (see green lines in Fig. 4). Here, M(t Bp ) is the magnetization of the sample, which is proportional to the area under the peak. M HP is the saturation magnetization for infinite polarizing time, t 0 is a time offset and T HP is the buildup time of the magnetization. The buildup time T HP depends also on the longitudinal relaxation time T 1, which is dependent on the B p field strength, as well as on the buildup time constant for the hyperpolarization of the substrate. T HP is dominated by the shorter of both processes. The fitted values for T HP of the 19F and 1H signals of the three substrates are listed in Table 2.

The results show that T HP can be different for the observed nuclei. Whereas for 3-fluoropyridine, T HP for 19F and 1H is within the confidence interval and has in comparison to the other two substrates a very long T HP , the other two substances show a different behavior. T HP for 1H of ethyl-5-fluoronicotinic acid is about 1 s shorter than for 19F, also within the confidence interval. The influence of the substituent (-COOCH2CH3) in ethyl-5-fluoronicotinic acid is even stronger and the T HP time for 19F of 3,5-bis(trifluoromethyl)pyridine was too short and could not be fitted (see Fig. 4). As can be seen in Fig. 2 (purple dots) the hyperpolarization process for 19F is independent from the field strength. For the sequence shown in Fig. 1d the hyperpolarization process starts during the data acquisition time of the previous shot directly after the 90° pulse resulting in a minimum hyperpolarization time of the data acquisition time. For short T HP the signal is already saturated. Therefore, a fit using Equation (1) will not lead to reasonable results. For the other substances, even though the hyperpolarization processes start also at the same position, they are not saturated for short t BP . Therefore, by varying t BP , Equation (1) leads to reasonable results.

In order to get more information on the hyperpolarization process, hyperpolarized high-resolution ULF spectra were additionally acquired using the simple FID sequence. The sequence parameters were set to the B p amplitude for maximal enhancement, and t Bp was roughly estimated to get the highest SNR for multiple averages. The sequence parameters are listed in Table 1 (rows 13 to 18). A detailed analysis of the 1H and 19F frequency regions in the ULF spectra show J-coupling resolved multiple resonances for the first two hyperpolarized substrates. For 3,5-bis(trifluoromethyl)pyridine only two singlets were observable (see Fig. 5).

To differentiate the hyperpolarization effects from the standard spectra, which were obtained at thermal equilibrium, simulations of NMR spectra without hyperpolarized signals were performed both at high and ultra-low fields using Bruker Topspin software (high-field) as well as VeSPA (low-field)54. The coupling constants were determined using thermal 1H and 19F spectra acquired at high-field (7 T) of all three non-hyperpolarized substrates dissolved in methanol-d4 (see supplementary material). Interestingly, the thermal simulations fit well even to the quite complex ULF hyperpolarized spectra (Fig. 5, green lines) except for a central resonance with large amplitude in the 1H spectrum. This line shows most likely hyperpolarized methanol. The 19F signal in the spectrum displays the 1H- 19F couplings, while 1H-1H as well as 1H- 19F couplings dominate the 1H spectrum. For 3,5-bis(trifluoromethyl)pyridine no J-couplings were detected. It may therefore be hypothesized that the effect of hyperpolarization significantly increases the signal at ULF but alters relatively little the overall spectral characteristics, except for the ratio of the amplitudes of several spectral lines. An analysis of the methanol resonance line in the 1H data supports the conclusion that the central line in the 1H spectral range is due to hyperpolarized methanol. This signal is also visible in the 1H ULF spectrum of 3,5-bis(trifluoromethyl)pyridine. However, as no J-couplings were detectable at high-field simulations, no simulations were performed for this substance. The broad 1H signal around the narrow central peak may be due to 1H resonances with shortened T2-time.

These encouraging results show that SABRE in combination with SQUID-based ULF NMR is well suited to increase the low SNR significantly and allow to measure otherwise non-detectable multinuclear spectra.

Aside from allowing measuring samples with low spin densities or nuclei with low sensitivity, the use of SABRE as a hyperpolarization technique enables a steady state generation of hyperpolarization. Compared to standard PHIP, SABRE allows repeating the experiment multiple times thus additionally increasing the SNR.

Our measurements also demonstrate that the theory behind SABRE is not yet completely understood11,22,23,52. J-coupling as the main mechanism for the polarization transfer cannot explain the differences of the B p field dependencies between the 19F and 1H signals of all three substrates. One would expect that the polarization is first transferred to a proton of the substrate and subsequently transferred to 19F (see Fig. 6a). But then the B p dependencies of the 19F and 1H signal should be correlated, which is not the case. It seems that other coupling mechanisms such as dipole–dipole coupling cannot be neglected, since the polarization transfer takes place in the vicinity of the catalyst. Therefore, the secular part of the intramolecular dipole–dipole coupling does not average to zero, as it is the case in isotropic liquids55. Thus, a direct polarization transfer reaction from 1H to 19F might become possible (see Fig. 6b).

A further explanation of the observed results is based on a proton-proton exchange reaction (see Fig. 6c). It was reported that the ortho-standing hydrogen nuclei can exchange via the Ir-catalyst with Ir-coordinated hydrogen (here from parahydrogen)56. This supports the conclusion that at least one hydrogen atom from parahydrogen was transferred to the fluorinated pyridine derivative and followed by an intramolecular polarization transfer to the vicinal bounded fluorine. A complete simulation of the hyperpolarized spectra may therefore have to include additional processes.

The presented results demonstrate that a SQUID-based NMR system is a promising setup for the investigation of hyperpolarization techniques that work optimally in the μT and mT range and below, such as SABRE or dynamic nuclear polarization. As described by Appelt et al., low-field MR detection can be used for the determination of relaxation times, diffusion and J-coupling constants57. Our results also support the hypothesis of alternative reaction pathways in SABRE that are based on an H/H exchange56,58. Furthermore, SQUID-based systems can be used for magnetic resonance imaging and the simultaneous detection of different MR-active nuclei. Combining continuous hyperpolarization methods with SQUIDs increases the sensitivity to a point where spatially-resolved MR methods or even MR imaging of heteronuclei of background-free molecular probes at ultra-low-field conditions will be feasible.

The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.

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44. 44.

Maslennikov, Y. CRYOTON Co. Ltd. http://cryoton.org (2017).

45. 45.

Seton, H. C., Hutchison, J. M. & Bussell, D. M. Liquid helium cryostat for SQUID-based MRI receivers. Cryogenics 45, 348–355 (2005).

46. 46.

Tellez-Juárez, M. C. et al. Hydrogen storage in activated carbons produced from coals of different ranks: Effect of oxygen content. International Journal of Hydrogen Energy 39, 4996–5002 (2014).

47. 47.

Weitzel, D. H. & White, L. E. Continuous analysis of ortho-parahydrogen mixtures. Review of Scientific Instruments 26, 290–292 (1955).

48. 48.

Weitzel, D. H. & Park, O. E. Iron catalyst for production of liquid parahydrogen. Review of Scientific Instruments 27, 57 (1956).

49. 49.

Weitzel, D. H., van Valin, C. C. & Draper, J. W. Design data for ortho-parahydrogen converters. In: Timmerhaus K.D. (eds) Advances in Cryogenic Engineering. Advances in Cryogenic Engineering, vol 3. Springer, Boston, MA (1960).

50. 50.

Weitzel, D. H. & White, L. E. Continuous analysis of ortho-parahydrogen mixtures. In: Timmerhaus K.D. (eds) Advances in Cryogenic Engineering. Advances in Cryogenic Engineering, vol 1. Springer, Boston, MA (1960).

51. 51.

Cowley, M. J. et al. Iridium N-heterocyclic carbene complexes as efficient catalysts for magnetization transfer from para -hydrogen. J. Am. Chem. Soc. 133, 6134–6137 (2011).

52. 52.

Pravdivtsev, A. N., Yurkovskaya, A. V., Zimmermann, H., Vieth, H.-M. & Ivanov, K. L. Transfer of SABRE-derived hyperpolarization to spin-1/2 heteronuclei. RSC Adv 5, 63615–63623 (2015).

53. 53.

Plaumann, M. et al. Parahydrogen-induced polarization transfer to 19F in perfluorocarbons for 19F NMR spectroscopy and MRI. Chemistry - A European Journal 19, 6334–6339 (2013).

54. 54.

VeSPA – Versatile Simulation, Pulses, and Analysis http://scion.duhs.duke.edu/vespa (2017).

55. 55.

