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Q: Facebook & iOS Login Bug since January 2012 - Even with ShareKit or SDK Since the last week in January we've seen most of our iOS apps, and those commercially available on the app store have problems logging in and posting to Facebook. We've also seen this with apps using ShareKit to post to FB.
I've filed bugs with FB and got nowhere other than that people confirm they are seeing the same thing.
Today I did some NSLog's of the URL requests and redirect URLs where the problem is showing itself...
Could someone please try and help?
BTW - Please note - all of this was working perfectly before Jan 26th 2012. There has been no code changes on our side since then.
So we have an empty UIWebView and we make the login request
2012-02-22 12:32:28.870 iPad App[1640:15203] REQUEST = https://graph.facebook.com/oauth/authorize?client_id=135916249814649&redirect_uri=http://www.facebook.com/connect/login_success.html&scope=publish_stream,user_photos&type=user_agent&display=touch>
2012-02-22 12:32:32.531 iPad App[1640:15203] REQUEST = https://m.facebook.com/dialog/permissions.request?app_id=135916249814649&display=touch&next=http%3A%2F%2Fwww.facebook.com%2Fconnect%2Flogin_success.html&type=user_agent&perms=publish_stream%2Cuser_photos&fbconnect=1>
2012-02-22 12:32:35.671 iPad App[1640:15203] REQUEST = http://m.facebook.com/login.php?app_id=135916249814649&cancel=http%3A%2F%2Fwww.facebook.com%2Fconnect%2Flogin_success.html%3Ferror_reason%3Duser_denied%26error%3Daccess_denied%26error_description%3DThe%2Buser%2Bdenied%2Byour%2Brequest.&fbconnect=1&next=https%3A%2F%2Fm.facebook.com%2Fdialog%2Fpermissions.request%3F_path%3Dpermissions.request%26app_id%3D135916249814649%26redirect_uri%3Dhttp%253A%252F%252Fwww.facebook.com%252Fconnect%252Flogin_success.html%26display%3Dtouch%26type%3Duser_agent%26perms%3Dpublish_stream%252Cuser_photos%26fbconnect%3D1%26from_login%3D1%26client_id%3D135916249814649&rcount=1&_rdr>
[Switching to process 1640 thread 0x15203]
At this point the FB login page is shown inside the iOS UIWebView. After correctly entering in a valid FB email address and password, the following occurs.
2012-02-22 12:32:56.632 iPad App[1640:15203] REQUEST = https://m.facebook.com/login.php?m=m&next=https%3A%2F%2Fm.facebook.com%2Fdialog%2Fpermissions.request%3F_path%3Dpermissions.request%26app_id%3D135916249814649%26redirect_uri%3Dhttp%253A%252F%252Fwww.facebook.com%252Fconnect%252Flogin_success.html%26display%3Dtouch%26type%3Duser_agent%26perms%3Dpublish_stream%252Cuser_photos%26fbconnect%3D1%26from_login%3D1%26client_id%3D135916249814649&refsrc=http%3A%2F%2Fm.facebook.com%2Flogin.php&refid=9>
2012-02-22 12:32:59.547 iPad App[1640:15203] REQUEST = https://m.facebook.com/#!/dialog/permissions.request?_path=permissions.request&app_id=135916249814649&redirect_uri=http%3A%2F%2Fwww.facebook.com%2Fconnect%2Flogin_success.html&display=touch&type=user_agent&perms=publish_stream%2Cuser_photos&fbconnect=1&from_login=1&client_id=135916249814649&refid=9>
2012-02-22 12:33:02.034 iPad App[1640:15203] REQUEST = https://m.facebook.com/dialog/permissions.request?_path=permissions.request&app_id=135916249814649&redirect_uri=http%3A%2F%2Fwww.facebook.com%2Fconnect%2Flogin_success.html&display=touch&type=user_agent&perms=publish_stream%2Cuser_photos&fbconnect=1&from_login=1&client_id=135916249814649&refid=9>
And here's the error that the UIWebView returns. Note - all this happens BEFORE the UIWebView is dismissed and BEFORE we even sent the POST to FB url.
2012-02-22 12:33:02.035 iPad App[1640:15203] ERROR = Error Domain=NSURLErrorDomain Code=-999 "The operation couldn't be completed. (NSURLErrorDomain error -999.)" UserInfo=0x8a136d0 {NSErrorFailingURLKey=https://m.facebook.com/#!/dialog/permissions.request?_path=permissions.request&app_id=135916249814649&redirect_uri=http%3A%2F%2Fwww.facebook.com%2Fconnect%2Flogin_success.html&display=touch&type=user_agent&perms=publish_stream%2Cuser_photos&fbconnect=1&from_login=1&client_id=135916249814649&refid=9, NSErrorFailingURLStringKey=https://m.facebook.com/#!/dialog/permissions.request?_path=permissions.request&app_id=135916249814649&redirect_uri=http%3A%2F%2Fwww.facebook.com%2Fconnect%2Flogin_success.html&display=touch&type=user_agent&perms=publish_stream%2Cuser_photos&fbconnect=1&from_login=1&client_id=135916249814649&refid=9}
=========
So, other than "FB is broke..." which they don't seem to be doing anything about nor addressing this issue as it's been almost a month and we have shouting and complaining customers and users... what else could be going on?
Some apps (our own and 3rd party) that use share kit are working, and some not...
Any help would be tremendously appreciated.
Thanks
A: This has been fixed.
FB changed not only some URL's but changed the way the Client Side Authentication worked and didn't bother to tell anyone.
If you look at the Javascript code for Client Side Authentication and emulate this in iOS it works fine.
(for the moment.)
Gods I hate Facebook so much...
| {
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{"url":"https:\/\/plainmath.net\/7502\/how-to-prove-the-following-tan-2x-plus-plus-tanx-secx-equal-plus-sin-cos-2x","text":"# How to prove the following: tan^2x+1+tanx secx = 1+sin x\/ cos^2x\n\nTrigonometric equation and identitie\nHow to prove the following:\n$$\\displaystyle{{\\tan}^{{2}}{x}}+{1}+{\\tan{{x}}}{\\sec{{x}}}={1}+\\frac{{\\sin{{x}}}}{{{\\cos}^{{2}}{x}}}$$\n\n$$\\displaystyle{\\tan{{x}}}\\cdot{\\sec{{x}}}=\\frac{{\\sin{{x}}}}{{{\\cos}^{{2}}{x}}}{\\quad\\text{or}\\quad}{\\sin{{x}}}\\cdot{{\\sec}^{{2}}{x}}.{{\\tan}^{{2}}{x}}+{1}={{\\sec}^{{2}}{x}}.$$\nSo we have $$\\displaystyle{{\\sec}^{{2}}{x}}+{{\\sec}^{{2}}{x}}{\\sin{{x}}}={{\\sec}^{{2}}{x}}{\\left({1}+{\\sin{{x}}}\\right)}=\\frac{{{1}+{\\sin{{x}}}}}{{{\\cos}^{{2}}{x}}}.$$\n(To prove $$\\displaystyle{{\\sec}^{{2}}{x}}={{\\tan}^{{2}}{x}}+{1}$$, start with $$\\displaystyle{{\\sin}^{{2}}{x}}+{{\\cos}^{{2}}{x}}={1}$$, then divide through by $$\\displaystyle{{\\cos}^{{2}}{x}}:{{\\tan}^{{2}}{x}}+{1}={{\\sec}^{{2}}{x}}$$ because secx is $$\\displaystyle\\frac{{1}}{{\\cos{{x}}}}$$.)","date":"2021-07-24 14:06:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9731645584106445, \"perplexity\": 2804.425939704033}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046150266.65\/warc\/CC-MAIN-20210724125655-20210724155655-00221.warc.gz\"}"} | null | null |
The next generation S-class will grow from just being a sedan into filling the rolls of the Maybachs and of the CL coupe. This is the first time the replacement for the CL-Class has been photographed with a somewhat lighter camouflage. The model is to be known as the S?class Coupe, filling a niche much like the C? and E-class two-door models.
Front end styling looks slightly different to the S-class. The headlights look a bit surprised, like slimmer versions of the ones on the SL. Engine-wise, we should still see the twin-turbo V8 and V12 units seen on current Mercedes models. The Germans are also working on a twin-turbo 3-litre V6 for the C400 which could fit very well into the range. The new S-class could debut as early as April's Shanghai Auto Show, though this coupe should arrive towards the end of the year or even later. | {
"redpajama_set_name": "RedPajamaC4"
} | 6,410 |
This page in Danish? Please visit: Sixt Foursquare information in Danish.
If you are a Foursquare user you can check in at Sixt rent a car in Denmark. When you check in with Sixt you can unlock Foursquare specials from Sixt rent a car.
We will add new specials on a continous basis – only for Foursquare users. And don't forget to add Sixt in Denmark as a friend on Foursquare. Check out our Foursquare page: www.foursquare.com/sixtdk.
To claim an unlocked special at Sixt rent a car in Denmark, please show the Sixt staff your mobile screen, where you have unlocked the special along with your reservation number or alternatively make a new reservation with the staff.
All specials (including upgrades) are depending on availability at the participating rental station.
The normal car rental terms and conditions of Sixt are valid. See them here (PDF).
Specials are only valid in connection with a Sixt reservation.
Foursquare specials cannot be combined with other specials / promotions / campaigns / discounts etc. incl. corporate rates.
The special is only valid the day that you have unlocked the special and only at the venue, that you have unlocked the special at – unless otherwise is stated in the Foursquare special. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,600 |
\section{Introduction}
Research has been done studying the geometry of internal representations and its effect on classification in deep neural networks \cite{Cohen644658}. However, this was not studied in RNNs where recurrent dynamics also play a significant role in the task completion. Therefore, our experiments were chosen with two goals in mind: finding the geometric properties of internal class representations and evaluating the effects of the recurrent dynamics on the classification accuracy.
