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# How to prepare solutions of different concentrations of potassium from stock solutions with different concentartions?

I need to prepare different concentrations of $\ce{K+}$ ($10$, $20$, $30$, $40$, $\pu{50 mM}$) by diluting a solution of $\pu{100 mM}$ $\ce{K+}$ with an isotonic buffer of $\pu{5 mM}$ $\ce{K+}$. I need $\pu{4 mL}$ of each concentration.

I really don't remember how to calculate this. Because I need to consider the $\ce{K+}$ within the isotonic buffer right, because they will also add amount of $\ce{K+}$? If I would have diluted the $\pu{100 mM}$ $\ce{K+}$ with just water, then I understand how to do.

• The only thing you need to remember in chemistry is conservation of matter. "Just water" is a solution where $\ce{K^+}$ concentration is simply zero. Apr 26, 2016 at 9:57
• Thanks for answearing, but I still dont understand how to prepare the 4 concentrations :/ Apr 26, 2016 at 10:17

## 1 Answer

If you mix two solutions of different concentrations with each other you can easily determine the resulting concentration from the general formula for it: $$c = \frac{n}{V}\tag1\label{concentration-general}$$

You have one solution with $c_1 = \pu{100 mM}$ and one with $c_2 = \pu{5 mM}$, therefore with $\eqref{concentration-general}$ you can determine the amount of substance for each, and the total amount of substance: $$n_\mathrm{tot} = c_1\cdot V_1 + c_2\cdot V_2\tag2\label{amount-of-substance-total}$$

Assuming there is not volume contraction, we can simply add them to know the total volume: $$V_\mathrm{total} = V_1 + V_2 \tag3\label{volume-total}$$

The total concentration therefore is: $$c_\mathrm{total} = \frac{n_\mathrm{total}}{V_\mathrm{total}} = \frac{c_1 V_1 + c_2 V_2}{V_1 + V_2}\tag4\label{concentration-total}$$

For your dilution series you will need to modify it, a $\eqref{volume-total}$ is fixed. $$V_\mathrm{total} = V_1 + V_2 = \text{const.}$$

Therefore we modify $\eqref{concentration-total}$: \begin{align} c_\mathrm{total} &= \frac{c_1 V_1 + c_2 (V_\mathrm{total} - V_1)}{V_\mathrm{total}} \tag{4a}\label{concentration-total-mod-a}\\ c_\mathrm{total} &= \frac{c_1 (V_\mathrm{total} - V_2) + c_2 V_2}{V_\mathrm{total}} \tag{4b}\label{concentration-total-mod-b}\\ \end{align}

Now you have one remaining unknown, solve for it: \begin{align} V_1 &= \frac{c_\mathrm{total} - c_2}{c_1 -c_2} V_\mathrm{total}\\ V_2 &= \frac{c_\mathrm{total} - c_1}{c_2 -c_1} V_\mathrm{total}\\ \end{align}

Unless you have very accurate equipment, however, I do suggest using larger quantities than $\pu{4 mL}$. \begin{array}{rrrrrr} c_\mathrm{total}/\pu{mM} & V_\mathrm{total}/\pu{mL} & c_1/\pu{mM} & c_2/\pu{mM} & V_1/\pu{mL} & V_2/\pu{mL} \\\hline 10 & 4 & 100 & 5 & 0.21 & 3.79 \\ 20 & 4 & 100 & 5 & 0.63 & 3.37 \\ 30 & 4 & 100 & 5 & 1.05 & 2.95 \\ 40 & 4 & 100 & 5 & 1.47 & 2.53 \\ 50 & 4 & 100 & 5 & 1.89 & 2.11 \\\hline \end{array}


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Oxidation number, also called oxidation state, the total number of electrons that an atom either gains or loses in order to form a chemical bond with another atom. The atom of the diatomic molecules like hydrogen, chlorine, oxygen, etc and metallic element like zinc, copper, sodium, etc is assigned zero oxidation number. *Response times vary by subject and question complexity. Chemistry. Tags: Question 6 . border: none !important; Redox (reduction–oxidation, pronunciation: / ˈ r ɛ d ɒ k s / redoks or / ˈ r iː d ɒ k s / reedoks) is a type of chemical reaction in which the oxidation states of atoms are changed. CaH2 2. Which Boots For Everest Base Camp Trek, Redox (reduction–oxidation, pronunciation: / ˈ r ɛ d ɒ k s / redoks or / ˈ r iː d ɒ k s / reedoks) is a type of chemical reaction in which the oxidation states of atoms are changed. jQuery(document).ready(function() { Edit. 3x = 8. x = $$+\frac{8}{3}$$ By stoichiometry, Fe 3 O 4 = $$\overset{+2-2}{FeO}.\overset{+3}{Fe_2}\overset{-2}{O_3}$$ $$\therefore$$ Oxidation number of Fe in Fe 3 O 4 is + 2 and +3. a year ago. 2. So, The oxidation state of oxygen is -2 except for a few exemptions. {"@context":"https://schema.org","@graph":[{"@type":"WebSite","@id":"http://backofficecenter.com/#website","url":"http://backofficecenter.com/","name":"GraphXSys.com","description":"GROW WITH EXCELLENCE","potentialAction":{"@type":"SearchAction","target":"http://backofficecenter.com/?s={search_term_string}","query-input":"required name=search_term_string"}},{"@type":"WebPage","@id":"http://backofficecenter.com/exizqj2z/#webpage","url":"http://backofficecenter.com/exizqj2z/","inLanguage":"en-US","name":"oxidation number of fe","isPartOf":{"@id":"http://backofficecenter.com/#website"},"datePublished":"2020-12-02T15:10:13+00:00","dateModified":"2020-12-02T15:10:13+00:00","author":{"@id":"http://backofficecenter.com/#/schema/person/"}}]} Free potassium has an oxidation number of +1. See Answer. Calculate oxidation number of Fe in Fe 3 O 4 and write a suitable justification of your answer. However at certain places I read that NO is neutral and Fe is in its +3 oxidation state. The highest known oxidation state is reported to be +9 in the tetroxoiridium(IX) cation (IrO + 4). Since there are two oxygen atoms in carbon dioxide, the total of the oxidation numbers corresponding to each oxygen is -4. var ajaxRemoveRevslider = function(obj) { +2 and +3 for Fe in Fe3O4. Hence, coordination no. It is a stoichiometric mixture of Ferrous (FeO) and Ferric (Fe 2 O 3) oxides combined as FeO.Fe 2 O 3.Therefore, the oxidation number found in such cases is average oxidation number. (914) 304 4263. Fluorine always has an oxidation number of -1. It is very likely that the coordinated Fe(III) is readily converted to Fe(V) through two-electron abstraction by PDS, and Fe(V) is responsible for the selective degradation of organic pollutants. Become a member and unlock all Study Answers. The relative stabilities of these oxidation states are markedly affected by coordination of different ligands. According to the rules to calculate oxidation number, which can be found in the previous subsection, the oxidation number of oxygen in its compounds (excluding peroxides) is -2. +2 and +3 for Fe in Fe3O4. Tags: Question 6 . } The oxidation state of iron in K 2 FeO 4 is... A. You can understand that the first iron has the number +three because it needs 3 groups ferrocyanide to make the compound. // EXTEND THE AJAX CONTENT LOADING TYPES WITH TYPE AND FUNCTION // obj.type : Post Type 280 times. That is enough to know it's B, but we'll do Fe … Live Weather Map Iraq, (a.addEventListener("DOMContentLoaded",n,!1),e.addEventListener("load",n,!1)):(e.attachEvent("onload",n),a.attachEvent("onreadystatechange",function(){"complete"===a.readyState&&t.readyCallback()})),(r=t.source||{}).concatemoji?d(r.concatemoji):r.wpemoji&&r.twemoji&&(d(r.twemoji),d(r.wpemoji)))}(window,document,window._wpemojiSettings); The oxidation number of CN= -1, so 6*CN has an oxidation number of -6 so therefore. Fe(ClO2)3 . }); 30 seconds . Oxidation involves an increase in oxidation state. The oxidation number of simple ions is equal to the charge on the ion. Oxidation number of O in its stable elemental form, O2: Oxidation numbers DRAFT. Lionesses Kill Lion, Lionesses Kill Lion, However, the role of organic ligands in microbial oxidation of solid-phase Fe(II) is not well understood. 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1. ## Solve the radical equation

Hi all,

I need help/directions on how to solve this problem:

Thank you much.