Levitt, M. H. Spin dynamics. Basics of nuclear magnetic resonance. 2nd ed. (John Wiley & Sons, Chichester, England, Hoboken, NJ, 2008).

56. 56.

Barskiy, D. A. et al. The feasibility of formation and kinetics of NMR signal amplification by reversible exchange (SABRE) at high magnetic field (9.4 T). Journal of the American Chemical Society 136, 3322–3325 (2014).

57. 57.

Appelt, S., Häsing, F. W., Kühn, H., Sieling, U. & Blümich, B. Analysis of molecular structures by homo- and hetero-nuclear J-coupled NMR in ultra-low field. Chemical Physics Letters 440, 308–312 (2007).

58. 58.

Permin, A. B. & Eisenberg, R. One-hydrogen polarization in hydroformylation promoted by platinum−tin and iridium carbonyl complexes. A new type of parahydrogen-induced effect. J. Am. Chem. Soc. 124, 12406–12407 (2002).

## Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft (BE 1824/12-1). We thank Joern Engelmann for fruitful discussions.

## Author information

Authors

### Contributions

K.B., M.R., T.M., D.K., R.K., H.M., K.S., J.B. and M.P designed research; K.B., M.R., C.B. and M.P. performed research; K.B., M.R., U.B., P.A., J.B. and M.P analyzed data; K.B. and M.P wrote the whole manuscript; and all authors reviewed the manuscript.

### Corresponding author

Correspondence to K. Buckenmaier.

## Ethics declarations

### Competing Interests

The authors declare that they have no competing interests.

Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Buckenmaier, K., Rudolph, M., Back, C. et al. SQUID-based detection of ultra-low-field multinuclear NMR of substances hyperpolarized using signal amplification by reversible exchange. Sci Rep 7, 13431 (2017). https://doi.org/10.1038/s41598-017-13757-7

• Accepted:

• Published:

• ### SQUID-based ultralow-field MRI of a hyperpolarized material using signal amplification by reversible exchange

• Seong-Joo Lee
• , Keunhong Jeong
• , Jeong Hyun Shim
• , Hyun Joon Lee
• , Sein Min
• , Heelim Chae
• , Sung Keon Namgoong
•  & Kiwoong Kim

Scientific Reports (2019)

• ### Metabolic and Molecular Imaging with Hyperpolarised Tracers

• Jason Graham Skinner
• , Luca Menichetti
• , Alessandra Flori
• , Anna Dost
• , Andreas Benjamin Schmidt
• , Markus Plaumann
• , Ferdia Aiden Gallagher
•  & Jan-Bernd Hövener

Molecular Imaging and Biology (2018)

format: plain


text:
# Select the correct option from the given alternatives: Let a, b, c be distinct non-negative numbers. If the vectors aiajckikandcicjbkai^+aj^+ck^,i^+k^ and ci^+cj^+bk^ lie in a plane, then c is - Mathematics and Statistics

MCQ

Select the correct option from the given alternatives:

Let a, b, c be distinct non-negative numbers. If the vectors "a"hat"i" + "a"hat"j" + "c"hat"k" , hat"i" + hat"k"  "and"  "c"hat"i" + "c"hat"j" + "b"hat"k" lie in a plane, then c is

#### Options

• the arithmetic mean of a and b

• the geometric mean of a and b

• the harmonic man of a and b

• 0

#### Solution

the geometric mean of a and b

Concept: Vectors and Their Types
Is there an error in this question or solution?


text:
×

# An ISI 2010 Problem

Let $$a_1>a_2>.......>a_r$$ be positive real numbers. Compute $$\lim_{n \to \infty} ({a_1}^n +{a_2}^n + ........+ {a_r}^n)^{\frac{1}{n}}$$.

[Please be kind enough to write the solution.]

Note by Maharnab Mitra
3 years, 8 months ago

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Let $L = \lim_{x \to \infty} (a_1^x + a_2^x + \dots + a_r^x)^{1/x}.$ Then \begin{align*} \log L &= \log \lim_{x \to \infty} (a_1^x + a_2^x + \dots + a_r^x)^{1/x} \\ &= \lim_{x \to \infty} \log (a_1^x + a_2^x + \dots + a_r^x)^{1/x} \\ &= \lim_{x \to \infty} \frac{\log (a_1^x + a_2^x + \dots + a_r^x)}{x}. \end{align*}

By L'Hopital's Rule, \begin{align*} \lim_{x \to \infty} \frac{\log (a_1^x + a_2^x + \dots + a_r^x)}{x} &= \lim_{x \to \infty} \frac{\frac{d}{dx} \log (a_1^x + a_2^x + \dots + a_r^x)}{\frac{d}{dx} x} \\ &= \lim_{x \to \infty} \frac{a_1^x \log a_1 + a_2^x \log a_2 + \dots + a_r^x \log a_r}{a_1^x + a_2^x + \dots + a_r^x}. \end{align*}

Then \begin{align*} \lim_{x \to \infty} \frac{a_1^x \log a_1 + a_2^x \log a_2 + \dots + a_r^x \log a_r}{a_1^x + a_2^x + \dots + a_r^x} &= \lim_{x \to \infty} \frac{(a_1^x \log a_1 + a_2^x \log a_2 + \dots + a_r^x \log a_r)/a_1^x}{(a_1^x + a_2^x + \dots + a_r^x)/a_1^x} \\ &= \lim_{x \to \infty} \frac{\log a_1 + (a_2/a_1)^x \log a_2 + \dots + (a_r/a_1)^x \log a_r}{1 + (a_2/a_1)^x + \dots + (a_r/a_1)^x} \\ &= \log a_1. \end{align*}

Therefore, $$L = a_1$$.

- 3 years, 8 months ago

Thanks :)

- 3 years, 8 months ago

Take a1 to power n common from bracket..it become lim(a1)[1 + (a2/a1)^n +....]^1/n .

It all terms terms become zero ..so answer will come to ...a1.

- 3 years, 8 months ago

Not immediately true. The limiting value of $$0 ^ 0$$ depends on what path you take.

Edit: Ooops, I spoke too quickly. The power needs to be logarithmic. Use $$f(x) = x, g(x) = \frac{ 1} {\log x}$$, then we have $$f(x) ^ { g(x) } = e$$.

Staff - 3 years, 8 months ago

$$\lim_{x \to 0} x^{x^0.2} = e^{\lim_{x \to 0} x^{\frac{1}{5}} ln (x)}$$

Now, $$\lim_{x \to 0} x^{\frac{1}{5}} ln (x)= \lim_{x \to 0} \frac{ ln (x)}{x^{\frac{-1}{5}}}$$ $$=\lim_{x \to 0} \frac{-x^{\frac{1}{5}}}{5}$$ (using $$L'Hospital$$'s rule) which goes to 0. Thus, the required limit is $$e^0=1$$

- 3 years, 8 months ago

Sorry, updated.

Staff - 3 years, 8 months ago

Sir, then what should the correct answer (or rather solution) to the problem be?

- 3 years, 8 months ago

Multiply and divide by a1 and proceed. Ans: a1

- 3 years, 7 months ago


text:
# A Seemingly Easy Indefinite Integral

So, I'm stuck with an integral. It asks- $$\int e^x \sec(x) dx$$

I tried integration by parts, tried substituting. Nothing worked. Wolframalpha gave me some peculiar stuff(something called 'Hypergeometric function') which I don't understand at all.

Any kind of help will be appreciated. Thank you.

• If Wolfram Alpha gives a hypergeometric function, presumably there is no elementary antiderivative. You can't integrate everything. Jan 4, 2020 at 18:37
• "seemingly" ??? The number of characters has no relation to the difficulty :-) Try $x^x$.
– user65203
Jan 4, 2020 at 20:04
• A hypergeometric $\phantom{}_2 F_1$ with complex parameters, hmm, such a joy! :D Jan 4, 2020 at 22:11


text:
For convenience we'll work in the hyperbolic upper half plane $H$. We are given a hyperbolic quadrilateral $Q$ with vertices $a,b,c,d$ and geodesic segment edges $[ a,b ]$ $[ b,c ]$ $[ c,d ]$ $[ d,a ]$. We are also given that $\angle abc=\angle bad=\frac{\pi}{2}$ and $\rho_H([a,d])=\rho_H([b,c])$ where $\rho_H([x,y])$ is the hyperbolic length of the geodesic segment from $x$ to $y$. Can we conclude that $\rho_H([a,b])<\rho_H([c,d])$?