\section{Methodology and Analysis tools}
We trained a vanilla RNN to complete the well known sequential MNIST classification task, where each input to the network is a sequence of 28 lines (rows) each having 28 pixels. The RNN has a single recurrent layer of 200 tanh neurons and its only parameters are the recurrent and input weights (i.e., with no bias parameters to affect the recurrent dynamics). The output layer is linear and the network is trained using the Adam optimization algorithm with a cross-entropy loss function. After training the network for 30 epochs we achieve an accuracy of a little over 93\% on the test data. While this accuracy is far from the state-of-the-art, it is sufficient to ensure that our network is able to generalize the information provided by the training data to the test data and correctly classify the images in a majority of cases. The code was implemented in Python using PyTorch and all the experiments were conducted using the trained network and the test data, which the network did not see during training.
All experiments reported below are conducted on networks trained in the above manner.
\par
\vspace{-0.05in}
For data analysis, PCA is used to estimate the linear dimensionality of the internal representations of both the entire data as a whole and of the classes. This is done by counting the number of principal components necessary to explain at least 90\% of the variance in the data (or in a class). Also, t-SNE \cite{tsne} is used to create a two dimensional representation of the network's internal states. In this representation, points that are close in the original space will be mapped close together and distant point will be mapped far from one another. The aim is to see how early in the classification process did the network group together similar inputs. Ultimately, we want to see how early does the network "know", in a sense, that a certain input is of a specific class.
\begin{figure*}[ht]
\vskip 0.2in
\begin{center}
\centerline{\includegraphics[width=1.82\columnwidth]{figures_5.png}}
\end{center}
\vskip -0.2in
\end{figure*}
\section{Experiments}
\textbf{Experiment 1:} The first experiment consists of modifying the test images so that the last $n$ lines of each image are blank (value 0), this was done for $n$ ranging from 1 to 27. The part-blank images were then used as inputs for the classification task. This experiment aims to determine how the recurrent dynamics alone help the classification task.
\par
\vspace{-0.05in}
\textbf{Experiment 2:} The second experiment consists of giving the network only the first $n$ lines of the images and then stopping, in contrast to Experiment 1.
This allows to probe the importance of early internal representations by the last linear layer and to see whether or not the network relies on a fixed sequence length for classification.
\par
\vspace{-0.05in}
\textbf{Experiment 3:} In the third experiment, blank pixel rows are added at the end of the full input sequences, increasing their length. This is in order to see how the network's dynamics affect the internal representations after the network was provided with all the available information.
\par
\vspace{-0.05in}
\section{Discussion}
Figure 1a shows that the network's accuracy in experiments 1 and 2 behaves in very different manners. The network dynamics seem to have a significant role in the classification process even when the inputs are blank, as is the case in experiment 1. Despite being shown the same number of real pixel rows, the accuracy in experiment 1 is greatly superior to experiment 2 for all amounts of shown rows. This also indicates that the network's classification abilities are highly dependent on the sequence length even when the amount of relevant information in the sequence is exactly the same.
Figure 1b further emphasizes the importance of the sequence length for proper classification since the accuracy dramatically drops as soon as additional blank rows of pixels are added to the input sequences. The reason we chose to plot the accuracy of the classification for up to 500 added blank rows is to display the highly dynamic nature of the network which seems to begin an oscillatory trajectory as can be deduced from the recurring pattern in the accuracy.
Figure 3 shows that the tendency of neural networks to rely on dimensionality expansions followed by dimensionality reductions in order to perform their tasks which was discussed by \cite{DBLP:journals/corr/abs-1906-00443} and \cite{FUSI201666} seems to be maintained for recurrent neural networks.\par
Finally the t-SNE visualization is especially evocative when the internal representations are colored according to their real digit class. As early as time step 4 (Figure 4) the network seems to create classification relevant clusters in the representation space. In particular, it seems to "know" that certain points are 6es (rightmost cluster) or either 6es or 2s (middle cluster) and it separates them from the rest of the points (leftmost cluster). This internal separation increases in precision throughout the time steps as is shown in Figures 5 and 6, but the initial cluster of "6es" is maintained suggesting that the initial separation of this cluster from the other points was correct.
\section{Conclusion}
Our results show that the network's internal representation is evocative of the real data classification early in the input's sequence so the task is carried out as soon as relevant information is available. Despite the separability of the internal representations and the clusters formed in the representation space, the task relevant information is only interpretable by the output layer after a fixed sequence length. If the sequence length varies the network's dynamics affects the hidden states in a way that greatly hinders accuracy.\par
The retrieval of "early classifications", where the network doesn't wait for the whole input sequence, could be of great impact in time sensitive real-world applications in which a decision is needed as soon as possible. Further work is required to determine how this information is retrievable (see related work \cite{NIPS2017_7188} and \cite{linprobes}). Also, machine learning visualization algorithms show great promise in exposing the inner mechanisms of neural networks and could greatly help in the understanding of these "black-box" algorithms. A great number of vizualisation, dimensionality reduction and manifold learning techniques exist and their applicability in neural network analysis should be evaluated more thoroughly.
\newpage
\onecolumn
| {
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{"url":"https:\/\/www.vedantu.com\/maths\/convert-decimal-to-fraction","text":"# Convert Decimal to Fraction\n\nHow to Convert a Decimal into a Fraction\n\nDecimal and Fraction are both the representation for rational numbers yet the two are very different from each other. Fractions are expressed as a division of two numbers. It has two parts named numerator which is the upper number and the denominator which is the bottom number. Usually, the denominator is a larger number as it represents the total parts of a whole and the numerator is smaller than the denominator as it represents the number of parts we consider. The basic example would be a pizza. The total number of pieces that a pizza has will be the denominator and the number of pieces that you ate will be the numerator. Now consider that a pizza had 8 slices and you ate 4 of them then the fraction would be $\\frac{4}{8}$\n\nA decimal number, on the other hand, has two parts which are separated by a decimal point, in simple word a \u201cdot\u201d. For example, in the decimal number 357.951, the digits to the left of the decimal point, are the whole number whereas the digits to the right of the decimal point are the fractional part.\n\nWhat is a Decimal Number?\n\nDecimal numbers are the numbers that fall between integers and non-integers. They are described as digits following a decimal point. Decimals use a system of numbers which is based on units of tens. In this system, the results in the spaces past the decimal point are as tenths, hundredths, thousandths, and so on.\n\nWhat is a Fraction?\n\nFractions are basically the ratios of two numbers. Often, these numbers are whole numbers, such as $\\frac{3}{4}$ or $\\frac{19}{40}$. Fractions are also a different way to represent division. For example,$\\frac{3}{4}$ can also mean \"three fourths\" or \"three divided by four.\"\n\nHow to Convert a Decimal into a Fraction\n\nTo convert a decimal into a fraction there is no need for decimal to fraction formula but just three simple steps that we need to follow.\n\nStep 1) Write the given decimal without the decimal point as a numerator.\n\nStep 2) Take 1 annexed with as many zeros as is the number of decimal places in the given decimal as a denominator.\n\nStep 3) Reduce the above fraction in the simplest form.\n\nFor example, 13.484 = $\\frac{13484}{1000}$ = $\\frac{3371}{250}$ = 13$\\frac{121}{250}$\n\nFor convenience, always try to make a decimal to fraction table and always keep it ready before you start solving the problems. Google more an example of decimal fraction and practice it. Always go through questions like how to change a decimal to a fraction? Or how to convert decimal number to a fraction or maybe how to convert a decimal into an improper fraction? These questions will help you to understand the topic better.\n\nRecurring Decimals\n\nWe know that $\\frac{3}{5}$ = 0.6, in this case, the dividend is exactly divisible after a few steps i.e., the remainder is zero. Such decimal numbers are called terminating decimals. However, there are fractions that give unending decimal values. Now look at this,\n\n$\\frac{2}{3}$ = 0.66666\u2026\u2026\u2026., in some fractions, the division does not stop and we obtain a certain block of digits which repeat over and over again. Such decimals numbers are called recurring decimals.\n\nSolved Examples\n\nExample 1) convert decimal to fraction:\n\ni) 0.8\n\nii) 4.75\n\niii) 0.056\n\niv) 32.7\n\nv) 0.4\n\nvi) 3.54\n\nvii) 0.36\n\nviii) 85.473\n\nix) 0.013\n\nx) 3.7\n\nxi) 57.4\n\nxii) 41.02\n\nxiii) 3.137\n\nxiv) 0.7\n\nxv) 0.06\n\nxvi) 0.012\n\nxvii) 0.25\n\nxviii) 31.08\n\nxix) 74.085\n\nxx) 0.225\n\nSolution 1) Conversion from decimal to fraction can be done as follows:\n\ni) 0.8 = $\\frac{8}{10}$ = $\\frac{4}{5}$\n\nii) 4.75 = $\\frac{475}{100}$ = $\\frac{19}{4}$ = 4$\\frac{3}{4}$\n\niii) 0.056 =\u00a0 $\\frac{56}{1000}$ =\u00a0 $\\frac{7}{125}$\n\niv) 32.7 =\u00a0 $\\frac{327}{10}$\n\nv) 0.4 =\u00a0 $\\frac{4}{10}$ =\u00a0 $\\frac{2}{5}$\n\nvi) 3.54 =\u00a0 $\\frac{354}{100}$ =\u00a0 $\\frac{177}{50}$\n\nvii) 0.36 =\u00a0 $\\frac{36}{100}$\u00a0 $\\frac{18}{50}$ =\u00a0 $\\frac{9}{25}$\n\nviii) 85.473 =\u00a0 $\\frac{85473}{1000}$\n\nix) 0.013 =\u00a0 $\\frac{13}{1000}$\n\nx) 3.7 =\u00a0 $\\frac{37}{10}$\n\nxi) 57.4 =\u00a0 $\\frac{574}{10}$\n\nxii) 41.02 =\u00a0 $\\frac{4102}{100}$\n\nxiii) 3.137 =\u00a0 $\\frac{3137}{1000}$\n\nxiv) 0.7 =\u00a0 $\\frac{7}{10}$\n\nxv) 0.06 =\u00a0 $\\frac{6}{100}$\n\nxvi) 0.012 =\u00a0 $\\frac{12}{1000}$\n\nxvii) 0.25 =\u00a0 $\\frac{25}{100}$ =\u00a0 $\\frac{1}{4}$\n\nxviii) 31.08 = 31+ $\\frac{8}{100}$ = 31 $\\frac{2}{25}$ = 31 $\\frac{2}{25}$\n\nxix) 74.085 = 74+ $\\frac{85}{1000}$ = 74 + $\\frac{17}{200}$ = 74$\\frac{17}{200}$\n\nxx) 0.225 = $\\frac{9}{40}$","date":"2020-09-20 23:55:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8756734728813171, \"perplexity\": 1303.1844499953188}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-40\/segments\/1600400198868.29\/warc\/CC-MAIN-20200920223634-20200921013634-00378.warc.gz\"}"} | null | null |
{"url":"http:\/\/m.blog.csdn.net\/hongjq\/article\/details\/122495","text":"### How to install rstp protocol to mozilla.\n\ngoogle a solution for Mozilla: have fun! yay...found a fix!!! here's the process for future reference. go to the firefox cofiguration by typing \"about:config\" in the adress bar. then right-click anywhere, and go to \"new-->string\". for the preference name, type... network.protocol-handler.app.rtsp for the value, just type in the path to the executable...but in my case, the exe cutable was located in \"\/usr\/local\/bin\", so i just typed it's name... hxplay now whenver i click the link, the helix player pops up and plays it, instead of getting the error :) i'll also rename the thread so it'll be easier for troubleshooters to find\n\n0 0","date":"2017-12-14 05:59:09","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8446135520935059, \"perplexity\": 6070.868745960428}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948541253.29\/warc\/CC-MAIN-20171214055056-20171214075056-00673.warc.gz\"}"} | null | null |
Q: Navbar links links staying outside of the dropdown menu I am trying to place my span classes inside my navbar header, however, for some reason, its staying outside the dropdown menu.