2. Just to clarify, do you mean $\sqrt3x-2$ or $\sqrt{3x}-2$ or $\sqrt{3x-2}$?

3. Oh, sorry, the third one.

The square root of 3x-2

4. Ok. Here's my attempt:

$x-\sqrt{3x-2}=4$
$x-4=\sqrt{3x-2}$
$(x-4)^2=\sqrt{3x-2}^2$
$x^2-8x+16=3x-2$
$x^2-11x+18=0$
$(x-9)(x-2)=0$
$x=2$ or $x=9$

x=2 is a solution for the negative result of , because when I square root a number, the result could be positive or negative. $(-2)^2=2^2$ so square rooting also gives two possible solutions. The reason I bring this up is that otherwise, when checking it does not seem to work.

5. Sorry about the latex errors, I've corrected them now.


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Journal cover Journal topic
Atmospheric Measurement Techniques An interactive open-access journal of the European Geosciences Union
Journal topic
Atmos. Meas. Tech., 11, 6163-6168, 2018
https://doi.org/10.5194/amt-11-6163-2018
Atmos. Meas. Tech., 11, 6163-6168, 2018
https://doi.org/10.5194/amt-11-6163-2018

Research article 13 Nov 2018

Research article | 13 Nov 2018

# Initial report on polar mesospheric cloud observations by Himawari-8

PMC observation by Himawari-8
Takuo T. Tsuda1, Yuta Hozumi1, Kento Kawaura1, Keisuke Hosokawa1, Hidehiko Suzuki2, and Takuji Nakamura3,4 Takuo T. Tsuda et al.
• 1Department of Computer and Network Engineering, University of Electro-Communications, Chofu, Japan
• 2Department of Physics, Meiji University, Kawasaki, Japan
• 3National Institute of Polar Research (NIPR), Tachikawa, Japan
• 4Department of Polar Science, Graduate University for Advanced Studies (SOKENDAI), Tachikawa, Japan
Abstract

We provide an initial report on polar mesospheric cloud (PMC) observations by the Japanese Geostationary Earth Orbit (GEO) meteorological satellite Himawari-8. Heights of the observed PMCs were estimated to be 80–82 km. Observed PMCs were active only during summertime in both the northern and southern polar regions. These observations are consistent with known PMC behavior. From its almost fixed location relative to the Earth, Himawari-8 is capable of continuously monitoring PMC every 10 min with three visible bands: blue (0.47 µm), green (0.51 µm), and red (0.64 µm). Thus, Himawari-8 can contribute to PMC research in the near future.

1 Introduction

Polar mesospheric clouds (PMCs) or noctilucent clouds (NLCs) consist of water-ice particles, which can be produced in summer at the mesopause region, mainly at high latitudes. The first report on PMCs was made by . Since then, various methods have been used to perform PMC observations. Optical observations by ground-based cameras, imagers, or lidars are often limited by weather conditions because a clear sky is required for such observations. Hence, satellite observations from space are valuable for more continuous observations, which enable significant systematic data coverage. Such systematic data coverage would be of benefit, for example, for monitoring long-term PMC activity, which may be related to global change (Thomas1996; von Zahn2003) because water-ice particle production can be enhanced by CO2 cooling and H2O increase, which may be induced by CO2 and CH4 increases (Roble and Dickinson1989).

A comprehensive review of PMC observations from satellites can be found in . In addition, the Aeronomy of Ice in the Mesosphere (AIM) satellite has been in operation, making PMC observations since 2007 . These observations include both limb- and nadir-viewing from low-Earth-orbit (LEO) satellites. By contrast, there are only few reports of PMC observations by limb-viewing Geostationary Earth Orbit (GEO) satellites (Gadsden2000a, b, 2001; Proud2015). The first PMC observations from a GEO satellite were reported using images by Meteosat First Generation (MFG) (Gadsden2000a, b, 2001), and their PMC images had ∼2.5km spatial resolution in a single visible band. Subsequently, extended such observations to Meteosat Second Generation (MSG), and he reported PMC observations using ∼1km spatial resolution images in a single visible band. This kind of GEO satellite can produce full-disk images including the Earth's limb, which would provide valuable opportunities for PMC observations by continuous limb-viewing from its almost fixed location relative to the Earth.

Figure 1(a) An original true-color image (composite of the three visible bands) around northern high latitudes at 21:00 UT on 9 July 2016. (b) Same as (a), but the color scale is 50× enhanced. (c) Same as (b), but latitudes and heights of the tangential points are overlaid.

In the present paper, we make an initial report on PMC observations from Himawari-8, the Japanese GEO meteorological satellite. Our PMC images from Himawari-8 have ∼1km spatial resolution in three visible bands. Japanese GEO meteorological satellites have a long history from 1977 (Himawari-1) to the present (Himawari-8). However, there was no PMC report from Japanese GEO satellite observations before this work. Therefore, in the present paper we examine basic features in PMC emissions observed by Himawari-8 and compare these with typical PMC characteristics, as a first step for our PMC research using Himawari-8 data.

Himawari-8 is the Japanese GEO meteorological satellite , which was successfully launched in October 2014. It has 16 observation bands, including three visible bands: blue (0.47 µm), green (0.51 µm), and red (0.64 µm). In the initial survey for PMCs, we used full-disk images in Portable Network Graphics (PNG) format, generated from the level-1a data, Himawari Standard Data (HSD). The PNG full-disk image is a true-color image, i.e., a composite of the three visible bands. Each color has an 8 bit resolution (i.e., values ranging from 0 to 255), describing emission intensities from 0 to 641.5092 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{sr}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\mathrm{µ}{\mathrm{m}}^{-\mathrm{1}}$ for the blue band, from 0 to 601.9766 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{sr}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\mathrm{µ}{\mathrm{m}}^{-\mathrm{1}}$ for the green band, and from 0 to 519.3457 $\mathrm{W}\phantom{\rule{0.125em}{0ex}}{\mathrm{m}}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}{\mathrm{sr}}^{-\mathrm{1}}\phantom{\rule{0.125em}{0ex}}\mathrm{µ}{\mathrm{m}}^{-\mathrm{1}}$ for the red band. The color value has a linear relation with the emission intensity for each band. The PNG full-disk image has a spatial resolution of ∼1km and is obtained every 10 min. The geometric accuracy of the images is typically less than 0.6 km, i.e., less than the ∼1km spatial resolution. More detailed information for the PNG images can be found in . For the present survey, we collected a year of PNG images for 2016, and focused our attention on the Earth's limb region, namely the middle and upper atmospheric regions.

Figure 2(a) Height–latitude distribution of emission intensity in the blue band at 21:00 UT on 9 July 2016. (b) Same as (a), but in the green band. (c) Same as (a), but in the red band.

3 Results and discussion

Figure 1 shows an example of PMC emissions observed at 21:00 UT on 9 July 2016 in the northern high latitudes. It is difficult to see any emissions in the limb region in the original true-color image (see Fig. 1a), but the 50× enhanced image makes it obvious that clear emissions exist in the limb region (see Fig. 1b). The appearance of these emissions is similar to that of PMC emissions in previous reports (Gadsden2000a, b, 2001; Proud2015). Here, we calculated tangential points in each line-of-sight (LOS) direction (i.e., for each pixel in the image). The image is described by the normalized geostationary projection, so the pixel corresponds to the LOS angle. From each LOS angle, we derived each vector along each LOS direction. Then, we considered intersections between each vector and the Earth-like ellipsoid of eccentricity defined by the World Geodetic System 1984 (WGS84). The equation for the intersections is expressed in quadratic form. Hence, if there is only a single solution of the equation, there is only a single intersection that corresponds to the tangential point. Thus, we solved the equation by changing the radius of the ellipsoid to produce only a single solution. Thus, we obtained information for the tangential points for each pixel in the image. The heights and latitudes of tangential points are overlaid in Fig. 1c. As shown in Fig. 1c, the typical height of the emissions was about 80 km.

For further details, Fig. 2 shows height–latitude distributions of emission intensities in the three visible bands. The emissions were mainly located at 80–82 km height at latitudes of 78–81 N. This emission height is consistent with typical reported PMC heights of 82–83 km (DeLand et al.2006). It should be noted that the tangential height may be an underestimation of the actual emission height because emissions may be coming from not only the tangential point but also foreground or background points. Our observations also show oblique, step-like structures with a spatial scale of ∼1km. These artifactual structures are because of the height resolution and the mapping calculation of the tangential point. Before application of the mapping calculation, images show step-like structures that are not oblique but horizontal, with a spatial scale of ∼1km because of ∼1km height or spatial resolution. These artifactual structures are because of a limitation of the height resolution. Such horizontal, step-like structures then become oblique, step-like structures through the mapping calculation from the pixel coordinate in the original images to the height–latitude coordinate in the tangential points.