It's clear that by a transformation by an isometry, we can put the segment $[a,b]$ on the imaginary axis and then the segments $[a,d]$ and $[b,c]$ will be segments of half circles with centre at $0$ and of equal length, so the points $c$ and $d$ lie on the same 'ray' extending from the origin. This clearly implies that $\alpha=\angle bcd=\angle cda$ and, by the Gauss-Bonnet Theorem, we may also deduce that $0<Area_H(Q)=2\pi-(\frac{pi}{2}+\frac{pi}{2}+\alpha+\alpha)$ and so conclude that $\alpha<\frac{\pi}{2}$ (although this can be fairly easily seen graphically).

I feel like the proof that $\rho_H([a,b])<\rho_H([c,d])$ should be obvious and hinges on the fact that $0<\alpha<\frac{\pi}{2}$, but playing around with hyperbolic trigonometric identities has just led to a rather large mess.

I should point out that this was the beginning of an attempt to prove something which I have now proved using a different method. I am, however, curious to see if this method would have worked.

-
Don't you mean "$\rho_H([c,d]) > \rho_H([a,b])$"? –  Blue Apr 21 '13 at 17:48
I do indeed. Thanks. –  Daniel Rust Apr 21 '13 at 17:51

I'll use upper-case letters for points, and lower-case letters for distances.

Edit. As @Daniel mentions in his own answer, the $|AD|=|BC|$ condition is unnecessary. I've revised my answer to be more general.

We have quadrilateral $\square ABCD$ with right angles at $A$ and $B$. Write $p :=|AB|$, $q =|CD|$, $r = |AD|$, $s=|BC|$ for the edge lengths; finally, let $t=|BD|$ be the length of a diagonal.

In right triangle $\triangle ABD$, the Law of Cosines and Law of Sines give us $$\cosh t = \cosh p \cosh r \qquad\qquad \sinh r = \sinh t \sin \angle ABD$$ In $\triangle CBD$, we have \begin{align} \cosh q &= \cosh s \cosh t - \sinh s \sinh t \cos \angle CBD \\ &= \cosh s (\cosh p \cosh r) - \sinh s ( \sinh t \sin \angle ABD ) \\ &= \cosh p \cosh r \cosh s - \sinh r \sinh s\\ &= ( 2 \sinh^2 \frac{p}{2}+1) \cosh r \cosh s - \sinh r \sinh s \\ &= 2 \sinh^2\frac{p}{2} \cosh r \cosh s + \cosh(r-s) \end{align} so that \begin{align} 2\sinh^2 \frac{q}{2} + 1 &= 2 \sinh^2\frac{p}{2} \cosh r \cosh s + \cosh(r-s) \\[6pt] 2\sinh^2 \frac{q}{2} &= 2 \sinh^2\frac{p}{2} \cosh r \cosh s + \cosh(r-s) - 1 \\[6pt] 2\sinh^2 \frac{q}{2} &= 2 \sinh^2\frac{p}{2} \cosh r \cosh s + 2\sinh^2\frac{r-s}{2} \\[6pt] \sinh^2 \frac{q}{2} &= \sinh^2\frac{p}{2} \cosh r \cosh s + \sinh^2\frac{r-s}{2} \end{align}

Clearly, then, $\sinh\frac{q}{2} \ge \sinh\frac{p}{2}$, so that $q\ge p$, with equality if and only if $r=s=0$.

-
This is great. Thanks. Didn't even need to use $\alpha <\frac{\pi}{2}$. –  Daniel Rust Apr 21 '13 at 18:02

I'll add an answer to my own question in case anyone would like to see an alternative proof. This is based on the method I eventually ended up using, which I alluded to in the question. It has the disadvantage of not being as elementary as Blue's proof as it makes use of a rather general lemma about pairs of non-intersecting geodesics. The lemma is stated by the following:

Lemma If $l_1$ and $l_2$ are two geodesics in $H$ which do not intersect, then there exists a unique pair $z\in l_1$ and $w\in l_2$ such that $\rho_H(z,w)=\min\{\rho_H(x,y)|\:x\in l_1, y\in l_2\}$ and further, the unique geodesic $l$ passing through $z$ and $w$ is the unique geodesic which meets both $l_1$ and $l_2$ at right angle

Using the construction mentioned in the question, $Q$ can be moved isometrically so that $a$ and $b$ lie on the imaginary axis and and the segments $[a,d]$ and $[b,c]$ are segments of semicircles $l_1$ and $l_2$ centered at the origin.

Given that $l_1$ and $l_2$ do not intersect, we may use the above lemma and, given that the imaginary axis intersects $l_1$ and $l_2$ at right angles at $a$ and $b$, it follows that $a$ and $b$ must be the unique minimising pair of elements. We also know that $c\in l_2$ and $d\in l_1$ and $c\neq b$, $d\neq a$ and so can conclude that $\rho_H(a,b)<\rho_H(c,d)$.

This has the added bonus that we do not need the hypothesis that $\rho_H(a,d)=\rho_H(b,c)$, only that one is non-zero.

-


text:
# mne.viz.circular_layout¶

mne.viz.circular_layout(node_names, node_order, start_pos=90, start_between=True, group_boundaries=None, group_sep=10)[source]

Create layout arranging nodes on a circle.

Parameters
node_names

Node names.

node_order

List with node names defining the order in which the nodes are arranged. Must have the elements as node_names but the order can be different. The nodes are arranged clockwise starting at “start_pos” degrees.

start_posfloat

Angle in degrees that defines where the first node is plotted.

start_betweenbool

If True, the layout starts with the position between the nodes. This is the same as adding “180. / len(node_names)” to start_pos.

group_boundaries

List of of boundaries between groups at which point a “group_sep” will be inserted. E.g. “[0, len(node_names) / 2]” will create two groups.

group_sepfloat

Group separation angle in degrees. See “group_boundaries”.

Returns
node_anglesarray, shape=(n_node_names,)

Node angles in degrees.


text:
Solves the differential equations with the optimised and fixed parameters from a previous successful call to mkinfit and plots the observed data together with the solution of the fitted model.

# S3 method for mkinfit
plot(
x,
fit = x,
obs_vars = names(fit$mkinmod$map),
xlab = "Time",
ylab = "Observed",
xlim = range(fit$data$time),
ylim = "default",
col_obs = 1:length(obs_vars),
pch_obs = col_obs,
lty_obs = rep(1, length(obs_vars)),
show_residuals = FALSE,
show_errplot = FALSE,
maxabs = "auto",
sep_obs = FALSE,
rel.height.middle = 0.9,
row_layout = FALSE,
lpos = "topright",
inset = c(0.05, 0.05),
show_errmin = FALSE,
errmin_digits = 3,
frame = TRUE,
...
)

plot_sep(
fit,
show_errmin = TRUE,
show_residuals = ifelse(identical(fit$err_mod, "const"), TRUE, "standardized"), ... ) plot_res( fit, sep_obs = FALSE, show_errmin = sep_obs, standardized = ifelse(identical(fit$err_mod, "const"), FALSE, TRUE),
...
)

plot_err(fit, sep_obs = FALSE, show_errmin = sep_obs, ...)

## Arguments

x Alias for fit introduced for compatibility with the generic S3 method. An object of class mkinfit. A character vector of names of the observed variables for which the data and the model should be plotted. Defauls to all observed variables in the model. Label for the x axis. Label for the y axis. Plot range in x direction. Plot range in y direction. Colors used for plotting the observed data and the corresponding model prediction lines. Symbols to be used for plotting the data. Line types to be used for the model predictions. Should the plot be added to an existing plot? Should a legend be added to the plot? Should residuals be shown? If only one plot of the fits is shown, the residual plot is in the lower third of the plot. Otherwise, i.e. if "sep_obs" is given, the residual plots will be located to the right of the plots of the fitted curves. If this is set to 'standardized', a plot of the residuals divided by the standard deviation given by the fitted error model will be shown. Should squared residuals and the error model be shown? If only one plot of the fits is shown, this plot is in the lower third of the plot. Otherwise, i.e. if "sep_obs" is given, the residual plots will be located to the right of the plots of the fitted curves. Maximum absolute value of the residuals. This is used for the scaling of the y axis and defaults to "auto". Should the observed variables be shown in separate subplots? If yes, residual plots requested by "show_residuals" will be shown next to, not below the plot of the fits. The relative height of the middle plot, if more than two rows of plots are shown. Should we use a row layout where the residual plot or the error model plot is shown to the right? Position(s) of the legend(s). Passed to legend as the first argument. If not length one, this should be of the same length as the obs_var argument. Passed to legend if applicable. Should the FOCUS chi2 error value be shown in the upper margin of the plot? The number of significant digits for rounding the FOCUS chi2 error percentage. Should a frame be drawn around the plots? Further arguments passed to plot. When calling 'plot_res', should the residuals be standardized in the residual plot?