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<script src="https://code.jquery.com/jquery-3.3.1.slim.min.js" integrity="sha384-q8i/X+965DzO0rT7abK41JStQIAqVgRVzpbzo5smXKp4YfRvH+8abtTE1Pi6jizo"
crossorigin="anonymous"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/popper.js/1.14.3/umd/popper.min.js" integrity="sha384-ZMP7rVo3mIykV+2+9J3UJ46jBk0WLaUAdn689aCwoqbBJiSnjAK/l8WvCWPIPm49"
crossorigin="anonymous"></script>
<script src="https://stackpath.bootstrapcdn.com/bootstrap/4.1.3/js/bootstrap.min.js" integrity="sha384-ChfqqxuZUCnJSK3+MXmPNIyE6ZbWh2IMqE241rYiqJxyMiZ6OW/JmZQ5stwEULTy"
crossorigin="anonymous"></script>
</body>
</html>
Also another problem that i noticed is within my div class, for some reason my div landing-next is displaying links inside my tags, i want them to show normal letters.
A: The three span tags to build the so-called hamburger menu is v3 code and it isn't available any longer. From the migration docs:
*
*.navbar-toggle is now .navbar-toggler and has different styles and inner markup (no more three <span>s).
There are several alternatives. You can use a vector font (e.g. FontAwesome with fa fa-bars) or, like Bootstrap in some examples, a
Data URI for SVG.
Regarding the links, you just forgot to close the <a>tags.
| {
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Q: openMP: use of global variable in child functions I've been trying to parallelize a C program using OpenMP, and it's like this:
#include<omp.h>
#include<stdio.h>
int test, result;
#pragma omp threadprivate(test, result)
void add(void)
{
result = result + test;
}
int main(void)
{
int i;
#pragma omp parallel for private(i)
for (test = 0; test < 5; test++) {
result = 0;
for (i = 1; i < 100; i++) {
add();
}
printf("Result in %dth test is: %d\n", test, result);
} //End of parallel region
return 0;
}
Compile and run it sequentially, I get the following output:
Result in 0th test is: 0
Result in 1th test is: 99
Result in 2th test is: 198
Result in 3th test is: 297
Result in 4th test is: 396
However, when I compile with -fopenmp and run it, I got all zeros:
Result in 0th test is: 0
Result in 1th test is: 0
Result in 3th test is: 0
Result in 4th test is: 0
Result in 2th test is: 0
Can anyone tell me what I did wrong in my program? I'm new in openMP.
Thanks!
A: Your program is not conforming, and yields undefined behavior. In particular if you check the latest standard you can find in section 2.7.1 the following restriction for the loop construct:
*
*The loop iteration variable may not appear in a threadprivate
directive
In your case what is likely to happen is a name clash between two variables named test: one should be a private int variable created for the loop iteration while the other is the global variable declared as threadprivate. But again, being undefined behavior, anything may happen.
| {
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Farmers and consumers in Maryland and throughout the U.S. are encouraged to participate in Buy Local Challenge Week.
Maryland's 5th annual Buy Local Challenge kicked off Thursday at Governor Martin O'Malley's mansion with a cookout decked out with—you guessed it—local foods. After the event, more than 3,000 Marylanders had signed a pledge to participate in Buy Local Challenge Week (July 23 to 31, 2011), in which they committed to consuming at least one local product each day of the week.
And Maryland, along with the rest of the U.S., continues to face the problem of farmland loss. According to the USDA's 2007 Agriculture Census, the amount of farmland in Maryland has dropped by nearly 42 percent since 1959 to slightly more than 2 million acres.
The goal of Buy Local Week, however, is to put farm-fresh products at the forefront of consumers' minds, giving extra visibility to local farms, increasing visitation to farmers' markets and getting people to think about where their food comes from.
Not only does buying from local farmers help the U.S. meet national health goals, it also helps to boost the local economy. If households spend just $12 a week for eight weeks on local products, $10.7 billion are reinvested directly to the nation's farms, says Bergmark, pulling from U.S. Census data. Plus, she says, local farmers are likely to reinvest that money into the local community. For example, when you buy your cheese from the local dairy farmer, he might use that money to purchase feed from a local farm store.
During this year's Buy Local Challenge, an emphasis is being placed on the workplace. Marylanders are encouraged to take the official Buy Local Challenge pledge as an office and can enter a photo contest, showcasing how they as an office intend to fulfill the pledge. The most unique entry from a Marylander will receive a catered lunch for up to 20 people and a gift basket of Maryland farm products.
The SMADC and its partner, the Maryland Agricultural Marketing Professionals, are also encouraging participants in the challenge nationwide. Interest in the challenge has already been generated in Virginia, Washington, Florida and Washington, D.C., Bergmark says. Those outside Maryland wanting to promote buying local in their workplace can download promotional materials from the Buy Local Challenge website.
"We hope people will buy local all year round and hope people will buy local while product is abundant across the U.S.," Bergmark says. | {
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#ifndef NEWSBOAT_FORMACTION_H_
#define NEWSBOAT_FORMACTION_H_
#include <memory>
#include <string>
#include <vector>
#include "history.h"
#include "keymap.h"
#include "listwidget.h"
#include "stflpp.h"
namespace newsboat {
class ConfigContainer;
class RssFeed;
class View;
typedef std::pair<std::string, std::string> QnaPair;
enum class CommandType {
QUIT,
SAVE,
GOTO,
TAG,
SET,
SOURCE,
DUMPCONFIG,
EXEC,
UNKNOWN, /// Unknown/non-existing command. Tokenized input is stored in Command.args
INVALID, /// differs from UNKNOWN in that no input was parsed
};
struct Command {
CommandType type;
std::vector<std::string> args;
};
class FormAction {
public:
FormAction(View*, std::string formstr, ConfigContainer* cfg);
virtual ~FormAction();
virtual void prepare() = 0;
virtual void init() = 0;
virtual void set_redraw(bool b)
{
do_redraw = b;
}
virtual const std::vector<KeyMapHintEntry>& get_keymap_hint() const = 0;
virtual std::string id() const = 0;
std::string get_value(const std::string& name);
void set_value(const std::string& name, const std::string& value);
void draw_form();
std::string draw_form_wait_for_event(unsigned int timeout);
void recalculate_widget_dimensions();
virtual void handle_cmdline(const std::string& cmd);
bool process_op(Operation op,
bool automatic = false,
std::vector<std::string>* args = nullptr);
virtual void finished_qna(Operation op);
void start_cmdline(std::string default_value = "");
void start_qna(const std::vector<QnaPair>& prompts,
Operation finish_op,
History* h = nullptr);
void set_parent_formaction(std::shared_ptr<FormAction> fa)
{
parent_formaction = fa;
}
std::shared_ptr<FormAction> get_parent_formaction() const
{
return parent_formaction;
}
virtual std::string title() = 0;
virtual std::vector<std::string> get_suggestions(
const std::string& fragment);
static void load_histories(const std::string& searchfile,
const std::string& cmdlinefile);
static void save_histories(const std::string& searchfile,
const std::string& cmdlinefile,
unsigned int limit);
std::string bookmark(const std::string& url,
const std::string& title,
const std::string& description,
const std::string& feed_title);
protected:
virtual bool process_operation(Operation op,
bool automatic = false,
std::vector<std::string>* args = nullptr) = 0;
virtual void set_keymap_hints();
void start_bookmark_qna(const std::string& default_title,
const std::string& default_url,
const std::string& default_feed_title);
static Command parse_command(const std::string& input,
std::string delimiters = " \r\n\t");
void handle_parsed_command(const Command& command);
bool handle_list_operations(ListWidget& list, Operation op);
View* v;
ConfigContainer* cfg;
Stfl::Form f;
bool do_redraw;
std::vector<std::string> qna_responses;
static History searchhistory;
static History cmdlinehistory;
std::vector<std::string> valid_cmds;
private:
void start_next_question();
bool handle_single_argument_set(std::string argument);
void handle_set(const std::vector<std::string>& args);
void handle_quit();
void handle_source(const std::vector<std::string>& args);
void handle_dumpconfig(const std::vector<std::string>& args);
void handle_exec(const std::vector<std::string>& args);
std::vector<QnaPair> qna_prompts;
Operation finish_operation;
History* qna_history;
std::shared_ptr<FormAction> parent_formaction;
};
} // namespace newsboat
#endif /* NEWSBOAT_FORMACTION_H_ */
| {
"redpajama_set_name": "RedPajamaGithub"
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{"url":"https:\/\/socratic.org\/questions\/59107ac4b72cff7430f3e681","text":"# What is the formula for calculating the distance between two points?\n\nMay 8, 2017\n\nThe formula for calculating the distance between two points on a plane is:\n\n$d = \\sqrt{{\\left(\\textcolor{red}{{x}_{2}} - \\textcolor{b l u e}{{x}_{1}}\\right)}^{2} + {\\left(\\textcolor{red}{{y}_{2}} - \\textcolor{b l u e}{{y}_{1}}\\right)}^{2}}$\n\nThe formula for calculating the distance between two points in 3D space is:\n\n$d = \\sqrt{{\\left(\\textcolor{red}{{x}_{2}} - \\textcolor{b l u e}{{x}_{1}}\\right)}^{2} + {\\left(\\textcolor{red}{{y}_{2}} - \\textcolor{b l u e}{{y}_{1}}\\right)}^{2} + {\\left(\\textcolor{red}{{z}_{2}} - \\textcolor{b l u e}{{z}_{1}}\\right)}^{2}}$","date":"2019-07-16 01:58:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 2, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9467675685882568, \"perplexity\": 77.47084124480527}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195524475.48\/warc\/CC-MAIN-20190716015213-20190716041213-00552.warc.gz\"}"} | null | null |
Hatzalah's rescue at NYC mayoral rally
Posted Thursday, July 18, 2013 12:00 am
One of the ambulances in Hatzalah's Far Rockaway-Five Towns region.