Figure 3(a) Arctic map showing footprints (red line) of the tangential points at tangential heights of 80–85 km. (b) Same as (a), but for an Antarctic map. It should be noted that the center longitude is the sub-satellite longitude of Himawari-8: 140.7 E.

We confirm that the PMC-like emission layer has a wavelike structure in the height–latitude cross section. For example, the heights of the layer were 80–81 km at ∼81 N, 81–82 km at ∼80 N, ∼81km at ∼79 N, and 81–82 km at ∼78 N. We attribute these fluctuations to atmospheric waves. Because the latitude range of 79–81 N corresponds to about a distance of ∼700km over the Arctic Ocean (see Fig. 3), the wavelength of the wavelike structure can be estimated to be ∼700km. Such PMC structures can be observed by several methods using observations from LEO satellites such as the Cloud Imaging and Particle Size experiment (CIPS) onboard AIM and the Optical Spectrograph and InfraRed Imager System (OSIRIS) onboard Odin. For example, AIM/CIPS can provide PMC nadir imaging, which is a powerful tool to observe horizontal information for PMC structures (Chandran et al.2010; Yue et al.2014; Zhao et al.2015). In addition, tomographic techniques using AIM/CIPS and Odin/OSIRIS can provide horizontal and vertical information for PMC structures. By contrast, Himawari-8 observation features high cadence and wide limb-viewing PMC observation from its almost fixed location relative to the Earth. The Himawari-8 full-disk image can be obtained every 10 min, and its field-of-view (FOV) coverage can be several thousands of kilometers (see Fig. 3). These features will provide valuable data, which can be complementary to PMC data from the LEO satellites.

Figure 4(a) Yearly variation in the total emission intensities for a region at heights of 70–90 km and latitudes of 60–90 N in 2016. Blue, green, and red lines correspond to the blue, green, and red visible bands, respectively. It should be noted that the upper limit of the latitude range is actually ∼81, that is, the highest latitude of the tangential points when the height range is set to 70–90 km. Vertical black lines indicate the northern summer solstice in 2016 (i.e., 20 June 2016), 20 days before the solstice (i.e., 31 May 2016), and 60 days after the solstice (i.e., 19 August 2016), in reference to the typical PMC period. (b) Same as (a), but for a region at heights of 70–90 km and latitudes of 60–90 S. Vertical black lines indicate the southern summer solstice in 2016 (i.e., 21 December 2016), 20 days before the solstice (i.e., 1 December 2016), and 60 days after the 2015 southern summer solstice (i.e., 20 February 2016), in reference to the typical PMC period.

Emission intensity was the strongest in the blue band, and the weakest in the red band. This result can be explained by Rayleigh or Mie scattering of sunlight by water-ice particles (i.e., PMCs). According to the Mie theory (Bohren and Huffman2007), Rayleigh scattering is Mie scattering in the limit where particle size is much smaller than the wavelength. In the Rayleigh scattering region, the scattering cross section at a fixed wavelength decreases with decreasing particle size, and the scattering cross section at a fixed particle size decreases with increasing wavelength. Most PMC particles are understood to have radii of 20–60 nm, along with a very small number of ∼200nm particles (DeLand et al.2006). Such particle size is smaller than the visible observation wavelengths: blue (0.47 µm), green (0.51 µm), and red (0.64 µm). Hence, the scattering would be close to Rayleigh scattering, though it may not be pure Rayleigh scattering. Therefore, the predominance of such small particles (20–60 nm) would account for the higher scattering intensities or stronger emission intensities at shorter wavelengths. Although additional quantitative evaluation would be helpful, the inferential framework described above would imply that the three visible images obtained by Himawari-8 may contain information on the size of PMC particles. In addition, Himawari-8 observation can cover an entire day of local time (i.e., 00:00–24:00 JST), which means that emission data at almost all scattering angles (i.e., 0–360) are available. Such information may be useful for particle size investigation, considering the wavelength dependence and the angle distribution in the Rayleigh to Mie scattering region. However, this concern is beyond the scope of the present paper.

To investigate seasonal variation of PMC emissions in 2016, we calculated total pixel values in specific regions (i.e., sum of pixel values in localized pixels). We set two regions; one is a region at heights of 70–90 km and latitudes of 60–90 N for the northern polar region (see Fig. 4a), whereas the other is a region at heights of 70–90 km and latitudes of 60–90 S for the southern polar region (see Fig. 4b). The height range of 70–90 km covers typical PMC occurrence heights, and the latitude range of 60–90 covers typical PMC occurrence latitudes. It should be noted that the upper limit of the latitude range is actually ∼81 (see Fig. 3), that is, the highest latitude of the tangential points when the height range is set to 70–90 km. Seasonal variation in the total pixel values is shown in Fig. 4. To remove within-day variation, data in Fig. 4 include only data at a single local time each day, 06:00 JST (21:00 UT). At that local time, 06:00 JST, the LOS of Himawari-8 is almost perpendicular to the sunward direction. This configuration would be beneficial, providing solar illumination to some extent while minimizing sun-induced noise, such as the stray light of direct sunlight, which can be a problem close to local midnight (00:00 JST) (Proud2015). As shown in Fig. 4a and b, PMC-like emissions, expressed as total pixel values, were active only during local summer months. The observed active periods would be similar to the typical PMC active period, from ∼20 days before summer solstice to ∼60 days after summer solstice (DeLand et al.2006).

As discussed in the above text, the heights and seasonal variations of the PMC emissions are consistent with the general characteristics of PMCs. These results suggest that the peculiar emissions observed by Himawari-8 are indeed PMCs. We further suggest that the availability of imaging in three visible bands constitutes a particular advantage of Himawari-8 for PMC study. This capability may provide valuable opportunities, for example, for obtaining information on the size of PMC particles. In addition, collaborations between Himawari-8 and LEO satellites such as AIM would allow a synergy of complementary capabilities. In particular, high time resolution data (imaging every 10 min) from Himawari-8, when combined with data from LEO satellites, can contribute to PMC research in the near future, e.g., diurnal PMC variation.

4 Summary

In this paper, we introduced new PMC observations by Himawari-8. These observations concerning PMC height and seasonal PMC activity are consistent with previously reported PMC characteristics (DeLand et al.2006). Although these are not new scientific findings, they serve as a brief but essential validation of Himawari-8 observational data as PMC emissions. The detection of emissions consistent with PMC emissions demonstrates that Himawari-8 is capable of PMC observations. Himawari-8 can indeed provide data on new aspects of PMCs, and it can provide valuable opportunities for new PMC research in the near future. In particular, Himawari-8 PMC observations would have the following advantages: (1) high time resolution (imaging every 10 min); (2) high spatial resolution (one pixel every ∼1km) in the height–latitude (or height–horizon) cross section; (3) three visible bands (blue: 0.47 µm, green: 0.51 µm, and red: 0.64 µm); (4) continuous monitoring from its almost fixed location relative to the Earth.

Data availability
Data availability.

The Himawari-8 data, provided by the Meteorological Satellite Center of the Japan Meteorological Agency (JMA), are available from the National Institute of Information and Communications Technology (NICT) Science Cloud (http://sc-web.nict.go.jp/himawari/, last access: 12 November 2018).

Author contributions
Author contributions.

TTT conducted the data analysis and wrote the first draft of the manuscript. YH and KK contributed the data analysis. All authors contributed toward revising and improving the manuscript. All authors read and approved the final paper.

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Special issue statement
Special issue statement.

This article is part of the special issue “Layered phenomena in the mesopause region (ACP/AMT inter-journal SI)”. It is a result of the LPMR workshop 2017 (LPMR-2017), Kühlungsborn, Germany, 18–22 September 2017.

Acknowledgements
Acknowledgements.