## Value

The function is called for its side effect.

## Details

If the current plot device is a tikz device, then latex is being used for the formatting of the chi2 error level, if show_errmin = TRUE.

## Examples

# One parent compound, one metabolite, both single first order, path from
# parent to sink included
# \dontrun{
SFO_SFO <- mkinmod(parent = mkinsub("SFO", "m1", full = "Parent"),
m1 = mkinsub("SFO", full = "Metabolite M1" ))#> Successfully compiled differential equation model from auto-generated C code.fit <- mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE)#> Warning: Observations with value of zero were removed from the datafit <- mkinfit(SFO_SFO, FOCUS_2006_D, quiet = TRUE, error_model = "tc")#> Warning: Observations with value of zero were removed from the dataplot(fit)plot_res(fit)plot_res(fit, standardized = FALSE)plot_err(fit)
# Show the observed variables separately, with residuals
plot(fit, sep_obs = TRUE, show_residuals = TRUE, lpos = c("topright", "bottomright"),
show_errmin = TRUE)
# The same can be obtained with less typing, using the convenience function plot_sep
plot_sep(fit, lpos = c("topright", "bottomright"))
# Show the observed variables separately, with the error model
plot(fit, sep_obs = TRUE, show_errplot = TRUE, lpos = c("topright", "bottomright"),
show_errmin = TRUE)# }


text:
# Troland

The troland (symbol Td), named after Leonard T. Troland, is a unit of conventional retinal illuminance. It is meant as a method for correcting photometric measurements of luminance values impinging on the human eye by scaling them by the effective pupil size. It is equal to retinal illuminance produced by a surface whose luminance is one nit when the apparent area of the entrance pupil of the eye is 1 square millimeter.[1]

The troland typically refers to the ordinary or photopic troland, which is defined in terms of the photopic luminance:

${\displaystyle \mathrm {T} =\mathrm {L} \times \mathrm {p} }$,

where L is the photopic luminance in cd m−2 and p is pupil area in mm2.

A scotopic troland is also sometimes defined:

${\displaystyle \mathrm {T'} =\mathrm {L'} \times \mathrm {p} }$,

where L′ is the scotopic luminance in cd m−2 and p is pupil area in mm2.

Although named "retinal luminance", Trolands should not be confused with actual photons flux incident on a retina Lr. Conversion between the two is dependent of the specific eye optical parameters and not done by a simple units conversion factor.

## Units conversion

Troland does not directly convert to other units, being a retinal luminance per unit area of a pupil.

However Troland is linked to retinal luminance in lux = lm/m^2 as follows. Assuming the corneal luminance L from an extended source, the pupil diameter p and the focal length of the eye F, the retinal luminance is:

Lr [lm / m^2] = pi * L / 4 / (f/#)^2 ~ pi * L * p^2 / 4 / F^2.

Multiplying by the pupil area : Trolands [cd / m^2 * mm^2] = L * pi * p^2 / 4 = F^2 * Lr ~ 289 * Lr.

Alternatively, the retinal luminance Lr [lm/m^2] = Trolands / 289 [mm^2]

As provided by a more accurate optical calculations, the conversion factor is 278 rather than 289 as demonstrated by simplified considerations above.

Sometimes (by convention only, although not rigorously accurate by definition), retinal luminance is expressed in [cd/m^2] = [lm/sr /m^2]. Assuming a Lambertian surface, 1 cd/m^2 = pi lm/m^2 = pi lux. That is, 1 [cd/m^2] = 289/pi [Troland] ~ 92 [Troland]

## Physical quantities

• luminance
• equivalent luminance [2]

## Unit system

centimeter-gram-second (cgs) [2]

## Basic unit dimensions

[length]^(-2) [luminous intensity] [2]

## Comparisons

• ≈ 0.8 × luminance of a kerosene candle (≈ 12000 cd/m^2 )
• ≈ luminance of a sperm candle (≈ 10000 cd/m^2 )
• ≈ luminance of an average daytime clear sky (≈ 8000 cd/m^2 )[2]


text:
Logical equivalence

In logic and mathematics, two statements are logically equivalent if they can prove each other (under a set of axioms),[1] or have the same truth value under all circumstances. In propositional logic, two statements are logically equivalent precisely when their truth tables are identical.[2] To express logical equivalence between two statements, the symbols ${\displaystyle \equiv }$, ${\displaystyle \Leftrightarrow }$ and ${\displaystyle \iff }$are often used.[3][4]


text:
11:46 AM
1

Definition of basis for a topology on $X$ given in the Foundation of Topology By Patty given by Let $(X,\mathscr T)$ be a topological space. A basis for $\mathscr T$ is a subcollection $\mathscr B$ of $\mathscr T$ with the property that if $U\in \mathscr T$ then $U=\emptyset$ or there is ...

0

Let $n\in \mathbb N$, and let $\mathscr P$ denote the collection of all polynomials in $n$ variables. For $p\in \mathscr P$, let $Z(p)=\{(x_1,x_2,...,x_n)|p(x_1,x_2,...,x_n)=0\}$. Show that $\{Z(p):p\in \mathscr P\}$ is a basis for the closed sets of some topology (Called the Zariski topology) on...

Please help me.

3 hours later…
2:44 PM
It seems to me that the notion of basis for closed sets seems to be used only seldom.
@Mathgeek Did you want to write there "subcollection $\mathscr D$ of $\mathscr U$" rather than "subcollection $\mathscr D$ of $\mathscr T$"?

1 hour later…
4:00 PM
@MartinSleziak I don't understand. where should I change?

4:11 PM
"Let $(X,\mathscr U)$ be a family of closed sets(closed sets of topology on $X$). A basis for $\mathscr U$ is a subcollection $\mathscr D$ of $\mathscr T$ with the property that ..."
@Mathgeek This is what your post says right after "my attempt".
If you say that $\mathscr D$ is a subcollection of $\mathscr T$, then you're actually saying that it consists of open sets.
I suppose that you want to say that $\mathscr D$ consist of closed sets.

1 hour later…
5:22 PM
@MartinSleziak Thank you very much I have corrected it :)


text:
## Amy in Randomland [book review]

Posted in Statistics with tags , , , , , , , , , , , , on August 15, 2022 by xi'an

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Book Review section in CHANCE.]

## conditioning on insufficient statistics in Bayesian regression

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , on October 23, 2021 by xi'an

“…the prior distribution, the loss function, and the likelihood or sampling density (…) a healthy skepticism encourages us to question each of them”

A paper by John Lewis, Steven MacEachern, and Yoonkyung Lee has recently appeared in Bayesian Analysis. Starting with the great motivation of a misspecified model requiring the use of a (thus necessarily) insufficient statistic and moving to their central concern of simulating the posterior based on that statistic.

Model misspecification remains understudied from a B perspective and this paper is thus most welcome in addressing the issue. However, when reading through, one of my criticisms is in defining misspecification as equivalent to outliers in the sample. An outlier model is an easy case of misspecification, in the end, since the original model remains meaningful. (Why should there be “good” versus “bad” data) Furthermore, adding a non-parametric component for the unspecified part of the data would sound like a “more Bayesian” alternative. Unrelated, I also idly wondered at whether or not normalising flows could be used in this instance..

The problem in selecting a T (Darjeeling of course!) is not really discussed there, while each choice of a statistic T leads to a different signification to what misspecified means and suggests a comparison with Bayesian empirical likelihood.

“Acceptance rates of this [ABC] algorithm can be intolerably low”

Erm, this is not really the issue with ABC, is it?! Especially when the tolerance is induced by the simulations themselves.

When I reached the MCMC (Gibbs?) part of the paper, I first wondered at its relevance for the mispecification issues before realising it had become the focus of the paper. Now, simulating the observations conditional on a value of the summary statistic T is a true challenge. I remember for instance George Casella mentioning it in association with a Student’s t sample in the 1990’s and Kerrie and I having an unsuccessful attempt at it in the same period. Persi Diaconis has written several papers on the problem and I am thus surprised at the dearth of references here, like the rather recent Byrne and Girolami (2013), Florens and Simoni (2015), or Bornn et al. (2019). In the present case, the  linear model assumed as the true model has the exceptional feature that it leads to a feasible transform of an unconstrained simulation into a simulation with fixed statistics, with no measure theoretic worries if not free from considerable efforts to establish the operation is truly valid… And, while simulating (θ,y) makes perfect sense in an insufficient setting, the cost is then precisely the same as when running a vanilla ABC. Which brings us to the natural comparison with ABC. While taking ε=0 may sound as optimal for being “exact”, it is not from an ABC perspective since the convergence rate of the (summary) statistic should be roughly the one of the tolerance (Fearnhead and Liu, Frazier et al., 2018).