By Malka Eisenberg
The excessive heat on Tisha B'Av Tuesday dropped a political intern to the pavement in Williamsburg, hijacking a press conference by mayoral hopeful Christine Quinn, igniting a debate over New York City's emergency preparedness, and bringing kudos to Hatzalah for its rapid response.
City Council Speaker Quinn waited 30 minutes for an ambulance to help an assistant to Councilwoman Diana Reyna. The 17-year-old intern collapsed and later fainted even though she was being tended to by a member of Quinn's security detail who is also a certified emergency medical technician.
When an ambulance did not arrive despite Quinn's calls to Fire Commissioner Salvatore Cassano and Police Commissioner Raymond Kelly, Hatzalah was called and promptly responded. They took the woman to Woodhull Medical Center in Bushwick.
"Nobody beats Hatzalah," Rayna told the Hatzalah volunteers. "Hatzalah is always there when we need them."
An email from Quinn to Hamodia thanked the volunteers. "The City is indebted to them for everything that they do every day to help New Yorkers in need of emergency medical attention."
Comptroller John C. Liu also weighed in, praising Hatzalah. "Many thanks to the Hatzalah volunteers who stepped forward at a moment's notice, even on their fasting day, when the emergency response system failed and they were most needed. New Yorkers are fortunate that these selfless emergency responders are willing to put it all on the line in order to ensure the health and well-being of those in need. We all owe you a debt of gratitude. A hearty Yasher Koach to you, Hatzalah."
Brooklyn Democratic Councilman David Greenfield, an Orthodox Jew, said that Hatzalah's quick appearance "is even more impressive considering that it occurred on Tisha B'Av, when its volunteers are in the middle of fasting. This incident reinforced what we already knew — our community is extremely fortunate to have capable and dedicated volunteers looking out for us around the clock."
Quinn had been holding a press conference in the near triple digit temperatures in front of P.S. 132 in Williamsburg, Brooklyn, in support of a trash dumping site on the Upper East Side — the East 91st Street Marine Transfer Station.
Since the intern was conscious and breathing and being tended to by an EMT, the city did not consider it a priority call, FDNY said.
"Every call for medical assistance is important and ambulance dispatching is prioritized so life-threatening calls — for a choking child, cardiac arrest or chest pains — take precedence over non-life-threatening injuries—when the patient is breathing, alert or communicating," FDNY said in a statement seeking to quell public concern heightened by significant media attention to the incident.
"That was the case here. In addition, the patient was being treated by a police officer, who is an EMT so care was being administered from the moment the incident occurred."
Rabbi Elozer Kanner, one of the coordinators of the local Chevra Hatzalah of the Rockaways and Nassau County, said that the ambulance that answered the call was "not my branch," but all branches of "Hatzalah are always happy to help anyone in need. The fact that resources sometimes are stretched thin and there is not a quick response can happen to any ambulance core.
"There was an EMT treating her and I'm sure he did whatever was appropriate."
Kanner added that it sometimes happens that all ambulances are on calls. "It can happen to anybody; I would not criticize anybody. It is hot weather, they are stretched to capacity on other calls. It was fortunate that Hatzalah could come. Having been in those shoes, I can understand being stretched to the limit and it can happen to any agency."
News sources indicated that there were 15 concurrent calls in that area and that 911 received about 4,000 calls due to the heat, more than the usual 3,200.
Meanwhile, following the state's recent closing of the emergency room at Long Island College Hospital in Downtown Brooklyn, Brooklyn ERs were overwhelmed, and ambulances were diverted from Brooklyn Hospital Center, creating backups at most Brooklyn facilities.
"It's a mess," a paramedic bringing a patient to New York Methodist Hospital in Park Slope told the Brooklyn Daily Eagle.
Ambulances linked up along Seventh Avenue outside Methodist "since the ambulance deck cannot always accommodate all of them," Methodist spokesperson Lyn Hill told the Eagle.
"Of course, it's dangerous," EMT Herby Dossous said. "As it is, we're overworked. You close two ERs and it's dangerous."
Regarding the problem at her rally in Williamsburg, Quinn asked: "[How long would it take] an ambulance to get anywhere else where there aren't television [cameras] and there aren't two elected officials?"
Scouts meet firefighters, Hatzalah team in Woodmere
Achiezer gala to honor Hatzalah doc
2 tragedies dim Chanukah | {
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Mary Lightbody
Rep. Lightbody:
The sixth of seven children, State Rep. Mary Lightbody grew up a country girl in Pepper Pike, Ohio. She earned her undergraduate degree from Harvard University and a doctorate in science education from The Ohio State University. She has spent the last 27 years teaching both in K-12 education and at the collegiate level in Central Ohio, most recently in a position at Ohio State's Newark campus.
Rep. Lightbody is a fearless advocate for her constituents in the 19th Ohio House District, fighting to make quality, affordable healthcare coverage available to all Ohioans as well as leading the way on policies that support working families—like reforming a broken education system and standing up for Ohio workers.
Rep. Lightbody married her husband, Rick Noss, in college. Shortly after, however, he was diagnosed with a chronic illness, which meant years of intense care and costly medical bills. Having gone through this experience, Rep. Lightbody knows just how much a sudden illness can affect a family. That's why she's fought to improve access to care and to bring down healthcare costs for Ohio families.
In addition to her full-time teaching career, Lightbody is the mother of three grown children, and has served as a deacon at her church, as a member of the Board of Trustees of the Westerville Public Library and in leadership positions with state and national science teachers' professional associations.
Rep. Lightbody currently lives in Plain Township, where she's lived for the past 30 years.
Commerce and Labor | {
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Plusieurs chapelles ont été nommées chapelle Saint-Jacques ou chapelle Saint-Jacques-le-Majeur en référence à saint Jacques :
la chapelle Saint-Jacques-le-Majeur d'Abriès-Ristolas en France ;
la chapelle Saint-Jacques de Bagnolet en France ;
la chapelle Saint-Jacques de Besançon en France ;
la chapelle Saint-Jacques de Bordeaux en France, dernier vestige de l'hôpital-prieuré Saint-Jacques de Bordeaux ;
la chapelle Saint-Jacques de Brech en France ;
la chapelle Saint-Jacques d'Orelle en France ;
la de Cogorno en Italie ;
la chapelle Saint-Jacques de Coussegrey en France ;
la de Macao en Chine ;
la chapelle Saint-Jacques de Maule en France ;
la chapelle Saint-Jacques de Menton en France ;
la chapelle Saint-Jacques de Saint-Léon à Merléac en France ;
la chapelle Saint-Jacques de Mutzig en France ;
la de Mosogno à Onsernone en Suisse ;
la chapelle Saint-Jacques de Saorge en France ;
la chapelle Saint-Jacques de Vignec en France.
Autres édifices
La chapelle Saint-Jacques-le-Mineur de Gap en France est nommée d'après l'autre apôtre Jacques.
Voir aussi
~
SaintJacques | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,969 |
{"url":"https:\/\/cms.math.ca\/10.4153\/CMB-2015-060-5","text":"location:\u00a0 Publications \u2192 journals \u2192 CMB\nAbstract view\n\nApproximation of a Function and its Derivatives by Entire Functions\n\nPublished:2015-12-22\nPrinted: Mar 2016\n\u2022 Paul M. Gauthier,\nD\u00e9partement de math\u00e9matiques et de statistique, Universit\u00e9 de Montr\u00e9al, CP-6128 Centreville, Montr\u00e9al H3C3J7\n\u2022 Julie Kienzle,\nD\u00e9partement de math\u00e9matiques et de statistique, Universit\u00e9 de Montr\u00e9al, CP-6128 Centreville, Montr\u00e9al H3C3J7\n Format: LaTeX MathJax PDF\n\nAbstract\n\nA simple proof is given for the fact that, for $m$ a non-negative integer, a function $f\\in C^{(m)}(\\mathbb{R}),$ and an arbitrary positive continuous function $\\epsilon,$ there is an entire function $g,$ such that $|g^{(i)}(x)-f^{(i)}(x)|\\lt \\epsilon(x),$ for all $x\\in\\mathbb{R}$ and for each $i=0,1\\dots,m.$ We also consider the situation, where $\\mathbb{R}$ is replaced by an open interval.\n Keywords: Carleman theorem Carleman theorem\n MSC Classifications: 30E10 - Approximation in the complex domain\n\n top of page | contact us | privacy | site map |","date":"2018-02-19 10:14:07","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.35004499554634094, \"perplexity\": 1920.7419871597315}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891812579.21\/warc\/CC-MAIN-20180219091902-20180219111902-00673.warc.gz\"}"} | null | null |
Q: grunt file glob file extension pattern I have a grunt task to traverse a folder structure for all jpg, jpeg and png files. Is there an easier way to do this compared to
images : {
expand : true,
flatten : true,
cwd : "develop/",
src : ["modules/**/*.jpg", "modules/**/*.jpeg", "modules/**/*.png"],
dest : "build/resources/img/"
},
According to the documentations, the { .. } bracket for a file allows multiple options to be compared with for the containing files. However, the following do not work for src array values:
"modules/**/*{jpg, jpeg, png}"
"modules/**/*.{jpg, jpeg, png}"
"modules/**/*{.jpg, .jpeg, .png}"
and so therefore I was forced to do a straight glob pattern for each file extension
A: There's extra spaces after the commas in your expression.