The authors are grateful to Ryota Yamamoto for providing information on PMC emissions in full-disk images from Himawari-8; this information enabled this work. Himawari-8 data were provided by the Meteorological Satellite Center of the Japan Meteorological Agency (JMA) through the National Institute of Information and Communications Technology (NICT) Science Cloud. This work was supported in part by MEXT/JSPS KAKENHI grants, JP15H05815, JP16H01171, JP16H06021, and JP17H02968, by the National Institute of Polar Research (NIPR) through the General Collaboration Project no. 28-2, and by the joint research program of the Institute for Space-Earth Environmental Research (ISEE), Nagoya University. Edited by: Robert Hibbins Reviewed by: Yasuhiro Murayama and two anonymous referees

References

Bessho, K., Date, K., Hayashi, M., Ikeda, A., Imai, T., Inoue, H., Kumagai, Y., Miyakawa, T., Murata, H., Ohno, T., Okuyama, A., Oyama, R., Sasaki, Y., Shimazu, Y., Shimoji, K., and Sumida, Y.: An introduction to Himawari-8/9 – Japan's new-generation geostationary meteorological satellites, J. Meteorol. Soc. Jpn., 94, 151–183, https://doi.org/10.2151/jmsj.2016-009, 2016. a, b

Bohren, C. F. and Huffman, D. R.: Absorption and scattering of light by small particles, WILEY-VCH Verlag GmbH & Co. KGaA, Germany, https://doi.org/10.1002/9783527618156, 2007. a

Chandran, A., Rusch, D. W., Merkel, A. W., Palo, S. E., Thomas, G. E., Taylor, M. J., Bailey, S. M., and Russell III, J. M.: Polar mesospheric cloud structures observed from the cloud imaging and particle size experiment on the Aeronomy of Ice in the Mesosphere spacecraft: Atmospheric gravity waves as drivers for longitudinal variability in polar mesospheric cloud occurrence, J. Geophys. Res., 115, D13102, https://doi.org/10.1029/2009JD013185, 2010. a

DeLand, M. T., Shettle, E. P., Thomas, G. E., and Olivero, J. J.: A quarter-century of satellite polar mesospheric cloud observations, J. Atmos. Sol.-Terr. Phy., 68, 9–29, https://doi.org/10.1016/j.jastp.2005.08.003, 2006. a, b, c, d, e

Gadsden, M.: Polar mesospheric clouds seen from geostationary orbit, J. Atmos. Sol. Terr. Phy., 62, 31–36, https://doi.org/10.1016/S1364-6826(99)00099-1, 2000a. a, b, c

Gadsden, M.: Structure in polar mesospheric clouds seen from a geostationary spacecraft, Geophys. Res. Lett., 27, 3671–3673, https://doi.org/10.1029/1999GL003682, 2000b. a, b, c

Gadsden, M.: Structure of polar mesospheric clouds seen from a geostationary satellite, Ads. Space Res., 27, 1697–1702, https://doi.org/10.1016/S0273-1177(01)00237-X, 2001. a, b, c

Hart, V. P., Taylor, M. J., Doyle, T. E., Zhao, Y., Pautet, P.-D., Carruth, B. L., Rusch, D. W., and Russell III, J. M.: Investigating Gravity Waves in Polar Mesospheric Clouds Using Tomographic Reconstructions of AIM Satellite Imagery, J. Geophys. Res.-Space Phys., 123, 955–973, https://doi.org/10.1002/2017JA024481, 2018. a

Hultgren, K., Gumbel, J., Degenstein, D., Bourassa, A., Lloyd, N., and Stegman, J.: First simultaneous retrievals of horizontal and vertical structures of Polar Mesospheric Clouds from Odin/OSIRIS tomography, J. Atmos. Sol. Terr. Phy., 104, 213–223, https://doi.org/10.1016/j.jastp.2013.06.013, 2013. a

Leslie, R. C.: Sky glows, Nature, 33, 245, https://doi.org/10.1038/032245a0, 1885. a

Proud, S.: Observation of polar mesospheric clouds by geostationary satellite sensors, IEEE Geosci. Remote S., 12, 1332–1336, https://doi.org/10.1109/LGRS.2015.2399532, 2015. a, b, c, d

Roble, R. G. and Dickinson, R. E.: How will changes in carbon dioxide and methane modify the mean structure of the mesosphere and thermosphere?, Geophys. Res. Lett., 16, 1441–1444, https://doi.org/10.1029/GL016i012p01441, 1989.  a

Russell, J. M., Bailey, S. M., Gordley, L. L., Rusch, D. W., Horányi, M., Hervig, M. E., Thomas, G. E., Randall, C. E., Siskind, D. E., Stevens, M. H., Summers, M. E., Taylor, M. J., Englert, C. R., Espy, P. J., McClintock, W. E., and Merkel, A. W.: The Aeronomy of Ice in the Mesosphere (AIM) mission: Overview and early science results, J. Atmos. Sol. Terr. Phy., 71, 289–299, https://doi.org/10.1016/j.jastp.2008.08.011, 2009. a

Thomas, G. E.: Is the polar mesosphere the miner's canary of global change?, Adv. Space Res., 18, 149–158, https://doi.org/10.1016/0273-1177(95)00855-9, 1996. a

von Zahn, U.: Are noctilucent clouds a “Miner's Canary” for global change?, Eos Trans. AGU, 84, 261–264, https://doi.org/10.1029/2003EO280001, 2003. a

Yue, J., Thurairajah, B., Hoffmann, L., Alexander, J., Chandran, A., Taylor, M. J., Russell III, J. M., Randall, C. E., and Bailey, S. M.: Concentric gravity waves in polar mesospheric clouds from the Cloud Imaging and Particle Size experiment, J. Geophys. Res.-Space Phys., 119, 5115–5127, https://doi.org/10.1002/2013JD021385, 2014. a

Zhao, Y., Taylor, M. J., Randall, C. E., Lumpe, J. D., Siskind, D. E., Bailey, S. M., and Russell III, J. M.: Investigating seasonal gravity wave activity in the summer polar mesosphere, J. Atmos. Sol. Terr. Phy., 127, 8–20, https://doi.org/10.1016/j.jastp.2015.03.008, 2015. a

Special issue


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What are the $x$- and $y$-components of vector $\vec { T } = 7\hat { j } -4\hat { i }$?

A

The $x$-component is 7 and the $y$-component is 4.

B

The $x$-component is 4 and the $y$-component is 7.

C

The $x$-component is 7 and the $y$-component is -4.

D

The $x$-component is -4 and the $y$-component is 7.

Select an assignment template


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Standard Deviation of the given sets

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Standard Deviation of the given sets [#permalink]  21 Jun 2016, 22:48
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Question Stats:

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Quantity A
The standard deviation of the set 10,20,30

Quantity B
The standard deviation of the set 10,20,20,20,20,20,30

Can someone please explain me how the standard deviation can be determined in this case.
Both sets have same mean, same range, I just don't get the concept of more spread out.
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Re: Standard Deviation of the given sets [#permalink]  22 Jun 2016, 01:33
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Definition

Lets understand the formula of Standard Deviation first (σ):
Variance = (SD)$$^2$$ =σ$$^2$$ = [∑$$( Xi - X )^2$$ / N ]

Variance is the average of the squared differences from the mean, which basically a measure of how spread out numbers are.

So lets take the first case:

Case 1

i = 3

Numbers = 10, 20 , 30.

Xm = $$\frac{10 + 20 + 30}{3}$$ = 20

Variance = $$\frac{(10-20)^2 + (20-20)^2 + (30-20)^2}{3}$$ = 66.667

SD = σ = $$\sqrt{Variance}$$ = 8.16

Case 2

i = 7

Numbers = 10, 20 , 20, 20 ,20 ,20 , 30.

Xm = 20 (Same as the previous one)

Variance = $$\frac{(10-20)^2 + (20-20)^2 + (20-20)^2 +(20-20)^2 +(20-20)^2 +(20-20)^2 +(30-20)^2}{7}$$ = 28.57

SD = σ = $$\sqrt{Variance}$$ = 5.34

So quantity A is greater.
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Re: Standard Deviation of the given sets [#permalink]  22 Jun 2016, 02:06
Thanm you for the response.
Isn't there a GRE way to solve this type of problems??