“[The Borel Paradox] shows that the concept of a conditional probability with regard to an isolated given hypothesis whose probability equals 0 is inadmissible.” A. Колмого́ров (1933)

As a side note for measure-theoretic purists, the derivation of the conditional of y given T(y)=T⁰ is arbitrary since the event has probability zero (ie, the conditioning set is of measure zero). See the Borel-Kolmogorov paradox. The computations in the paper are undoubtedly correct, but this is only one arbitrary choice of a transform (or conditioning σ-algebra).

## Bayesian sufficiency

Posted in Books, Kids, Statistics with tags , , , , , , , , , on February 12, 2021 by xi'an

“During the past seven decades, an astonishingly large amount of effort and ingenuity has gone into the search fpr resonable answers to this question.” D. Basu

Induced by a vaguely related question on X validated, I re-read Basu’s 1977 great JASA paper on the elimination of nuisance parameters. Besides the limitations of competing definitions of conditional, partial, marginal sufficiency for the parameter of interest,  Basu discusses various notions of Bayesian (partial) sufficiency.

“After a long journey through a forest of confusing ideas and examples, we seem to have lost our way.” D. Basu

Starting with Kolmogorov’s idea (published during WW II) to impose to all marginal posteriors on the parameter of interest θ to only depend on a statistic S(x). But having to hold for all priors cancels the notion as the statistic need be sufficient jointly for θ and σ, as shown by Hájek in the early 1960’s. Following this attempt, Raiffa and Schlaifer then introduced a more restricted class of priors, namely where nuisance and interest are a priori independent. In which case a conditional factorisation theorem is a sufficient (!) condition for this Q-sufficiency.  But not necessary as shown by the N(θ·σ, 1) counter-example (when σ=±1 and θ>0). [When the prior on σ is uniform, the absolute average is Q-sufficient but is this a positive feature?] This choice of prior separation is somewhat perplexing in that it does not hold under reparameterisation.

Basu ends up with three challenges, including the multinomial M(θ·σ,½(1-θ)·(1+σ),½(1+θ)·(1-σ)), with (n¹,n²,n³) as a minimal sufficient statistic. And the joint observation of an Exponential Exp(θ) translated by σ and of an Exponential Exp(σ) translated by -θ, where the prior on σ gets eliminated in the marginal on θ.

## Monte Carlo Markov chains

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , , on May 12, 2020 by xi'an

Darren Wraith pointed out this (currently free access) Springer book by Massimiliano Bonamente [whose family name means good spirit in Italian] to me for its use of the unusual Monte Carlo Markov chain rendering of MCMC.  (Google Trend seems to restrict its use to California!) This is a graduate text for physicists, but one could nonetheless expect more rigour in the processing of the topics. Particularly of the Bayesian topics. Here is a pot-pourri of memorable quotes:

“Two major avenues are available for the assignment of probabilities. One is based on the repetition of the experiments a large number of times under the same conditions, and goes under the name of the frequentist or classical method. The other is based on a more theoretical knowledge of the experiment, but without the experimental requirement, and is referred to as the Bayesian approach.”

“The Bayesian probability is assigned based on a quantitative understanding of the nature of the experiment, and in accord with the Kolmogorov axioms. It is sometimes referred to as empirical probability, in recognition of the fact that sometimes the probability of an event is assigned based upon a practical knowledge of the experiment, although without the classical requirement of repeating the experiment for a large number of times. This method is named after the Rev. Thomas Bayes, who pioneered the development of the theory of probability.”

“The likelihood P(B/A) represents the probability of making the measurement B given that the model A is a correct description of the experiment.”

“…a uniform distribution is normally the logical assumption in the absence of other information.”

“The Gaussian distribution can be considered as a special case of the binomial, when the number of tries is sufficiently large.”

“This clearly does not mean that the Poisson distribution has no variance—in that case, it would not be a random variable!”

“The method of moments therefore returns unbiased estimates for the mean and variance of every distribution in the case of a large number of measurements.”

“The great advantage of the Gibbs sampler is the fact that the acceptance is 100 %, since there is no rejection of candidates for the Markov chain, unlike the case of the Metropolis–Hastings algorithm.”

Let me then point out (or just whine about!) the book using “statistical independence” for plain independence, the use of / rather than Jeffreys’ | for conditioning (and sometimes forgetting \ in some LaTeX formulas), the confusion between events and random variables, esp. when computing the posterior distribution, between models and parameter values, the reliance on discrete probability for continuous settings, as in the Markov chain chapter, confusing density and probability, using Mendel’s pea data without mentioning the unlikely fit to the expected values (or, as put more subtly by Fisher (1936), “the data of most, if not all, of the experiments have been falsified so as to agree closely with Mendel’s expectations”), presenting Fisher’s and Anderson’s Iris data [a motive for rejection when George was JASA editor!] as a “a new classic experiment”, mentioning Pearson but not Lee for the data in the 1903 Biometrika paper “On the laws of inheritance in man” (and woman!), and not accounting for the discrete nature of this data in the linear regression chapter, the three page derivation of the Gaussian distribution from a Taylor expansion of the Binomial pmf obtained by differentiating in the integer argument, spending endless pages on deriving standard properties of classical distributions, this appalling mess of adding over the conditioning atoms with no normalisation in a Poisson experiment

$P(X=4|\mu=0,1,2) = \sum_{\mu=0}^2 \frac{\mu^4}{4!}\exp\{-\mu\}$,

botching the proof of the CLT, which is treated before the Law of Large Numbers, restricting maximum likelihood estimation to the Gaussian and Poisson cases and muddling its meaning by discussing unbiasedness, confusing a drifted Poisson random variable with a drift on its parameter, as well as using the pmf of the Poisson to define an area under the curve (Fig. 5.2), sweeping the improperty of a constant prior under the carpet, defining a null hypothesis as a range of values for a summary statistic, no mention of Bayesian perspectives in the hypothesis testing, model comparison, and regression chapters, having one-dimensional case chapters followed by two-dimensional case chapters, reducing model comparison to the use of the Kolmogorov-Smirnov test, processing bootstrap and jackknife in the Monte Carlo chapter without a mention of importance sampling, stating recurrence results without assuming irreducibility, motivating MCMC by the intractability of the evidence, resorting to the term link to designate the current value of a Markov chain, incorporating the need for a prior distribution in a terrible description of the Metropolis-Hastings algorithm, including a discrete proof for its stationarity, spending many pages on early 1990’s MCMC convergence tests rather than discussing the adaptive scaling of proposal distributions, the inclusion of numerical tables [in a 2017 book] and turning Bayes (1763) into Bayes and Price (1763), or Student (1908) into Gosset (1908).

[Usual disclaimer about potential self-plagiarism: this post or an edited version of it could possibly appear later in my Books Review section in CHANCE. Unlikely, though!]

## Rao-Blackwellisation, a review in the making

Posted in Statistics with tags , , , , , , , , , , on March 17, 2020 by xi'an

Recently, I have been contacted by a mainstream statistics journal to write a review of Rao-Blackwellisation techniques in computational statistics, in connection with an issue celebrating C.R. Rao’s 100th birthday. As many many techniques can be interpreted as weak forms of Rao-Blackwellisation, as e.g. all auxiliary variable approaches, I am clearly facing an abundance of riches and would thus welcome suggestions from Og’s readers on the major advances in Monte Carlo methods that can be connected with the Rao-Blackwell-Kolmogorov theorem. (On the personal and anecdotal side, I only met C.R. Rao once, in 1988, when he came for a seminar at Purdue University where I was spending the year.)


text:
# LaTeX compilation of simple document fails [closed]

I am just starting to work with LaTeX for the first time, so I apologize if my question is trivial.

I recently got a first `.tex` document from my supervising tutor, with the request we compile it twice via `pdflatex` so that references will work.