This should do it: "modules/**/*.{jpg,jpeg,png}"
| {
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Łubki is a village in the administrative district of Gmina Wojciechów, within Lublin County, Lublin Voivodeship, in eastern Poland. It lies approximately west of Wojciechów and west of the regional capital Lublin.
References
Villages in Lublin County | {
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SHAREHOLDER ALERT: Pomerantz Law Firm Investigates Claims On Behalf of Investors of Core Scientific, Inc. - CORZ
/EIN News/ -- NEW YORK, Nov. 28, 2022 (GLOBE NEWSWIRE) -- Pomerantz LLP is investigating claims on behalf of investors of Core Scientific, Inc. ("Core" or the "Company") (NASDAQ: CORZ). Such investors are advised to contact Robert S. Willoughby at newaction@pomlaw.com or 888-476-6529, ext. 7980.
The investigation concerns whether Core and certain of its officers and/or directors have engaged in securities fraud or other unlawful business practices.
[Click here for information about joining the class action]
On March 3, 2022, Culper Research issued a report alleging, among other things, that Core Scientific "has wildly oversold both its mining and hosting businesses, which it cobbled together in a series of questionable transactions before dumping onto the market via SPAC." Moreover, the Company had "waived the 180-day lockup on over 282 million shares, making them free to be dumped just 5 trading days" from the time of the report, showing that "insiders have abandoned any pretense of care for minority shareholders."
On this news, Core Scientific's stock price fell $0.72 per share, or 9.4%, to close at $6.98 per share on March 3, 2022.
Then, on September 28, 2022, Celsius Network LLC and related entities filed a motion to enforce the automatic stay and for civil contempt in bankruptcy proceedings alleging that Core Scientific "has knowingly and repeatedly violated the automatic stay provisions" by refusing to perform its contractual obligations, threatening to terminate the companies' agreement, and adding improper surcharges.
On this news, Core Scientific's stock price fell $0.15 per share, or 10.3%, to close at $1.30 per share on September 29, 2022.
Finally, on October 27, 2022, Core Scientific disclosed that "given the uncertainty regarding the Company's financial condition, substantial doubt exists about the Company's ability to continue as a going concern," and that it is exploring alternatives to its capital structure. Moreover, the Company held 24 bitcoins, compared to 1,051 bitcoins as of September 30, 2022.
On this news, Core Scientific's stock price fell $0.79 per share, or 78.1%, to close at $0.22 per share on October 27, 2022.
The Pomerantz Firm, with offices in New York, Chicago, Los Angeles, and Paris is acknowledged as one of the premier firms in the areas of corporate, securities, and antitrust class litigation. Founded by the late Abraham L. Pomerantz, known as the dean of the class action bar, the Pomerantz Firm pioneered the field of securities class actions. Today, more than 80 years later, the Pomerantz Firm continues in the tradition he established, fighting for the rights of the victims of securities fraud, breaches of fiduciary duty, and corporate misconduct. The Firm has recovered numerous multimillion-dollar damages awards on behalf of class members. See www.pomerantzlaw.com.
Robert S. Willoughby
Pomerantz LLP
rswilloughby@pomlaw.com
Distribution channels: Consumer Goods, Law
Greenwave Expects Optimal Market Conditions in 2023
Great Southern Bancorp, Inc. to Hold 34th Annual Meeting of Stockholders
BlueSphere Bio Appoints Erkut Bahceci, M.D., as Chief Medical Officer | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 286 |
{"url":"https:\/\/indico.particle.mephi.ru\/event\/22\/contributions\/1317\/","text":"# IV international conference on particle physics and astrophysics\n\n22-26 October 2018\nHotel Intourist Kolomenskoye 4*\nEurope\/Moscow timezone\n\n## Femtoscopic scales of particle-emitting source in small and large systems\n\n26 Oct 2018, 17:10\n15m\nMoskvorechye 1 hall (Hotel Intourist Kolomenskoye 4*)\n\n### Moskvorechye 1 hall\n\n#### Hotel Intourist Kolomenskoye 4*\n\nKashyrskoye shosse, 39B, Moscow, Russia, 115409\nPlenary\/section talk Nuclear physics: heavy ion\n\nVarvara Semenova\n\n### Description\n\nThe femtoscopy technique allows one to measure the spatial and temporal scales of the particle-emitting source produced at high energy collisions. In non-central ultra-relativistic heavy-ion collisions, emitting source may be tilted in the reaction plane. The orientation of freeze-out distributions is interesting because it provides complementary information about quark-gluon matter properties. In the experiment, the tilt can be extracted by measuring femtoscopic radii as a function of the pair angle with respect to the first-order event plane.\n\nIn this talk, we will present results of azimuthally sensitive femtoscopic analysis of Au+Au collisions at 200 GeV using UrQMD and vHLLE models. We will also present the transverse momentum and multiplicity dependence of identical pion and kaon femtoscopic radii from d+Au, $^{3}$He+Au collisions at 200 GeV obtained from the UrQMD model.\n\n### Presentation Materials\n\n###### Your browser is out of date!\n\nUpdate your browser to view this website correctly. Update my browser now\n\n\u00d7","date":"2019-07-21 11:51:13","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5143515467643738, \"perplexity\": 9820.865759507535}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526948.55\/warc\/CC-MAIN-20190721102738-20190721124738-00344.warc.gz\"}"} | null | null |
using Atomiv.Core.Common.Http;
using Atomiv.Template.Core.Application.Commands.Customers;
using Atomiv.Template.Core.Application.Queries.Customers;
using System.Threading.Tasks;
namespace Atomiv.Template.Web.RestClient.Interface
{
public interface ICustomerControllerClient : IHttpControllerClient
{
#region Commands
Task<ObjectClientResponse<CreateCustomerCommandResponse>> CreateCustomerAsync(CreateCustomerCommand request, HeaderData header);
Task<ObjectClientResponse<DeleteCustomerCommandResponse>> DeleteCustomerAsync(DeleteCustomerCommand request, HeaderData header);
Task<ObjectClientResponse<EditCustomerCommandResponse>> EditCustomerAsync(EditCustomerCommand request, HeaderData header);
#endregion
#region Queries
Task<ObjectClientResponse<BrowseCustomersQueryResponse>> BrowseCustomersAsync(BrowseCustomersQuery request, HeaderData header);
Task<ObjectClientResponse<FilterCustomersQueryResponse>> FilterCustomersAsync(FilterCustomersQuery request, HeaderData header);
Task<ObjectClientResponse<ViewCustomerQueryResponse>> ViewCustomerAsync(ViewCustomerQuery request, HeaderData header);
#endregion
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 286 |
\section{Introduction}
\label{sec:Intro}
Laser spectroscopy of light muonic atoms and ions, where a single negative muon
orbits a bare nucleus, holds the promise for a vastly improved determination of
nuclear parameters, compared to the more traditional methods of elastic electron
scattering and precision laser spectroscopy of regular electronic atoms.
The CREMA collaboration has so far determined the charge radii of the proton and
the deuteron, by measuring several transitions in muonic hydrogen (\ensuremath{\mu}{\rm p}\xspace) \cite{Pohl:2010:Nature_mup1,Antognini:2013:Science_mup2,Antognini:2013:Annals} and muonic deuterium (\ensuremath{\mu}{\rm d}\xspace) \cite{Pohl:2016:mud,Krauth:2016:mud}. Interestingly, both values differ by as much as six standard deviations from the respective CODATA-2014 values \cite{Mohr:2016:CODATA14}, which contain data from laser spectroscopy in atomic hydrogen/deuterium and electron scattering.
This discrepancy has been coined ``proton radius puzzle" \cite{Pohl:2013:ARNPS,Carlson:2015:Puzzle,Hill:2017:PRP}.
However, the discrepancy exists for the deuteron, too.
Interestingly, for the proton and the deuteron, the muonic isotope shift is compatible with the electronic one from the 1S-2S transition in H and D \cite{Parthey:2010:PRL_IsoShift,Jentschura:2011:IsoShift}.
The respective radii are
\begin{align}
\ensuremath{r_\mathrm{p}} (\ensuremath{\mu}{\rm p}\xspace) =& ~0.84087(\hphantom{0}26)^\mathrm{exp}(29)^\mathrm{th} \nonumber \\
=& ~0.84087(\hphantom{0}39)\,\ensuremath{\mathrm{fm}}\xspace &\text{\cite{Pohl:2010:Nature_mup1,Antognini:2013:Science_mup2}} \label{eq:mup}\\
\ensuremath{r_\mathrm{p}} (\rm CODATA'14) =& ~0.87510(610)\,\ensuremath{\mathrm{fm}}\xspace &\text{\cite{Mohr:2016:CODATA14}}\\[10pt]
\ensuremath{r_\mathrm{d}} (\ensuremath{\mu}{\rm d}\xspace) =& ~2.12562(\hphantom{0}13)^\mathrm{exp}(77)^\mathrm{th} \nonumber \\
=& ~2.12562(\hphantom{0}78)\,\ensuremath{\mathrm{fm}}\xspace &\text{\cite{Pohl:2016:mud}} \label{eq:mud}\\
\ensuremath{r_\mathrm{d}} (\rm CODATA'14) =& ~2.14130(250)\,\ensuremath{\mathrm{fm}}\xspace. &\text{\cite{Mohr:2016:CODATA14}}
\end{align}
Very recently, the CREMA collaboration has measured a total of five transitions in muonic helium-3 and -4 ions \cite{Antognini:2011:Conf:PSAS2010}, which have been analyzed now.
These measurements will help to improve our understanding of nuclear model theories \cite{Machleidt:2011:nuclforces,NevoDinur:2016:TPE} and shed more light on the proton radius puzzle. Several ideas exist to solve the puzzle \cite{Antognini:2016:PRP}, some within the standard model \cite{Miller:2013:pol,Jentschura:2015:virtPart} and others proposing muon specific forces beyond the standard model \cite{Tucker-Smith:2011,Batell:2011:PV_muonic_forces,Karshenboim:2014:darkForces,Carlson:2015:BSM}.