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Re: Standard Deviation of the given sets [#permalink]  22 Jun 2016, 04:03
Expert's post
If by GRE way you mean quick intuition, then yes. Standard deviation is simply a measure of spread of the population. In the two cases you stated above you can see in the second case many values are basically a repetition of the mean value which suggests that it will have a way lower SD. So someone who knows the concept of SD will take 5-10 secs to answer this question.
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Re: Standard Deviation of the given sets [#permalink]  22 Jun 2016, 04:35
So if I'm getting you right,
if we have

Set A : 10,20,30
Set B : 10,12,13,20,26,29,30

The SD of Set B will be greater then SD of Set A.
Please correct me if I'm wrong.
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Re: Standard Deviation of the given sets [#permalink]  22 Jun 2016, 05:28
Expert's post
Nope. No conclusion can be derived from the above sets because in this case you have to do all the calculation to get it right, there is no place of applying any intuition and this type of question has very small chance of appearing in actual GRE exam. However, the first question you posted looks like a good GRE question (albeit an easy one).
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Re: Standard Deviation of the given sets [#permalink]  22 Jun 2016, 06:30
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I agree totally with soumya regarding the example just poste by the student.

In this latter scenario you can not say which SD is higher unless you perform calculation

However, regarding the main topic, conceptually you can achive the solution without any calculation.

We do have set A 10,20,30

Set B 10,20,20,20,20,20,30

Now if you consider this simple concept

Quote:
If every element in the data set is equal, they all equal the mean, each deviation from the mean is zero, and the standard deviation is zero. This is the lowest possible standard deviation for any set to have.

From this you can infer that the SD of the first set is a little bit higher of the second one because it has LESS 20' in there. Considering that in both sets 10 and 30 are equal, because just present the gist of the problem boils down to the presence of the 20'. In the second set we have MORE 20'. As such, the SD is more "diluted", thinner.

The first set has a SD higher. For this reason A is the answer.

Hope this helps
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Re: Standard Deviation of the given sets [#permalink]  22 Jun 2016, 19:54
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afu2cool wrote:
So if I'm getting you right,
if we have

Set A : 10,20,30
Set B : 10,12,13,20,26,29,30

The SD of Set B will be greater then SD of Set A.
Please correct me if I'm wrong.

There is no shortcut to look at a set and get its SD. You have to use the formula.

But if two sets have the same mean and range as you example is then, you can look at how many values are nearer to mean in each set, more the numbers nearer indicates smaller SD.
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Re: Standard Deviation of the given sets [#permalink]  24 Jun 2016, 11:06
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For the purposes of the GRE, it's sufficient to think of Standard Deviation as the Average Distance from the Mean. Here's what I mean:

Consider these two sets: Set A {7,9,10,14} and set B {1,8,13,18}
The mean of set A = 10 and the mean of set B = 10
How do the Standard Deviations compare? Well, since the numbers in set B deviate the more from the mean than do the numbers in set A, we can see that the standard deviation of set B must be greater than the standard deviation of set A.

Alternatively, let's examine the Average Distance from the Mean for each set.

Set A {7,9,10,14}
Mean = 10
7 is a distance of 3 from the mean of 10
9 is a distance of 1 from the mean of 10
10 is a distance of 0 from the mean of 10
14 is a distance of 4 from the mean of 10
So, the average distance from the mean = (3+1+0+4)/4 = 2

B {1,8,13,18}
Mean = 10
1 is a distance of 9 from the mean of 10
8 is a distance of 2 from the mean of 10
13 is a distance of 3 from the mean of 10
18 is a distance of 8 from the mean of 10
So, the average distance from the mean = (9+2+3+8)/4 = 5.5

IMPORTANT: I'm not saying that the Standard Deviation of set A equals 2, and I'm not saying that the Standard Deviation of set B equals 5.5 (They are reasonably close however).

What I am saying is that the average distance from the mean can help us see that the standard deviation of set B must be greater than the standard deviation of set A.
More importantly, the average distance from the mean is a useful way to think of standard deviation. This model is a convenient way to handle most standard deviation questions on the GRE.

More here:

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Re: Standard Deviation of the given sets [#permalink]  25 Jun 2016, 01:21
Expert's post
Great explanation @GreenlightTestprep
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Re: Standard Deviation of the given sets [#permalink]  21 Sep 2017, 22:19
What a way to throw light on this concept. U turned my light green about SD.
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Re: Standard Deviation of the given sets [#permalink]  21 Sep 2017, 23:22
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Pushkar96 wrote:
What a way to throw light on this concept. U turned my light green about SD.

see the video above by Brent from Greelighttestprep

Regards
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Re: Standard Deviation of the given sets   [#permalink] 21 Sep 2017, 23:22
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# Code review of forward invocation

Here's the situation. I'm writing a simple game and I had two main actors: GameController and GridView.

• GridView is a UIView subclass displaying a grid with which the user interacts. It defines its custom delegate protocol (GridViewDelegate) for handling callbacks such as gridView:didSelectTile and others.

• GameController is the game controller (duh!) which instantiate a GridView and set itself as a delegate, therefore implementing the GridViewDelegate protocol.

So far so good, then I decided to add HUD components on top of the grid (a score, a timer and other stuff). The most reasonable choice seemed to wrap such HUD components along with the grid into a new UIView subclass called GameBoard.

And here comes the design issue: I need the controller to talk to the GridView and I think there's two reasonable options here.

1. Expose a gridView property and do something like

[self.gameBoard.gridView doStuff];

2. Forward the invocation made to GameBoard directly to GridView overriding the forwardInvocation: method of GameBoard

The first options looks like the most convenient, but I cannot get myself into liking it, due to the Law of Demeter.

So I decided to go for the second approach and do something like

// The GameBoard serves a proxy between the GameContoller and the GridView
- (void)forwardInvocation:(NSInvocation *)anInvocation {
if ([self.gridView respondsToSelector:anInvocation.selector]) {
[anInvocation invokeWithTarget:self.gridView];
} else {
[super forwardInvocation:anInvocation];
}
}

// This method is necessary for the above forwardInvocation to work
- (NSMethodSignature *)methodSignatureForSelector:(SEL)aSelector {
NSMethodSignature * signature = [super methodSignatureForSelector:aSelector];
if (!signature) {
signature = [self.gridView methodSignatureForSelector:aSelector];
}
return signature;
}

It works as expected, but I'd like to have a second opinion on my design choice.

I generally tend to avoid overusing of the dynamic features of Objective-C, but it looked like an elegant way to achieve my result fulfilling the Law of Demeter.

-

In this case I probably prefer to think about gridView as legitimate subview of GameBoard. I really doubt Demetra suffering when you doing something like [self.view.titleLabel sizeToFit].
So my advice is to use [self.gameBoard.gridView doStuff]; as simplest and straightforward solution.
By the way forwardInvocation will not be typechecked at all. Typesystem of Objective-C already too weak, don't make your life even harder. If you insist—I can advice to define @protocol for GameBoard with methods of GridView. –  Sёmka Novikov Jan 23 '13 at 4:09


text:
# Ordering Board Game Geek GeekLists by Rank

A quick script.

I play a lot of board games. With the whole COVID-19 mess, I’ve been playing a lot on Board Game Arena, which is a wonderful site. But one thing that’s a bit lacking is having ratings / metadata about games to great games I’ve just not heard about before. Where can you get lots of that data? Board Game Geek!

The problem though, is merging the two. So, how do we do it? Well, this time around, I’m going to start with this GeekList that someone else maintains of BGA games on BGG. Which has the games, but no ranks. And apparently there are no ways to rank those by BGG (for some reason). But that’s easy when you know a bit of scripting!

# A Generic Brute Force Backtracking Solver

One of the projects I’ve had vaguely in the back of my head is a sort of generic puzzle solver. I really love puzzles, but of the pencil and paper and video game varieties. So I think it would be awesome to write out a definition of a puzzle (say how to play Sudoku), give it input, and have it give me an answer back.

Well, I finally got around to trying it!