After manually installing the TeX Live 2012 distribution, pdflatex fails to compile the document. From what I can understand, the problem is a some kind of missing font:

``````This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)
entering extended mode
(./sem-aus.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, nohyphenation, loaded.
(/usr/share/texmf-texlive/tex/latex/base/article.cls
Document Class: article 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/size10.clo))
(/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
(/usr/share/texmf-texlive/tex/latex/base/utf8.def
(/usr/share/texmf-texlive/tex/latex/base/t1enc.dfu)
(/usr/share/texmf-texlive/tex/latex/base/ot1enc.dfu)
(/usr/share/texmf-texlive/tex/latex/base/omsenc.dfu)))
(/usr/share/texmf-texlive/tex/latex/base/fontenc.sty
(/usr/share/texmf-texlive/tex/latex/base/t1enc.def)
kpathsea: Running mktextfm ecrm1000
/usr/share/texmf/web2c/mktexnam: Could not map source abbreviation for ecrm1000.
/usr/share/texmf/web2c/mktexnam: Need to update ?
mktextfm: Running mf-nowin -progname=mf \mode:=ljfour; mag:=1; nonstopmode; input ecrm1000
This is METAFONT, Version 2.718281 (TeX Live 2009/Debian)

kpathsea: Running mktexmf ecrm1000
! I can't find file `ecrm1000'.
<*> ...ljfour; mag:=1; nonstopmode; input ecrm1000

Please type another input file name
! Emergency stop.
<*> ...ljfour; mag:=1; nonstopmode; input ecrm1000

Transcript written on mfput.log.
grep: ecrm1000.log: No such file or directory
mktextfm: `mf-nowin -progname=mf \mode:=ljfour; mag:=1; nonstopmode; input ecrm1000' failed to make ecrm1000.tfm.
kpathsea: Appending font creation commands to missfont.log.

relax
l.100 \fontencoding\encodingdefault\selectfont

? x
No pages of output.
Transcript written on sem-aus.log.
``````

I am pretty sure I set up TeX Live correctly; I also included the distribution in my PATH variable so that the binaries should be found. What is going wrong? I am pretty clueless and thankful for any hints.

Update 1: I installed the TeX Live distribution into `/usr/share/texlive/2012/i386-linux`, and I see the path `/usr/share/texmf-texlive/tex/latex/base/` appear instead - could this be a sign that the `pdflatex` command is looking in the wrong places? Again, since I have (yet) to learn what the files are used for, I don't know if this hunch makes any sense.

Update 2: I realized the `pdflatex` command invoked some TeX Live 2009 binary that was roaming around in my system. After removing those packages (hopefully without anything remaining), I will try again and see what happens.

Update 3: Through some weird bash caching mechanism, it was still invoking the old binary even though the PATH was set correctly. I removed all of the `texlive` packages from my package management system, and finally got it working.

-

## closed as too localized by Martin Schröder, Marco Daniel, Kurt, barbara beeton, GuidoMay 27 '13 at 18:16

This question is unlikely to help any future visitors; it is only relevant to a small geographic area, a specific moment in time, or an extraordinarily narrow situation that is not generally applicable to the worldwide audience of the internet. For help making this question more broadly applicable, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

Welcome to TeX.SX! You can have a look on our starter guide to familiarize yourself further with our format. –  Jubobs May 27 '13 at 9:35
You probably need to install some additional packages which contains the fonts. Btw: TeXLive 2009 is very outdated. Consider to install a current texlive 2012 or 2013pretest (tug.org/texlive). –  Ulrike Fischer May 27 '13 at 9:36
@UlrikeFischer I already installed texlive2012! Your comment made me realize that the binary which was executed is somehow in some old texlive distribution that came preinstalled with my system. I'm removing them and trying again. Thank you. –  MHaaZ May 27 '13 at 9:41

A quick google search on `ecrm1000` gives good results:
Both give the same solution: you need to install `texlive-latex-recommended` using your package manager
He would need to if he only had his distribution's version of TeX Live installed on his system. But he also has TeX Live 2012, so the problem is making sure that TL2012 is used when running commands like `latex` and `pdflatex` (etc., etc.), and not the OS-installed version of TeX Live, which needs the `texlive-latex-recommended`. –  jon May 27 '13 at 16:15
for me installing `texlive-fonts-recommended` did it (which doesn't seem to be a dependency of `texlive-latex-recommended` –  lewurm Nov 4 at 23:33


text:
# In a 1*mL volume of solution that is 0.10*mol*L^-1 with respect to CaCl_2(aq), what is the NUMBER of "ions in solution?"

Jun 11, 2017

There are $3 \times {10}^{-} 4 \cdot m o l \times {N}_{A}$ ions in solution.......

#### Explanation:

Now $\text{concentration}$ is given by the quotient:

$\text{concentration"="Moles of solute"/"Volume of solution}$

And thus to get the $\text{moles of solute}$ we takes the product

$\text{Volume of solution "xx" concentration}$.......

And so here,

$\text{moles of calcium chloride}$ $=$

$1 \cdot m L \times {10}^{-} 3 L \cdot m {L}^{-} 1 \times 0.10 \cdot m o l \cdot {L}^{-} 1 = {10}^{-} 4 \cdot m o l$

WITH RESPECT TO THE SALT $C a C {l}_{2}$.

Why did we emphasize $\text{the salt}$? Because EACH formula unit of $C a C {l}_{2}$ contributes 3 ions to the solution, i.e.....

$C a C {l}_{2} \left(s\right) \stackrel{{H}_{2} O}{\rightarrow} C {a}^{2 +} + 2 C {l}^{-}$

And so we have 3xx10^-4*molxx6.022xx10^23*mol^-1=?? ions??


text:
# Nuclear based wireless power?

I would like to ask if there is any Nuclear process which can be used to produce a wireless power ?

Is there any nuclear process can be radiate to the environment with no harm to human beings?

# How can I use the photo electric effect with some high Intensity radiation ?

-
Neutrinos are pretty harmless, i guess... – Asphir Dom Feb 27 '13 at 10:31
@AsphirDom but interact so weakly that they are useless as a power source. – anna v Feb 28 '13 at 5:28
Try sending hydrogen through mail. See en.wikipedia.org/wiki/Sneakernet (mailing of digital information). In case you're interested there's background radiation en.wikipedia.org/wiki/Background_radiation – raindrop Feb 28 '13 at 6:40
@Raindrop can you write it as an answer? – 0x90 Feb 28 '13 at 6:57


text:
# Tag Info

## Hot answers tagged app

4

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4

bugstatus-completed See the most recent Stack Overflow election, for example. It says, "These are your new moderators", but... who are they exactly? This image makes it seem like codeMagic was a new moderator, when in fact that's not the case as can be seen at http://stackoverflow.com/election/6

3

status-completedI asked this as a question, but apparently the convention is to report bugs/feature requests in the announcement thread. It appears, that the MathJax extension mhchem is currently not enabled in the StackPrinter app. This is rather inconvenient for most of the chemistry.se questions and answers. It would therefore be nice if this feature ...

3

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2

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2

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1

Here's a bug: When you enter a site, we can do meta.stackexchange. However, entering meta.stackexchange.com breaks EVERYTHING! Yes, everything. Please fix this.

1

Here is a small script (stackprinter) which uses Stack Printer from the command-line: #!/bin/sh type wkhtmltopdf || { echo >&2 "wkhtmltopdf command is required. Aborting."; exit 1; } : ${1?"Usage:$0 (url)"} URL=$1 && shift HOST=$(echo $URL | egrep -o "([.[:alnum:]]*?)\.(com|net|org)" | head -c-5) QID=$(echo \$URL | grep -o "\d.*/" | head ...

1

It seems that the HTML entities are escaped, in the results. Can this be fixed?

1

I have forked your source code, and enabled Retina display support. I have also recreated all the icons. Check it out on GitHub.

1

Outside world until the constructor returns. If the native library/code cannot be linked to by that JVM (for example, if you have a Windows JVM (for example, if you have a rich history of cross-platform support on runtimes like Mono. You may like to pose a few more possible for readability. Instead of declaring ProductOfDouble at the ...

Only top voted, non community-wiki answers of a minimum length are eligible


text:
## Testing Hypothesis

Testing hypothesis may be the most used and most misunderstood statistics tool. When we do even a simple fitting, and want to evaluate the fitting result, we have to use the hypothesis testing. One common quantity used is the reduced chi-squared.

A hypothesis testing means given an observation and hypothesis, Is the hypothesis NOT true? right, hypothesis test never tell us the trueness  of the hypothesis, but the wrongness of it. The core of the test is “Can we reject the null hypothesis?