These ideas lead to predictions which can be tested with precise charge radius determinations in muonic helium ions.
The measurement of the charge radius in both, helium-3 and helium-4 ions will in addition help understand the discrepancy between several measurements of the helium isotope shift in electronic helium \cite{Shiner:1995:heliumSpec,Rooij:2011:HeSpectroscopy,CancioPastor:2012:PRL108,Patkos:2016:HeIso,Patkos:2017:HeIsoII} which yield the difference of the squared charge radii (see Fig.\,\ref{fig:iso_shift}).
\begin{figure*}
\centering
\includegraphics[width=0.7\linewidth]{he_isotope2}
\caption{Difference of squared helion-to-alpha particle charge radii as obtained from laser spectroscopy of transitions in regular, electronic helium-3 and helium-4 atoms \cite{Shiner:1995:heliumSpec,Rooij:2011:HeSpectroscopy,CancioPastor:2012:PRL108} when combined with accurate theory (*\cite{Patkos:2017:HeIsoII}, **\cite{Patkos:2016:HeIso}). A $4\,\sigma$ discrepancy persists.
Also shown are the individual theory uncertainties which enter $\rh^2-r_\alpha^2$ (\ensuremath{\mu^4}{\rm He}\ensuremath{^+}\xspace: \cite{Diepold:2016:muHe4theo}, \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace: this work), as well as the expected uncertainty from our laser spectroscopy of the Lamb shift in muonic helium ions.
Note that the combination of the two theoretical uncertainties should contain correlations which will partly cancel in the total uncertainty.
}
\label{fig:iso_shift}
\end{figure*}
Several other experiments are on the way to contribute to the puzzle in the future \cite{Antognini:2016:PRP} by precision spectroscopy measurements in electronic hydrogen \cite{Vutha:2012:H2S2P,Beyer:2013:AdP_2S4P,Peters:2013:AdP} and He$^+$ \cite{Herrmann:2009:He1S2S,Kandula:2010:XUV_comb_metrology}, as well as by electron scattering at very low $Q^2$ \cite{Mihovilovic:2013:ISR_exp_MAMI,Gasparian:2014:PRad} and muon-scattering \cite{Gilman:2013:MUSE}.
The He$^+$ spectroscopy, in combination with our measurement in muonic helium ions, will be able to determine the Rydberg constant independently from hydrogen and deuterium. This is particularly interesting as the proton charge radius and the Rydberg constant are highly correlated which means that a change in the Rydberg constant could also resolve the puzzle \cite{Beyer:2013:AdP_2S4P}.
The determination of the helion charge radius from muonic helium spectroscopy
requires accurate knowledge of the corresponding theory.
Similar to muonic hydrogen \cite{Antognini:2013:Annals}, deuterium \cite{Krauth:2016:mud}, and helium-4 ions \cite{Diepold:2016:muHe4theo}, we feel therefore obliged to summarize the current knowledge on the state of theory contributions to the Lamb shift, fine-, and hyperfine structure in muonic helium-3 ions.
The accuracy to be expected from the experiment will be on the order of 20\,GHz, which corresponds to $\sim 0.08\,\ensuremath{\mathrm{meV}}\xspace$~\footnote{$1\,\ensuremath{\mathrm{meV}}\xspace~\widehat=~241.799\,\mathrm{GHz}$}. In order to exploit the experimental precision, theory should, ideally, be accurate to a level of
\begin{equation}
\sigma_\mathrm{theory}\sim \mathcal{O}( 0.01\,\mathrm{meV}).
\label{eq:uncertainty}
\end{equation}
This would result in a nearly hundred-fold better accuracy in the helion rms charge radius \rh ~compared to the value from electron scattering of
\begin{equation}
\rh = 1.973(14)\,\ensuremath{\mathrm{fm}}\xspace,
\end{equation}
deduced by Sick \cite{Sick:2014:HeZemach}.
A more precise value has been given by Angeli~{\it et\,al.}~\cite{Angeli:2013:radii}, which should be discarded. Their value is based on a charge radius extraction from \ensuremath{\mu^4}{\rm He}\ensuremath{^+}\xspace by Carboni~{\it et\,al.}~\cite{Carboni:1978:LS_mu4he} and on the isotope shift measurement from Shiner~{\it et\,al.}~\cite{Shiner:1995:heliumSpec}. The Carboni measurement has however shown to be wrong \cite{Hauser:1992:LS_search}, and the more recent measurement of the electronic isotope shift by van Rooij {\it et\,al.}\ \cite{Rooij:2011:HeSpectroscopy} disagrees by $4\,\sigma$ from the Shiner one \cite{Shiner:1995:heliumSpec}, see Fig.\,\ref{fig:iso_shift}.
We anticipate here that the total uncertainty in the theoretical calculation of the Lamb shift transition amounts to 0.52\,\ensuremath{\mathrm{meV}}\xspace (corresponding to a relative uncertainty of $\sim$0.03\%), neglecting the charge radius contribution to be extracted from the \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace measurement. This value is completely dominated by the two-photon exchange contributions which are difficult to calculate but have seen wonderful progress in recent years \cite{NevoDinur:2016:TPE,Hernandez:2016:POLupdate,Carlson:2016:tpe}. The total uncertainty of the pure QED contributions (without the two-photon exchange) amounts to 0.04\,\ensuremath{\mathrm{meV}}\xspace and is thus in the desired order of magnitude. Note that while the theory uncertainty from the two-photon exchange in \ensuremath{r_\mathrm{p}}\ is of similar size as the experimental uncertainty (Eq.\,(\ref{eq:mup})), already for \ensuremath{\mu}{\rm d}\xspace the theory uncertainty is vastly dominant (Eq.\,(\ref{eq:mud})). Experiments with muonic atoms are thus a sensitive tool to determine the two-photon exchange contributions.
\section{Overview}
\label{sec:overview}
The $n=2$ energy levels of the muonic helium-3 ion are sketched in Fig.\,\ref{fig:energy_level}.
The helion has nuclear spin $I=1/2$, just as the proton. Hence the level scheme is very similar to the one of muonic hydrogen. However, the helion magnetic moment $g=-2.127\,625\,308(25)$ \cite{Mohr:2016:CODATA14} (here given in units of the nuclear magneton) is negative, which swaps the ordering of the hyperfine levels.
\begin{figure*}[t]
\centering
\includegraphics[width=0.75\linewidth]{he3_levels.pdf}
\caption{The 2S and 2P energy levels in the muonic helium-3 ion.
The inset on the right displays the shift $\Delta$
of the 2P levels due to the mixing of levels with same quantum number $F$, as described in Sec.\,\ref{sec:2Plevels}.
The figure is not to scale.}
\label{fig:energy_level}
\end{figure*}
A note on the sign convention of the Lamb shift contributions used in this article: The 2S level is shifted below the 2P levels due to the Lamb shift. This means that, fundamentally, the 2S Lamb shift should be given a \textit{negative} sign.\\
However, following long-established conventions we assign the {\em measured} $\ensuremath{2\textrm{S}_{1/2}}\xspace \rightarrow \ensuremath{2\textrm{P}_{1/2}}\xspace$ energy difference a \textit{positive} sign, i.e. E(2P) -- E(2S) $>$ 0. This is in accord with almost all publications we review here and we will mention explicitly when we have inverted the sign with respect to the original publications where the authors calculated level shifts.\\
Moreover, we obey the traditional definition of the Lamb shift as the terms beyond the Dirac equation and the leading order recoil corrections, i.e.\ excluding effects of the hyperfine structure. In particular, this means that the mixing of the hyperfine levels (Sec.\,\ref{sec:2Plevels}) does {\em not} influence the Lamb shift.
The Lamb shift
is dependent on the rms charge radius of the nucleus and is treated in Sec.\,\ref{sec:LS}. We split the Lamb shift contributions into \textit{nuclear structure-independent} contributions and \textit{nuclear structure-dependent} ones. The latter are composed out of one-photon exchange diagrams which represent the finite size effect and two-photon exchange diagrams which contain the polarizability contributions.
In Sec.\,\ref{sec:HFS}, we treat the 2S hyperfine structure, which depends on the Zemach radius. It also has two-photon exchange contributions. However, these have not been calculated yet and can only be estimated with a large uncertainty.
In Sec.\,\ref{sec:2Plevels}, we compile the 2P level structure which includes fine- and hyperfine splitting, and the mixing of the hyperfine levels \cite{Brodsky:1967:zeemanspectrum}.
For the theory compilation presented here, we use the calculations from many sources mentioned in the following. The names of the authors of the respective groups are ordered alphabetically.
The first source is E.~Borie who was one of the first to publish detailed calculations of many terms involved in the Lamb shift of muonic atoms. Her most recent calculations for \ensuremath{\mu}{\rm p}\xspace, \ensuremath{\mu}{\rm d}\xspace, \ensuremath{\mu^4}{\rm He}\ensuremath{^+}\xspace, and \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace are all found in her Ref.\,\cite{Borie:2012:LS_revisited_AoP}. Several updated versions of this paper are available on the arXiv. In this work we always refer to \cite{Borie:2014:arxiv_v7} which is version-7, the most recent one at the time of this writing.
The second source is the group of Elekina, Faustov, Krutov, and Martynenko {\it et\,al.}~(termed ``Martynenko group" in here for simplicity). The calculations we use in here are found in Krutov {\it et\,al.}~\cite{Krutov:2014:JETP120_73} for the Lamb shift, in Martynenko {\it et\,al.}~\cite{Martynenko:2010:2SHFS_muHe,Martynenko:2008:muheHFS} and Faustov {\it et\,al.}~\cite{Faustov:2014:radrec} for the 2S hyperfine structure, and Elekina {\it et\,al.}~\cite{Elekina:2010:2Pmu3He} for the 2P fine- and hyperfine structure.
Jentschura and Wundt calculated some Lamb shift contributions in their Refs.\,\cite{Jentschura:2011:SemiAnalytic,Jentschura:2011:PRA84_012505}. They are referred to as ``Jentschura'' for simplicity.