# Categorizing r/Fantasy Book Bingo Books

I’ve been working through the r/Fantasy 2021 Book Bingo this year:

# 2021 Book Bingo

 Attack on Titan, Vol. 1(Hard Mode)5 SFF Short Stories (Hard: An entire anthology or collection) The Poppy War(Hard Mode)Set in Asia (Hard: by an Asian author) The Changeling(Hard Mode)r/Fantasy A to Z Genre Guide (Hard: by a BIPOC author) The House in the Cerulean Sea(Hard Mode)Found Family (Hard: featuring an LGBTQ+ character) The Scorpio Races(Hard Mode)First person POV (Hard: Multiple) The Wyrmling Horder/Fantasy Book Club (Hard: with participation)Replaced with: Sequel: Not the First Book in the Series (2017) The Borrowers Afield(Hard Mode)New to you author (Hard: haven’t heard much about) Mexican Gothic(Hard Mode)Gothic Fantasy (Hard: not in the Book Riot article) Transmetropolitan, Vol. 1: Back on the Street(Hard Mode)Backlist book (Hard: published before 2000) Red Sister(Hard Mode)Revenge-seeking character (Hard: revenge as the major book plot) Six Wakes(Hard Mode)Mystery plot (Hard: not primary world urban fantasy) Wild Sign(Hard Mode)Comfort read (Hard: that isn’t a reread) Tales of Nezura: Book 1: The Zevolra(Hard Mode)Debut novel (Hard: published in 2021) Hellblazer, Vol. 1: Original Sins(Hard Mode)Cat squasher (500+ pages; Hard: 800+ pages) Daemon Voices(Hard Mode)SFF-related nonfiction (Hard: published in the last 5 years) Cece Rios and the Desert of Souls(Hard Mode)Latinx or Latin American author (Hard: with fewer than 1000 Goodreads ratings) Black Rain and Paper Cranes(Hard Mode)Self published (Hard: with fewer than 50 Goodreads ratings) Annihilation(Hard Mode)Forest setting (Hard: for the entire book) Gideon the Ninth(Hard Mode)Genre mashup (Hard: of three or more genres) The Midnight Library(Hard Mode)Has chapter titles of more than one word (Hard: for every chapter) An Alchemy of Masques and Mirrors(Hard Mode)___ of ___ (Hard: and ___) Project Hail Mary(Hard Mode)First contact (Hard: that doesn’t lead to war) Black SunTrans or Nonbinary (Hard: protagonist) The Long Way to a Small, Angry Planet(Hard Mode)Debut author (Hard: with an AMA) A Great and Terrible Beauty(Hard Mode)Witches (Hard: as the main protagonist)

One more fairly standard tech interview problem (for better or for worse, you’re likely to see one of these if you go for a programming job):

Given a linked list and an element x. Partition the list so that all elements less than x are before elements greater than or equal to x, but do not otherwise change the order of the elements.

# Dynamic Programming over a Matrix

Another LeetCode problem. Given an MxN matrix of numbers, find the longest path of strictly increasing numbers. So for example in this matrix: 994 668 211 You can start with the 1 in the bottom center, go left to the two, then up to the 6, and 9. That’s the longest path, so return a 4. In this 3x3 case, it’s really easy to just brute force. Calculate all possible paths.

# Phone Words--In English!

Okay, let’s take this one step further. Rather than generating just phone words, let’s actually generate phone words. Someone has provided a list of words in English as a package, so we’ll add a filter to add that to our comprehension: from english_words import english_words_set def letterCombinations(self, digits: str) -> List[str]: if not digits: return [] letters = { '1': '', '2': 'abc', '3': 'def', '4': 'ghi', '5': 'jkl', '6': 'mno', '7': 'pqrs', '8': 'tuv', '9': 'wxyz', '0': ' ', } return [ word for product in itertools.


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# (5r-3)(r-8)=0

## Simple and best practice solution for (5r-3)(r-8)=0 equation. Check how easy it is, and learn it for the future. Our solution is simple, and easy to understand, so dont hesitate to use it as a solution of your homework.

If it's not what You are looking for type in the equation solver your own equation and let us solve it.

## Solution for (5r-3)(r-8)=0 equation:

(5r-3)(r-8)=0
We multiply parentheses ..
(+5r^2-40r-3r+24)=0
We get rid of parentheses
5r^2-40r-3r+24=0
We add all the numbers together, and all the variables
5r^2-43r+24=0
a = 5; b = -43; c = +24;Δ = b2-4acΔ = -432-4·5·24Δ = 1369The delta value is higher than zero, so the equation has two solutionsWe use following formulas to calculate our solutions:$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}$$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}$$\sqrt{\Delta}=\sqrt{1369}=37$
$r_{1}=\frac{-b-\sqrt{\Delta}}{2a}=\frac{-(-43)-37}{2*5}=\frac{6}{10} =3/5$
$r_{2}=\frac{-b+\sqrt{\Delta}}{2a}=\frac{-(-43)+37}{2*5}=\frac{80}{10} =8$

`


text:
# Plot the mean torque values against the proportion of binder added (%v/w) using the data recorded... 1 answer below »

Question 1
Plot the mean torque values against the proportion of binder added (%v/w) using the data recorded in Table 1 of the laboratory worksheet.
Determine the maximum torque value (T¬max) and the amount of water corresponding to Tmax (WTmax) from the plotted curve.
Discuss on the different states of liquid saturation, i.e., pendular, funicular, capillary and droplet and their effect on the torque.

Attachments:


text:
# 9.2. Parsing#

You could code your own lexer and parser from scratch. But many languages include tools for automatically generating lexers and parsers from formal descriptions of the syntax of a language. The ancestors of many of those tools are lex and yacc, which generate lexers and parsers, respectively; lex and yacc were developed in the 1970s for C.

As part of the standard distribution, OCaml provides lexer and parser generators named ocamllex and ocamlyacc. There is a more modern parser generator named menhir available through opam; menhir is “90% compatible” with ocamlyacc and provides significantly improved support for debugging generated parsers.

## 9.2.1. Lexers#

Lexer generators such as lex and ocamllex are built on the theory of deterministic finite automata, which is typically covered in a discrete math or theory of computation course. Such automata accept regular languages, which can be described with regular expressions. So, the input to a lexer generator is a collection of regular expressions that describe the tokens of the language. The output is an automaton implemented in a high-level language, such as C (for lex) or OCaml (for ocamllex).

That automaton itself takes files (or strings) as input, and each character of the file becomes an input to the automaton. Eventually the automaton either recognizes the sequence of characters it has received as a valid token in the language, in which case the automaton produces an output of that token and resets itself to being recognizing the next token, or rejects the sequence of characters as an invalid token.

## 9.2.2. Parsers#

Parser generators such as yacc and menhir are similarly built on the theory of automata. But they use pushdown automata, which are like finite automata that also maintain a stack onto which they can push and pop symbols. The stack enables them to accept a bigger class of languages, which are known as context-free languages (CFLs). One of the big improvements of CFLs over regular languages is that CFLs can express the idea that delimiters must be balanced—for example, that every opening parenthesis must be balanced by a closing parenthesis.

Just as regular languages can be expressed with a special notation (regular expressions), so can CFLs. Context-free grammars are used to describe CFLs. A context-free grammar is a set of production rules that describe how one symbol can be replaced by other symbols. For example, the language of balanced parentheses, which includes strings such as (()) and ()() and (()()), but not strings such as ) or ((), is generated by these rules:

• $$S \rightarrow (S)$$

• $$S \rightarrow SS$$

• $$S \rightarrow \epsilon$$

The symbols occurring in those rules are $$S$$, $$($$, and $$)$$. The $$\epsilon$$ denotes the empty string. Every symbol is either a nonterminal or a terminal, depending on whether it is a token of the language being described. $$S$$ is a nonterminal in the example above, and ( and ) are terminals.

In the next section we’ll study Backus-Naur Form (BNF), which is a standard notation for context-free grammars. The input to a parser generator is typically a BNF description of the language’s syntax. The output of the parser generator is a program that recognizes the language of the grammar. As input, that program expects the output of the lexer. As output, the program produces a value of the AST type that represents the string that was accepted. The programs output by the parser generator and lexer generator are thus dependent upon on another and upon the AST type.

## 9.2.3. Backus-Naur Form#

The standard way to describe the syntax of a language is with a mathematical notation called Backus-Naur form (BNF), named for its inventors, John Backus and Peter Naur. There are many variants of BNF. Here, we won’t be too picky about adhering to one variant or another. Our goal is just to have a reasonably good notation for describing language syntax.

BNF uses a set of derivation rules to describe the syntax of a language. Let’s start with an example. Here’s the BNF description of a tiny language of expressions that include just the integers and addition:

e ::= i | e + e
i ::= <integers>

These rules say that an expression e is either an integer i, or two expressions with the symbol + appearing between them. The syntax of “integers” is left unspecified by these rules.

Each rule has the form

metavariable ::= symbols | ... | symbols

A metavariable is variable used in the BNF rules, rather than a variable in the language being described. The ::= and | that appear in the rules are metasyntax: BNF syntax used to describe the language’s syntax. Symbols are sequences that can include metavariables (such as i and e) as well as tokens of the language (such as +). Whitespace is not relevant in these rules.