There are one-tailed and two-tailed testing, as a result, the p-value has different meanings.

https://en.wikipedia.org/wiki/One-_and_two-tailed_tests

https://en.wikipedia.org/wiki/P-value

The p-value is the probability that the model agree with the observation. when the p-value too small, smaller than the confident level, the null hypothesis Rejected. But if the p-value is very large, in a 1-tailed test, we cannot say the null hypothesis is true, but we can say the null hypothesis CANNOT be rejected.

In 2-tailed test, there are two p-values, corresponding to each tail.

https://en.wikipedia.org/wiki/Confidence_interval

https://en.wikipedia.org/wiki/Type_I_and_type_II_errors

## Variance and Sigma in finding resolution

I have been mis-understood for a while.

First of all, the variance of a distribution is not equal its sigma, except for Normal distribution.

In an observation, there should be an intrinsic variance, for example, some hole size, or physical windows. And there is a resolution from the detection. As a result, we observed an overall effect of the intrinsic variance and the detector resolution. In an data analysis, one of the goal is the find out the resolution.

Lets denote the random variable of the intrinsic variance is

$X \sim D(\mu, \sigma^2)$

and the resolution of the detection is an other random variable of Normal distribution,

$Y \sim N(0, s^2)$

Then, we observed,

$Z = X + Y \sim D(\mu, \sigma^2 + s^2)$,

according to the algebra of distribution.

If the $\sigma >> s$ and the the intrinsic distribution is NOT a gaussian, say, it is a uniform distribution, then, the observed distribution is NOT a gaussian. One why to get the resolution is to do de-convolution. Since we are not interesting on the intrinsic distribution but the resolution, thus we can simply use the variance of the intrinsic distribution and the variance of the observed distribution to extract the resolution.

When $\sigma <= s$, the observed distribution is mostly a gaussian-like. and we can approximate the observed variance as the squared-sigma of a gaussian fit.

for example, in deducing the time resolution using time-difference method with a help of a tracking, that a narrow width of the position was gated.

The narrow width of the position is equivalent to a uniform distribution of time-difference. Thus, the time resolution is deduced using the observed variance and the variance of the uniform distribution. For the width of the position is $\Delta X$, the width of the time difference is

$\Delta t = \Delta X / c/\beta$,

Thus,

$Var(resol.) = Var(obser.) - Var(\Delta t)$

The variance of an uniform distribution is 1/12 of the square of the width.

$Var(\Delta t) = (\Delta t)^2/ 12 \neq (\Delta t)^2$

The effect of the factor 1/12 is very serious when the resolution is similar with the width of the time difference. But can be neglected when $Var(resol.) >> Var( \Delta t)$.

Because of missing this 1/12 factor, the resolution will be smaller than actual resolution.

Here is an example, I generated 10,000 data, the intrinsic distribution is a uniform distribution from 0 to 100, the resolution is a gaussian of sigma of 20. The result distribution is a convolution of them.

When find resolution using projection from a 2-D distribution, we should be careful about the projected 1-D distribution and its variance. For example, a uniform 2-D disk projected to 1-D distribution, the 1-D distribution is not uniform but a half circle,

$pdf =f(x) =\sqrt{(r^2-x^2)}$

the variance is $(0.5 r)^2$.

the formula to calculate variance is

$Var(X) = \int x^2 f(x) dx / \int f(x) dx$,

where $f(x)$ is the pdf.

ToDo, estimate the error of the deduced resolution, with number of data. For small number of data, the error should be large. but how large?

## Goodness of Fit

We assumed each data point is taking from a distribution with mean $\mu$ and variance $\sigma^2$

$Y\sim D(\mu, \sigma^2)$

in which, the mean can be a function of X.

For example, we have a data $Y_i$, it has relation with an independent variable $X_i$. We would like to know the relationship between $Y_i$ and $X_i$, so we fit a function $y = f(x)$.

After the fitting (least square method), we will have so residual for each of the data

$e_i = y_i - Y_i$

This residual  should be follow the distribution

$e \sim D(0, \sigma_e^2)$

The goodness of fit, is a measure, to see the distribution of the residual, agree with the experimental error of each point, i.e. $\sigma$

Thus, we would like to divide the residual with $\sigma$ and define the chi-squared

$\chi^2 = (\sum (e_i^2)/\sigma_{e_i}^2 )$.

we can see, the distribution of

$e/\sigma_e \sim D(0, 1)$

and the sum of this distribution would be the chi-squared distribution. It has a mean of the degree of freedom $DF$. Note that the mean and the peak of the chi-squared distribution is not the same that the peak at  $DF-1$.

In the case we don’t know the error, then, the sample variance of the residual is out best estimator of the true variance. The unbiased sample variance is

$\sigma_s^2 = Var(e)/DF$,

where $DF$ is degree of freedom. In the cause of $f(x) = a x + b$, the $DF = n-1$, because there is 1 degree o freedom used in x. And because the 1  with the b is fixed, it provides no degree of freedom.

## Weighted mean and error

We have n values of $x_i$ and error $\sigma_i$,

With a weighting $w_i$, the uncorrelated weighted mean and error is

$X= \sum x_i w_i / \sum w_i$

$S^2 = \sum w_i^2 \sigma_i^2 / (\sum w_i)^2$

when combining data, the weighting is

$w_i = 1/\sigma_i^2$

and the weighted error becomes

$S^2 = \sum{\frac{1}{\sigma_i^2}} / (\sum{\frac{1}{\sigma_i^2}})^2 = 1 / \sum \frac{1}{\sigma_i^2}$

Example.

we measured a quantity n times, we can assume the intrinsic error of the data is fixed. Thus,

$w_i = 1/n$

$X = \sum x_i / n$

$S^2 = \sum \sigma_0^2/n^2 = \sigma_0^2 /n$

Therefore, when we take more and more data, the error is proportional to $1/\sqrt{n}$.

In normal distribution, the sample of size n, the estimator of the sample mean and sample variance are

$X =\sum x_i/n$

$S^2 = \sum (x_i-X)^2 / (n-1)$

Don’t mix up the sample variance and intrinsic error, although they are very similar.

To explain the formula of the weighted variance, we have to go to the foundation of the algebra of distribution.

For a random variable follow a distribution with mean $\mu$ and variance $\sigma^2$,

$X \sim D(\mu, \sigma^2)$

Another random variable built on it,

$Z=aX+b \sim D(a \mu + b, a^2\sigma^2)$

The adding of two independent random variables is

$Z=aX + bY \sim D(a \mu_X + b \mu_Y, a^2\sigma_X^2 + b^2 \sigma_Y^2)$

But there is a catch, when the $\mu_X = \mu_Y$ and $\sigma_X = \sigma_Y$, The rule does not apply. But lets look back, if the mean and variance are the same, the two distribution does not really independent.

## Maximum Likelihood

In data analysis, especially the number of data is small, in order to found out the parameter of the distribution, which fit the data the best, maximum likelihood method is a mathematical tool to do so.

The ideal can be found in Wikipedia. For illustration, I generate 100 data points from a Gaussian distribution with mean = 1, and sigma = 2.

In Mathematica,

Data = RandomVariate[NormalDistribution[1, 2], 100]
MaxLikeliHood = Flatten[Table[{
mean,
sigma,
Log[Product[PDF[NormalDistribution[mean, sigma], Data[[i]]] // N, {i, 1,100}]],
},
{mean, -3, 3, 1}, {sigma, 0.5, 3.5, 0.5}], 1]

This calculate the a table of mean form -3 to 3, step 1, sigma from 0.5 to 3.5 step 0.5. To find the maximum of the LogProduct in the table:

maxL=MaxLikeliHood[[1 ;; -1, 3]];
Max[%]
maxN = Position[maxL[[1 ;; -1, 3]], %]
maxL[[Flatten[maxN]]]

The result is

{{1,2.,-217.444}}

which is the correct mean and sigma.

## estimate the mean reaction rate given that no event in time interval

The count follows Poisson Distribution:

$P(n|\lambda) = \frac{(\lambda T)^n}{n!} Exp(-\lambda T)$

this means, the probability of number of count, n, happened in time interval T given that the mean rate is $\lambda$.

now, we measured no count in time interval T, what is the mean rate for given confident interval?

if the count is zero, then the probability is:

$P(0|\lambda) = Exp(-\lambda T )$

and we can treat this as a new probability density function that, the probability of count rate is $\lambda$ in time interval T.

normalized this pdf.

$P(\lambda) = T Exp(-\lambda T )$

we can see, for $\lambda = 0$, the probability is 1. of course, if the count rate is 0, then no count is 100 %. therefore, we want to estimate the maximum count rate it will be to give “zero count “.