The group of Ivanov, Karshenboim, Korzinin, and Shelyuto is referred to ``Karshenboim group'' for simplicity. Their calculations are found in Korzinin {\it et\,al.}~\cite{Korzinin:2013:PRD88_125019} and in Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509} for Lamb shift and fine structure contributions.
The group of Bacca, Barnea, Hernandez, Ji, and Nevo Dinur, situated at TRIUMF and Hebrew University, has performed \textit{ab initio} calculations on two-photon exchange contributions of the Lamb shift. Their calculations are found in Nevo Dinur {\it et\,al.}~\cite{NevoDinur:2016:TPE} and Hernandez {\it et\,al.}~\cite{Hernandez:2016:POLupdate}. For simplicity we refer to them as ``TRIUMF-Hebrew group''.
A recent calculation of the two-photon exchange using scattering data and dispersion relations has been performed by Carlson, Gorchtein, and Vanderhaeghen \cite{Carlson:2016:tpe}.
Item numbers \# in our tables
follow the nomenclature in Refs.~\cite{Antognini:2013:Annals,Krauth:2016:mud}.
In the tables, we usually identify the ``source'' of all values
entering ``our choice'' by the first letter of the (group of) authors
given in adjacent columns (e.g.\ ``B'' for Borie).
We denote as average ``avg.'' in the tables the center of the band covered by
all values $v_i$ under consideration,
with an uncertainty of half the spread, i.e.\
\begin{equation}
\label{eq:avg}
\begin{aligned}
\mathrm{avg.} ~ = & ~ &
\frac{1}{2}\,\big[ {\rm MAX}(v_i) + {\rm MIN}(v_i) \big] \\[1ex]
& \pm &
\frac{1}{2}\,\big[ {\rm MAX}(v_i) - {\rm MIN}(v_i) \big].
\end{aligned}
\end{equation}
If individual uncertainties are provided by the authors we add these in quadrature.
We would like to point out that uncertainties due to uncalculated higher order terms are
often not indicated explicitly by the authors. In the case some number is given, we include it in our sum. But in general our method can not account for uncertainty estimates of uncalculated higher order terms.
Throughout the paper,
$Z$ denotes the nuclear charge with $Z=2$ for the helion and alpha particle,
$\alpha$ is the fine structure constant,
$m_r = 199\,m_e$ is the reduced mass of the muon-nucleon system.
``VP'' is short for ``vacuum polarization'',
``SE'' is ``self-energy'',
``RC'' is ``recoil correction''.
``Perturbation theory'' is abbreviated as ``PT'', and SOPT and TOPT denote
$2^{\rm nd}$ and $3^{\rm rd}$ order perturbation theory, respectively.
\section{Lamb shift in muonic helium-3}
\label{sec:LS}
\subsection{Nuclear structure-independent contributions}
\label{sec:LS:QED}
Nuclear structure-independent contributions have been calculated by Borie, Martynenko group, Karshenboim group, and Jentschura. The contributions are listed in Tab.\,\ref{tab:LS:QED}, labeled with \#$i$.
The leading contribution is the one-loop electron vacuum polarization (eVP) of order $\alpha(Z\alpha)^2$, the so-called Uehling term (see Fig.\,\ref{fig:uehling}). It accounts for 99.5\% of the radius-independent part of the Lamb shift, so it is very important that this contribution is well understood. There are two different approaches to calculate this term.
\begin{figure}
\begin{center}
\begin{fmffile}{uehling}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{3}
\fmfbottomn{b}{3}
\fmf{plain,tension=1.0}{t1,t3}
\fmf{plain,width=3 }{b1,b3}
\fmf{photon}{t2,c1}
\fmf{photon}{c3,b2}
\fmfpoly{smooth, pull=?, tension=0.3}{c0,c1,c2,c3}
\fmffreeze
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfv{label.angle=180,label=$e$}{c2}
\end{fmfgraph*}
\end{fmffile}
\end{center}
\caption{Item \#1, the leading order 1-loop electron vacuum polarization
(eVP), also called Uehling term.}
\label{fig:uehling}
\end{figure}
Borie \cite{Borie:2014:arxiv_v7} (p.\,4, Tab.) and the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,I) use relativistic Dirac wavefunctions to calculate a relativistic Uehling term (item \#3).
A relativistic recoil correction (item \#19) has to be added to allow comparison to nonrelativistic calculations (see below).
Borie provides the value of this correction explicitly in \cite{Borie:2014:arxiv_v7} Tab.\,6, whereas the Karshenboim group only gives the total value which includes the correction, thus corresponding to ($\#3+\#19$).
Nonrelativistic calculations of the Uehling term (item \#1) exist from the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,1, Tab.\,1) and Jentschura \cite{Jentschura:2011:PRA84_012505}, which are in very good agreement. Additionally, a relativistic correction (item \#2) has to be applied. This relativistic correction already accounts for relativistic recoil effects (item \#19).
Item \#2 has been calculated by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,7+10, Tab.\,1), Borie \cite{Borie:2014:arxiv_v7} (Tab.\,1), Jentschura \cite{Jentschura:2011:PRA84_012505,Jentschura:2011:SemiAnalytic} (Eq.\,17), and Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509}, which agree well within all four groups, however do not have to be included in Borie's and Korzinin {\it et\,al.}'s value because their relativistic Dirac wavefunction approach already accounts for relativistic recoil effects.
Both approaches agree well within the required uncertainty. As \textit{our choice} for the Uehling term with relativistic correction ($\#1+\#2$) or ($\#3+\#19$) we take the average
\begin{equation}
\Delta E (\mathrm{Uehling\,+\,rel.~corr.}) = 1642.3962\pm0.0018\,\ensuremath{\mathrm{meV}}\xspace.
\end{equation}
Item \#4, the second largest contribution in this section, is the two-loop eVP of order $\alpha^2(Z\alpha)^2$, the so-called K\"all\'en-Sabry term \cite{KallenSabry:1955} (see Fig.\,\ref{fig:item_4}). It has been calculated by Borie \cite{Borie:2014:arxiv_v7} (p.\,4, Tab.) and the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,2, Tab.\,1) which agree within 0.0037\,\ensuremath{\mathrm{meV}}\xspace. As \textit{our choice} we take the average.
Item \#5 is the one-loop eVP in two Coulomb lines of order $\alpha^2(Z\alpha)^2$ (see Fig.\,\ref{fig:item_5}). It has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,6), the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,9, Tab.\,1), and Jentschura \cite{Jentschura:2011:SemiAnalytic} (Eq.\,13) of whom the latter two obtain the same result, which differs from Borie by 0.0033\,\ensuremath{\mathrm{meV}}\xspace. As \textit{our choice} we adopt the
average.
The Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,I) has calculated the sum of item \#4 and \#5, the two-loop eVP (K\"all\'en-Sabry) and one-loop eVP in two Coulomb lines (Fig.\,\ref{fig:item_4} and \ref{fig:item_5}).
Good agreement between all groups is observed.
\begin{figure
\begin{center}
\begin{minipage}{0.22\columnwidth}
(a)\hfill\hspace{5px}
\vspace{5px}\\
\begin{fmffile}{item_4}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain,tension=1.0}{t1,t7}
\fmf{plain,width=3 }{b1,b7}
\fmf{phantom,tension=0.0001}{t4,c1,x1,c2,c4,x2,c5,b4}
\fmf{photon,tension=0}{t4,c1}
\fmf{photon,tension=0}{c2,c4}
\fmf{photon,tension=0}{c5,b4}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmf{plain,left,tension=0}{c4,c5,c4}
\fmffreeze
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=180,label=$e$}{l1}
\fmfpoly{phantom}{c4,l3,c5,l4}
\fmfv{label.angle=180,label=$e$}{l3}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.22\columnwidth}
(b)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{item_4b}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{3}
\fmfbottomn{b}{3}
\fmf{plain}{t1,t2,t3}
\fmf{plain,width=3}{b1,b2,b3}
\fmf{photon}{t2,c0}
\fmf{photon}{c4,b2}
\fmfpoly{smooth,pull=?,tension=0.5}{c0,c1,c2,c3,c4,c5,c6,c7}
\fmffreeze
\fmf{photon}{c2,c6}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfv{label.angle=180,label=$e$}{c1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{1cm}
\begin{minipage}{0.22\columnwidth}
(c)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{item_4c}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{3}
\fmfbottomn{b}{3}
\fmf{plain}{t1,t2,t3}
\fmf{plain,width=3}{b1,b2,b3}
\fmf{photon}{t2,c0}
\fmf{photon}{c6,b2}
\fmfpoly{smooth,pull=?,tension=0.8}{c0,c1,c2,c3,c4,c5,c6,c7,c8,c9,c10,c11}
\fmffreeze
\fmf{photon}{c7,c11}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfv{label.angle=180,label=$e$}{c1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\vspace{-10px}
\caption{Item \#4, the two-loop eVP
(K\"allen-Sabry) contribution.
This is Fig.\,1 (b,c,d) from the Martynenko group~\cite{Krutov:2014:JETP120_73}.
}
\label{fig:item_4}
\end{figure}
\begin{figure
\begin{center}
\mbox{~}\vfill
\begin{fmffile}{item_5}
\begin{fmfgraph*}(100,70)
\fmfstraight
\fmftopn{t}{8}
\fmfbottomn{b}{8}
\fmf{plain,tension=1.0}{t1,t8}
\fmf{plain,width=3 }{b1,b8}
\fmf{phantom,tension=0.0001}{t3,c1,c2,b3}
\fmf{photon,tension=0}{t3,c1}
\fmf{photon,tension=0}{c2,b3}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmf{phantom,tension=0.0001}{t6,c3,c4,b6}
\fmf{photon,tension=0}{t6,c3}
\fmf{photon,tension=0}{c4,b6}
\fmf{plain,left,tension=0}{c3,c4,c3}
\fmffreeze
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=110,label.dist=10,label=$e$}{l1}
\fmfpoly{phantom}{c3,l3,c4,l4}
\fmfv{label.angle=100,label.dist=10,label=$e$}{l3}
\end{fmfgraph*}
\end{fmffile}
\end{center}
\caption{Item \#5, the one-loop eVP in 2-Coulomb lines.}
\label{fig:item_5}
\end{figure}
Item \#6+7 is the third order eVP of order $\alpha^3(Z\alpha)^2$. It has been calculated by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,$4+11+12$, Tab.\,1) and the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,I). Borie \cite{Borie:2014:arxiv_v7} (p.\,4) adopts the value from Karshenboim {\it et\,al.}. Martynenko {\it et\,al.}\ and Karshenboim {\it et\,al.}\ differ by $0.004\,\ensuremath{\mathrm{meV}}\xspace$, which is in agreement considering the uncertainty of $0.003\,\ensuremath{\mathrm{meV}}\xspace$ given by the Martynenko group. As \textit{our choice} we adopt the average and obtain an uncertainty of 0.0036\,\ensuremath{\mathrm{meV}}\xspace
via Gaussian propagation of uncertainty.