Sometimes we might want to easily refer to individual occurrences of metavariables. We do that by appending some distinguishing mark to the metavariable(s). For example, we could rewrite the first rule above as

e ::= i | e1 + e2

or as

e ::= i | e + e'

Now we can talk about e2 or e' rather than having to say “the e on the right-hand side of +”.

If the language itself contains either of the tokens ::= or |—and OCaml does contain the latter—then writing BNF can become a little confusing. Some BNF notations attempt to deal with that by using additional delimiters to distinguish syntax from metasyntax. We will be more relaxed and assume that the reader can distinguish them.

## 9.2.4. Example: SimPL#

As a running example, we’ll use a very simple programming language that we call SimPL. Here is its syntax in BNF:

e ::= x | i | b | e1 bop e2
| if e1 then e2 else e3
| let x = e1 in e2

bop ::= + | * | <=

x ::= <identifiers>

i ::= <integers>

b ::= true | false

Obviously there’s a lot missing from this language, especially functions. But there’s enough in it for us to study the important concepts of interpreters without getting too distracted by lots of language features. Later, we will consider a larger fragment of OCaml.

We’re going to develop a complete interpreter for SimPL. You can download the finished interpreter here: simpl.zip. Or, just follow along as we build each piece of it.

### 9.2.4.1. The AST#

Since the AST is the most important data structure in an interpreter, let’s design it first. We’ll put this code in a file named ast.ml:

type bop =
| Mult
| Leq

type expr =
| Var of string
| Int of int
| Bool of bool
| Binop of bop * expr * expr
| Let of string * expr * expr
| If of expr * expr * expr

There is one constructor for each of the syntactic forms of expressions in the BNF. For the underlying primitive syntactic classes of identifiers, integers, and booleans, we’re using OCaml’s own string, int, and bool types.

Instead of defining the bop type and a single Binop constructor, we could have defined three separate constructors for the three binary operators:

type expr =
...
| Add of expr * expr
| Mult of expr * expr
| Leq of expr * expr
...

But by factoring out the bop type we will be able to avoid a lot of code duplication later in our implementation.

### 9.2.4.2. The Menhir Parser#

Let’s start with parsing, then return to lexing later. We’ll put all the Menhir code we write below in a file named parser.mly. The .mly extension indicates that this file is intended as input to Menhir. (The ‘y’ alludes to yacc.) This file contains the grammar definition for the language we want to parse. The syntax of grammar definitions is described by example below. Be warned that it’s maybe a little weird, but that’s because it’s based on tools (like yacc) that were developed quite awhile ago. Menhir will process that file and produce a file named parser.ml as output; it contains an OCaml program that parses the language. (There’s nothing special about the name parser here; it’s just descriptive.)

There are four parts to a grammar definition: header, declarations, rules, and trailer.

Header. The header appears between %{ and %}. It is code that will be copied literally into the generated parser.ml. Here we use it just to open the Ast module so that, later on in the grammar definition, we can write expressions like Int i instead of Ast.Int i. If we wanted we could also define some OCaml functions in the header.

%{
open Ast
%}

Declarations. The declarations section begins by saying what the lexical tokens of the language are. Here are the token declarations for SimPL:

%token <int> INT
%token <string> ID
%token TRUE
%token FALSE
%token LEQ
%token TIMES
%token PLUS
%token LPAREN
%token RPAREN
%token LET
%token EQUALS
%token IN
%token IF
%token THEN
%token ELSE
%token EOF

Each of these is just a descriptive name for the token. Nothing so far says that LPAREN really corresponds to (, for example. We’ll take care of that when we define the lexer.

The EOF token is a special end-of-file token that the lexer will return when it comes to the end of the character stream. At that point we know the complete program has been read.

The tokens that have a <type> annotation appearing in them are declaring that they will carry some additional data along with them. In the case of INT, that’s an OCaml int. In the case of ID, that’s an OCaml string.

After declaring the tokens, we have to provide some additional information about precedence and associativity. The following declarations say that PLUS is left associative, IN is not associative, and PLUS has higher precedence than IN (because PLUS appears on a line after IN).

%nonassoc IN
%nonassoc ELSE
%left LEQ
%left PLUS
%left TIMES

Because PLUS is left associative, 1 + 2 + 3 will parse as (1 + 2) + 3 and not as 1 + (2 + 3). Because PLUS has higher precedence than IN, the expression let x = 1 in x + 2 will parse as let x = 1 in (x + 2) and not as (let x = 1 in x) + 2. The other declarations have similar effects.

Getting the precedence and associativity declarations correct is one of the trickier parts of developing a grammar definition. It helps to develop the grammar definition incrementally, adding just a couple tokens (and their associated rules, discussed below) at a time to the language. Menhir will let you know when you’ve added a token (and rule) for which it is confused about what you intend the precedence and associativity should be. Then you can add declarations and test to make sure you’ve got them right.

After declaring associativity and precedence, we need to declare what the starting point is for parsing the language. The following declaration says to start with a rule (defined below) named prog. The declaration also says that parsing a prog will return an OCaml value of type Ast.expr.

%start <Ast.expr> prog

Finally, %% ends the declarations section.

%%

Rules. The rules section contains production rules that resemble BNF, although where in BNF we would write “::=” these rules simply write “:”. The format of a rule is

name:
| production1 { action1 }
| production2 { action2 }
| ...
;

The production is the sequence of symbols that the rule matches. A symbol is either a token or the name of another rule. The action is the OCaml value to return if a match occurs. Each production can bind the value carried by a symbol and use that value in its action. This is perhaps best understood by example, so let’s dive in.

The first rule, named prog, has just a single production. It says that a prog is an expr followed by EOF. The first part of the production, e=expr, says to match an expr and bind the resulting value to e. The action simply says to return that value e.

prog:
| e = expr; EOF { e }
;

The second and final rule, named expr, has productions for all the expressions in SimPL.

expr:
| i = INT { Int i }
| x = ID { Var x }
| TRUE { Bool true }
| FALSE { Bool false }
| e1 = expr; LEQ; e2 = expr { Binop (Leq, e1, e2) }
| e1 = expr; TIMES; e2 = expr { Binop (Mult, e1, e2) }
| e1 = expr; PLUS; e2 = expr { Binop (Add, e1, e2) }
| LET; x = ID; EQUALS; e1 = expr; IN; e2 = expr { Let (x, e1, e2) }
| IF; e1 = expr; THEN; e2 = expr; ELSE; e3 = expr { If (e1, e2, e3) }
| LPAREN; e=expr; RPAREN {e}
;

• The first production, i = INT, says to match an INT token, bind the resulting OCaml int value to i, and return AST node Int i.

• The second production, x = ID, says to match an ID token, bind the resulting OCaml string value to x, and return AST node Var x.

• The third and fourth productions match a TRUE or FALSE token and return the corresponding AST node.

• The fifth, sixth, and seventh productions handle binary operators. For example, e1 = expr; PLUS; e2 = expr says to match an expr followed by a PLUS token followed by another expr. The first expr is bound to e1 and the second to e2. The AST node returned is Binop (Add, e1, e2).

• The eighth production, LET; x = ID; EQUALS; e1 = expr; IN; e2 = expr, says to match a LET token followed by an ID token followed by an EQUALS token followed by an expr followed by an IN token followed by another expr. The string carried by the ID is bound to x, and the two expressions are bound to e1 and e2. The AST node returned is Let (x, e1, e2).

• The last production, LPAREN; e = expr; RPAREN says to match an LPAREN token followed by an expr followed by an RPAREN. The expression is bound to e and returned.

The final production might be surprising, because it was not included in the BNF we wrote for SimPL. That BNF was intended to describe the abstract syntax of the language, so it did not include the concrete details of how expressions can be grouped with parentheses. But the grammar definition we’ve been writing does have to describe the concrete syntax, including details like parentheses.

There can also be a trailer section after the rules, which like the header is OCaml code that is copied directly into the output parser.ml file.

### 9.2.4.3. The Ocamllex Lexer#

Now let’s see how the lexer generator is used. A lot of it will feel familiar from our discussion of the parser generator. We’ll put all the ocamllex code we write below in a file named lexer.mll. The .mll extension indicates that this file is intended as input to ocamllex. (The ‘l’ alludes to lexing.) This file contains the lexer definition for the language we want to lex. Menhir will process that file and produce a file named lexer.ml as output; it contains an OCaml program that lexes the language. (There’s nothing special about the name lexer here; it’s just descriptive.)