Now we have to introduce the Confident Interval (CL). this is the chance that the “interest” is true. in here, our interest is the “count rate is smaller then some value “. Thus, the total probability is:

$1- CL = \int_0^(\lambda_0) P(\lambda) d\lambda$

$\lambda = \frac{1}{T} ln(\frac{1}{CL})$

Lets give an example. suppose we count nothing in 10 second. what is the count rate for 95% Confident Interval?

plug in the equation and give $\lambda = 0.005$.

Now, we can see how the Confident Interval means. 95% means, in 100 measurement of 10 second interval, there is 95% that no count. thus, in 1000 second time interval, there is at most 5 count. which is same to say  $\lambda = 0.005$.

or, if we put the  $\lambda = 0.005$ into the Poisson distribution.

$P(0|0.005) = Exp(-0.05) = 0.951$

this is same that it has 95% give 0 count.

## On NMR noise reduction

there are many ways to reduce the signal-to-noise ratio in NMR.

here, we will show the simplest one – by measuring it many times.

the reason behind this is the different of statistic nature between signal and noise.

for signal, the correlation between each measurement is strong. or to say, if you measured the signal is around 10 this time, it will probable measure it around 10 next time.

however, the noise is different, it is random in nature. so, in average, it will equal to zero.

in mathematics, we use the standard deviation $\sigma$ and covariance $cov$ to show it.

$\sigma(S_T)=\sigma(\sum S_i) = \sqrt{ \sum ( \sigma( S_i)^2 + 2cov(S_i,S_j) ) }$

$\sigma(N_T)=\sigma(\sum N_i) = \sqrt{ \sum ( \sigma( N_i)^2 + 2cov(N_i,N_j) ) }$

since the signal should be correlated, $cov(S_i , S_j) = \sigma(S_i) \sigma(S_j)$ thus,

$\sigma(S_T) = n \sigma(S_i)$

but the noise should have no correlation, $cov(N_i, N_j) = 0$

$\sigma(N_T) = \sqrt{n} \sigma(N_i)$

Thus, the signal-to-noise level will be

$\sigma( S/N _T ) = \sqrt{n} \sigma (S/N)$

For 100 times measurements, the signal-to-noise ratio will be 10 times stronger.


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Question

# Is it true that every relation which is symmetric and transitive is also reflexive ? Give reasons.

Verified
156.3k+ views
Hint: - If you have to check whether a relation is reflexive or not in this question , first you have to assume a relation which is symmetric and transitive both then you have to check for reflexive(that means you have to check the number is in relation with itself or not).

Clearly aRb $\Rightarrow$bRa that is if a and b are both odd then b and a are also both odd, so it is a symmetric relation.
But this relation is not reflexive because $2 \in {\text{I}}$ but it can’t be in relation with 2 that is with itself to be in reflexive relation, because 2 is even number, and condition for to be in relation is number has to be a odd number.


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# Solving the matrix ODE $M\ddot x + C\dot x + Kx = \sin(\omega t)e_i$

Consider the following ODE, where $$x$$ is a vector, $$M, C, K$$ real square matrices and $$e_i$$ a basis vector (for example $$[1\ 0\ \dots 0]^\top$$).:

$$M\ddot x + C\dot x + Kx = \sin(\omega t)e_i$$

with $$x(0)=x_0$$ and $$\dot x(0)=v_0$$, $$\omega>0$$.

It's a common linear second order (matrix) ODE. The one-dimensional case is very common (damped oscillator with harmonic excitation). Is it possible to find a simple solution in the matrix case?

It is easy to derive a particular solution: assume $$x(t) = Re\big[x_1 e^{i\omega t}\big]$$, then $$x_1 = (-\omega^2 M +i\omega C + K)^{-1} e_i.$$

For the homogeneous solution, as in the one-dimensional case: if $$x=e^{\lambda t}x_2$$, $$(\lambda^2 M + \lambda C + K)x_2 = 0$$ and there should be a discussion on the value of $$\lambda$$. But in the general case, $$M$$, $$C$$ and $$K$$ are not simultaneously diagonizable... So, is there a trick?

I managed to solve the ODE using the first-order form, but I am wondering if it is possible to take advantage of the less general form of the original problem to simplify the result:

$$\begin{bmatrix} \dot x \\ \ddot x \end{bmatrix} = \underbrace{\begin{bmatrix} 0 & I \\ -M^{-1}K & -M^{-1}C \end{bmatrix}}_{A} \underbrace{\begin{bmatrix} x \\ \dot x\end{bmatrix} }_{X(t)} + \underbrace{ \sin(\omega t)\begin{bmatrix} O \\ -M^{-1}e_i \end{bmatrix}}_{b(t)}:= \dot X(t) = A X(t) + b(t)$$

The solutions of $$X'(t) = AX(t) + b(t)$$ are $$X = X_h + X_p$$ with $$X_h = e^{tA}X_1$$ and $$X_p = e^{tA}\int e^{-sA}b(s) ds$$, or $$X(t) = e^{tA}\big( X_0 + \int_0^t e^{-sA}b(s)ds\big)$$.

$$b(t) =\sin(\omega t) \underbrace{\begin{bmatrix} O \\ M^{-1} e_i \end{bmatrix}}_{H} := \sin(\omega t)H$$ so $$\int_0^t e^{-sA}b(s)ds = Re\left\lbrace \int_0^t e^{-sA+i(\omega s-\pi/2)I}ds\right\rbrace H = Re\left\lbrace e^{-i\pi/2}\int_0^t e^{sA_2}ds \right\rbrace H = Re\left\lbrace e^{-i\pi/2}A_2^{-1}(e^{A_2 t} - I)H \right\rbrace\quad \text{with }A_2 = -A+i\omega I$$ In the end: $$X(t)=e^{tA}\Big(X_0 - Re\left\lbrace i A_2^{-1}(e^{tA_2}-I)\right\rbrace H \Big)\quad \text{with } A_2 = -A + i \omega I$$

and of course, $$X_0 = [x_0, \ v_0]^\top$$. It does not really look like the solutions of the linear second-order ODE in this form!


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# 7th Class Mathematics Linear Equations Notes - Linear Equation in one variable

Notes - Linear Equation in one variable

Category : 7th Class

LINEAR EQUATION IN ONE VARIABLE

FUNDAMENTALS

• Variable: A symbol which takes various values is known as a variable. Normally it is denoted by letters x, y etc.
• Constant: A symbol having a fixed numerical value is called a constant.

Symbols used to denote a constant are generally, 'c', 'k' etc...

• Coefficient: In the product of a variable and a constant, each is called the coefficient of the other. Sometimes, symbols like a, b, 1, m etc..., are used to denote the coefficients.
• Expression: An expression can be defined as a combination of constant, variables and coefficients by some or all of the four fundamental mathematical operations $(+,\,\,-,\,\,\times \,\,\text{and}\,\,\div ).$

e.g., 2x - 6; here, 2 is the coefficient of $'x';\,\,'x'$ is the variable and $-6$is the constant. Similarly in $ay+b;\,\,a$ is the coefficient of $y;\,\,'y'$ is the variable and $(+b)$ is the constant.

• Equation: A statement of equality of two algebraic expression involving a variable is called an equation.
• Simple linear equation: An equation which contains only one variable of degree 1 is called a simple linear equation.

e.g.,      (i) $5x-1=6x+m$                                  (ii) $3\left( x-4 \right)=5$

(iii)$2y+5=\frac{y}{6}-2$                                  (iv) $\frac{t-1}{6}+\frac{2t}{7}=a$

• Solution of an equation: That value of the variable, which when substituted in the given equation, makes the two sides L.H.S. (Left Hand Side) and R.H.S. (Right hand sided) of the equation equal is called the solution or root of that equation.

e.g., $2x+6=3x-10\Rightarrow 6+10=3x-2x\Rightarrow 16=x$

Verification

Substituting $x=16$ we have LHS $=2\times 16+6=38$ & RHS $=\text{ }3\times 16-10=38$

$\therefore$$x=16$is a solution of the above equation.

• Rules for solving an equation

(a) Same number can be added to both sides of an equation.

(b) Same number can be subtracted from both sides of an equal.

(c) Both sides of an equation can be multiplied by the same non - zero number

(d) Both sides of an equation can be divided by the same non - zero number

(e) Cross multiplication: If$\frac{ax+b}{cx+d}=\frac{p}{q}$, then $q\left( ax+b \right)=p\left( cx+d \right).$

This process is called cross multiplication.

#### Other Topics

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