There are four parts to a lexer definition: header, identifiers, rules, and trailer.

Header. The header appears between { and }. It is code that will simply be copied literally into the generated lexer.ml.

{
open Parser
}

Here, we’ve opened the Parser module, which is the code in parser.ml that was produced by Menhir out of parser.mly. The reason we open it is so that we can use the token names declared in it, e.g., TRUE, LET, and INT, inside our lexer definition. Otherwise, we’d have to write Parser.TRUE, etc.

Identifiers. The next section of the lexer definition contains identifiers, which are named regular expressions. These will be used in the rules section, next.

Here are the identifiers we’ll use with SimPL:

let white = [' ' '\t']+
let digit = ['0'-'9']
let int = '-'? digit+
let letter = ['a'-'z' 'A'-'Z']
let id = letter+

The regular expressions above are for whitespace (spaces and tabs), digits (0 through 9), integers (nonempty sequences of digits, optionally preceded by a minus sign), letters (a through z, and A through Z), and SimPL variable names (nonempty sequences of letters) aka ids or “identifiers”—though we’re now using that word in two different senses.

FYI, these aren’t exactly the same as the OCaml definitions of integers and identifiers.

The identifiers section actually isn’t required; instead of writing white in the rules we could just directly write the regular expression for it. But the identifiers help make the lexer definition more self-documenting.

Rules. The rules section of a lexer definition is written in a notation that also resembles BNF. A rule has the form

rule name =
parse
| regexp1 { action1 }
| regexp2 { action2 }
| ...

Here, rule and parse are keywords. The lexer that is generated will attempt to match against regular expressions in the order they are listed. When a regular expression matches, the lexer produces the token specified by its action.

Here is the (only) rule for the SimPL lexer:

rule read =
parse
| white { read lexbuf }
| "true" { TRUE }
| "false" { FALSE }
| "<=" { LEQ }
| "*" { TIMES }
| "+" { PLUS }
| "(" { LPAREN }
| ")" { RPAREN }
| "let" { LET }
| "=" { EQUALS }
| "in" { IN }
| "if" { IF }
| "then" { THEN }
| "else" { ELSE }
| id { ID (Lexing.lexeme lexbuf) }
| int { INT (int_of_string (Lexing.lexeme lexbuf)) }
| eof { EOF }

Most of the regular expressions and actions are self-explanatory, but a couple are not:

• The first, white { read lexbuf }, means that if whitespace is matched, instead of returning a token the lexer should just call the read rule again and return whatever token results. In other words, whitespace will be skipped.

• The two for ids and ints use the expression Lexing.lexeme lexbuf. This calls a function lexeme defined in the Lexing module, and returns the string that matched the regular expression. For example, in the id rule, it would return the sequence of upper and lower case letters that form the variable name.

• The eof regular expression is a special one that matches the end of the file (or string) being lexed.

Note that it’s important that the id regular expression occur nearly last in the list. Otherwise, keywords like true and if would be lexed as variable names rather than the TRUE and IF tokens.

### 9.2.4.4. Generating the Parser and Lexer#

Now that we have completed parser and lexer definitions in parser.mly and lexer.mll, we can run Menhir and ocamllex to generate the parser and lexer from them. Let’s organize our code like this:

- <some root folder>
- dune-project
- src
- ast.ml
- dune
- lexer.mll
- parser.mly

In src/dune, write the following:

(library
(name interp))

(menhir
(modules parser))

(ocamllex lexer)

That organizes the entire src folder into a library named Interp. The parser and lexer will be modules Interp.Parser and Interp.Lexer in that library.

Run dune build to compile the code, thus generating the parser and lexer. If you want to see the generated code, look in _build/default/src/ for parser.ml and lexer.ml.

### 9.2.4.5. The Driver#

Finally, we can pull together the lexer and parser to transform a string into an AST. Put this code into a file named src/main.ml:

open Ast

let parse (s : string) : expr =
let lexbuf = Lexing.from_string s in
let ast = Parser.prog Lexer.read lexbuf in
ast

This function takes a string s and uses the standard library’s Lexing module to create a lexer buffer from it. Think of that buffer as the token stream. The function then lexes and parses the string into an AST, using Lexer.read and Parser.prog. The function Lexer.read corresponds to the rule named read in our lexer definition, and the function Parser.prog to the rule named prog in our parser definition.

Note how this code runs the lexer on a string; there is a corresponding function from_channel to read from a file.

We could now use parse interactively to parse some strings. Start utop and load the library declared in src with this command:

\$ dune utop src

Now Interp.Main.parse is available for use:

# Interp.Main.parse "let x = 3110 in x + x";;
- : Interp.Ast.expr =
Interp.Ast.Let ("x", Interp.Ast.Int 3110,
Interp.Ast.Binop (Interp.Ast.Add, Interp.Ast.Var "x", Interp.Ast.Var "x"))

That completes lexing and parsing for SimPL.


text:
# Cauchy sequence

In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. To be more precise by dropping a finite number of elements from the start of the sequence we can make the distance between any two remaining elements arbitrarily small.

Cauchy sequences require the notion of distance so they can only be defined in a metric space. Generalizations to more abstract uniform spaces exist in the form of Cauchy filter and Cauchy net.

They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to the definition of convergence.

 Contents

## Cauchy sequence in a metric space

### Formal definition

Formally, a Cauchy sequence is a sequence

[itex]x_1, x_2, x_3, \ldots [itex]

in a metric space (M, d) such that for every positive real number r > 0, there is an integer N such that for all integers m,n > N, the distance

[itex]d(x_m, x_n)[itex]

is less than r. Roughly speaking, the terms of the sequence are getting closer and closer together in a way that suggests that the sequence ought to have a limit in M. Nonetheless, this may not be the case.

### Completeness

A metric space X in which every Cauchy sequence has a limit (in X) is called complete.

#### Example: real numbers

The real numbers are complete, and the standard construction of the real numbers involves Cauchy sequences of rational numbers.

#### Counter-example: rational numbers

The rational numbers Q are not complete (for the usual distance): There are sequences of rationals that converge (in R) to irrational numbers; these are Cauchy sequences having no limit in Q.

For example:

• The sequence defined by x0 = 1, xn+1 = (xn + 2/xn)/2 consists of rational numbers (1, 3/2, 17/12,...), which is clear from the definition; it converges to the irrational square root of two, see Babylonian method of computing square root.
• The values of the exponential, sine and cosine functions, exp(x), sin(x), cos(x), are irrational for any rational value of x≠0, but are defined as limit of a rational sequence which is their Maclaurin series.

### Other properties

Every convergent sequence is a Cauchy sequence, and every Cauchy sequence is bounded. If [itex]f \colon M \rightarrow N[itex] is a uniformly continuous map between the metric spaces M and N and (xn) is a Cauchy sequence in M, then [itex](f(x_n))[itex] is a Cauchy sequence in N. If [itex](x_n)[itex] and [itex](y_n)[itex] are two Cauchy sequences in the rational, real or complex numbers, then the sum [itex](x_n + y_n)[itex] and the product [itex](x_n y_n)[itex] are also Cauchy sequences.

## Cauchy sequences in topological vector spaces

There is also a concept of Cauchy sequence for a topological vector space X: Pick a local base B for X about 0; then (xk) is a Cauchy sequence if for all members V of B, there is some number N such that whenever n,m > N, xn - xm is an element of V. If the topology of X is compatible with a translation-invariant metric d, the two definitions agree.

## Cauchy sequences in groups

There is also a concept of Cauchy sequence in a group G: Let H=(Hr) be a decreasing sequence of normal subgroups of G of finite index. Then a sequence (xn) in G is said to be Cauchy (w.r.t. H) iff for any r there is N such that ∀m,n > N, xn xm-1 ∈ Hr.

The set C of such Cauchy sequences forms a group (for the componentwise product), and the set C0 of null sequences (s.th. ∀r, ∃N, ∀n > N, xn∈Hr) is a normal subgroup of C. The factor group C/C0 is called the completion of G w.r.t. H.

One can then show that this completion is isomorphic to the inverse limit of the sequence (G/Hr).

If H is a cofinal sequence (i.e. any normal subgroup of finite index contains some Hr), then this completion is canonical in the sense that it is isomorphic to the inverse limit of (G/H)H, where H varies over all normal subgroups of finite index. For further details, see ch. I.10 in Lang's "Algebra".

## References

• Art and Cultures
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Space and Astronomy