Item \#29 is the second order eVP of order $\alpha^2(Z\alpha)^4$. It has been calculated by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,$8+13$, Tab.\,1) and the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,VIII). Their values did agree in the case of \ensuremath{\mu}{\rm d}\xspace, however for \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace they differ by 0.004\,\ensuremath{\mathrm{meV}}\xspace.
This difference is twice as large as the value from Martynenko {\it et\,al.}\ but this contribution is small, so the uncertainty is not at all dominating. We reflect the difference by adopting the average as \textit{our choice}.
Items \#9, \#10, and \#9a are the terms of the Light-by-light (LbL) scattering contribution (see Fig.\,\ref{fig:lbl}). The sum of the LbL terms is calculated by the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,I). Borie \cite{Borie:2014:arxiv_v7} also lists the value from Karshenboim {\it et\,al.}.
Item \#9 is the \textit{Wichmann-Kroll} term, or ``1:3'' LbL, which is of order $\alpha(Z\alpha)^4$. This item has also been calculated by Borie \cite{Borie:2014:arxiv_v7} (p.\,4) and the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,5, Tab.\,1) who obtain the same result.
Item \#10 is the \textit{virtual Delbr\"uck} or ``2:2'' LbL, which is of order $\alpha^2(Z\alpha)^3$.
Item \#9a is the \textit{inverted Wichmann-Kroll} term, or ``3:1'' LbL, which is of order $\alpha^3(Z\alpha)^2$. The sum of the latter two is also given by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,6, Tab.\,1).
As \textit{our choice} we use the one from Karshenboim {\it et\,al.}, who are the first and only group to calculate all three LbL contributions. The groups are in agreement when taking into account the uncertainty of 0.0006\,\ensuremath{\mathrm{meV}}\xspace given by Karshenboim {\it et\,al.}.
\begin{figure
\begin{center}
\begin{minipage}{0.22\columnwidth}
(a)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{lblone}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftop{i1,v1,o1}
\fmfbottom{i2,v2,v3,v4,o2}
\fmf{plain}{i1,v1,o1}
\fmf{plain,width=3}{i2,v2,v3,v4,o2}
\fmf{photon}{v1,c0}
\fmf{photon}{c2,v3}
\fmfpoly{default, pull=?, tension=0.3}{c0,c1,c2,c3}
\fmffreeze
\fmf{photon}{c1,v2}
\fmf{photon}{c3,v4}
\fmfv{label.angle=180,label=$e$}{c1}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{25px}
\begin{minipage}{0.22\columnwidth}
(b)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{lbltwo}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftop{i1,v1,v2,o1}
\fmfbottom{i2,v3,v4,o2}
\fmf{plain}{i1,v1,v2,o1}
\fmf{plain,width=3}{i2,v3,v4,o2}
\fmf{photon}{v1,c0}
\fmf{photon}{v2,c3}
\fmf{photon}{v3,c1}
\fmf{photon}{v4,c2}
\fmfpoly{default, pull=?, tension=0.5}{c0,c1,c2,c3}
\fmfv{label.angle=120,label=$e$}{c1}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{25px}
\begin{minipage}{0.22\columnwidth}
(c)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{lblthree}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftop{i1,v1,v3,v4,o1}
\fmfbottom{i2,v2,o2}
\fmf{plain}{i1,v1,v3,v4,o1}
\fmf{plain,width=3}{i2,v2,o2}
\fmf{photon}{v3,c0}
\fmf{photon}{c2,v2}
\fmfpoly{default, pull=?, tension=0.3}{c0,c1,c2,c3}
\fmffreeze
\fmf{photon}{v1,c1}
\fmf{photon}{v4,c3}
\fmfv{label.angle=180,label=$e$}{c1}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\vspace{-10px}
\caption{The three contributions to Light-by-light scattering:
(a) Wichmann-Kroll or ``1:3'' term, item~\#9,
(b) Virtual Delbr\"uck or ``2:2'' term, item~\#10, and
(c) inverted Wichmann-Kroll or ``3:1'' term, item~\#9a$^\dagger$.}
\label{fig:lbl}
\end{figure}
Item \#20 is the contribution from muon self-energy ($\mu$SE) and muon vacuum polarization ($\mu$VP) of order
$\alpha(Z\alpha)^4$ (see Fig.\,\ref{fig:onephotonSE}). This item constitutes the third largest term in this section~\footnote{In ordinary hydrogen-like atoms this term is the leading order Lamb shift contribution: The leptons in the loop are the same as the orbiting lepton. This term can thus be rescaled from well-known results in hydrogen.}. This item has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,2, Tab.\,6) and the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,24, Tab.\,1). They differ by 0.001\,\ensuremath{\mathrm{meV}}\xspace. As \textit{our choice} we adopt the average.
\begin{figure
\begin{center}
\hfill~(a)\hspace{60px}~\hspace{25px}~(b)\hspace{60px}~\hfill\vspace{5px}\\
\begin{fmffile}{item_20}
\begin{fmfgraph*}(70,60)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain}{t1,t7}
\fmf{plain,width=3}{b1,b7}
\fmf{photon,tension=0.1,left=1.0}{t3,t5}
\fmf{photon}{t4,b4}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\hspace{40px}
\begin{fmfgraph*}(70,60)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain}{t1,t7}
\fmf{plain,width=3}{b1,b7}
\fmf{photon}{t4,c1}
\fmf{photon}{c2,b4}
\fmfpoly{smooth, pull=?, tension=0.3}{c1,l1,c2,l2}
\fmfv{label.angle=180,label=$\mu$}{l1}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\end{fmffile}
\end{center}
\caption{Item \#20, the muon-self energy (a) and the muon vacuum polarization (b), $\alpha(Z\alpha)^4$. }
\label{fig:onephotonSE}
\end{figure}
Items \#11, \#12, \#30, \#13, and \#31 are all corrections to VP or $\mu$SE and of order $\alpha^2(Z\alpha)^4$.
Item \#11 is the $\mu$SE correction to eVP (see Fig.\,\ref{fig:item_11}). It has been calculated by all four groups. Martynenko {\it et\,al.}\ calculate this term (Eq.\,99) in \cite{Krutov:2014:JETP120_73}, however in their table (No.\,28) they use the more exact calculation from Jentschura. Jentschura \cite{Jentschura:2011:SemiAnalytic} (Eq.\,29), and the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,VIII a) are in excellent agreement. Borie \cite{Borie:2014:arxiv_v7} (Tab.\,16) differs significantly because she only calculates a part of this contribution in her App.\,C. This value does not enter her sum and thus is also not considered in here. On p.\,12 of \cite{Borie:2014:arxiv_v7} she states that this value should be considered as an uncertainty. As \textit{our choice} we adopt the number from Jentschura and Karshenboim {\it et\,al.}.
\begin{figure
\begin{center}
\begin{minipage}{0.22\columnwidth}
(a)\hfill\mbox{~}
\vspace{8px}\\
\begin{fmffile}{item_11a}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain,tension=1.0}{t1,t7}
\fmf{plain,width=3 }{b1,b7}
\fmf{photon,tension=0.1,left=1.0}{t3,t5}
\fmf{phantom,tension=0.0001}{t4,c1,c2,b4}
\fmf{photon,tension=0}{t4,c1}
\fmf{photon,tension=0}{c2,b4}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmffreeze
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=180,label=$e$}{l1}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{25px}
\begin{minipage}{0.22\columnwidth}
(b)\hfill\mbox{~}
\vspace{8px}\\
\begin{fmffile}{item_11b}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain,tension=1.0}{t1,t7}
\fmf{plain,width=3 }{b1,b7}
\fmf{photon,tension=0.1,left=1.0}{t2,t4}
\fmf{phantom,tension=0.0001}{t5,c1,c2,b5}
\fmf{photon,tension=0}{t5,c1}
\fmf{photon,tension=0}{c2,b5}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmffreeze
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=180,label=$e$}{l1}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{25px}
\begin{minipage}{0.22\columnwidth}
(c)\hfill\mbox{~}
\vspace{8px}\\
\begin{fmffile}{item_11c}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain,tension=1.0}{t1,t7}
\fmf{plain,width=3 }{b1,b7}
\fmf{photon,tension=0.1,left=1.0}{t4,t6}
\fmf{phantom,tension=0.0001}{t3,c1,c2,b3}
\fmf{photon,tension=0}{t3,c1}
\fmf{photon,tension=0}{c2,b3}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmffreeze
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=180,label=$e$}{l1}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\vspace{-10px}
\caption{Item~\#11, muon self-energy corrections to the electron vacuum polarization
$\alpha^2 (Z\alpha)^4$.
This figure is Fig.\,2 from Jentschura~\cite{Jentschura:2011:AnnPhys1}.
It corresponds to Fig.~6(a) from Karshenboim~\cite{Korzinin:2013:PRD88_125019}.}
\label{fig:item_11}
\end{figure}
Item \#12 is the eVP in $\mu$SE (see Fig.\,\ref{fig:item_12}). This item has been calculated by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,27, Tab.\,1) and the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,VIII d), which are in perfect agreement. On p.\,10 of \cite{Borie:2014:arxiv_v7} Borie mentions that she included the ``fourth order electron loops'' in ``muon Lamb shift, higher order'' term, which is our item \#21. As we include item \#21 from Borie, we will not on top include item \#12.
\begin{figure
\begin{center}
%
%
\begin{minipage}{0.5\columnwidth}
\mbox{~}\vfill
\centering
\begin{fmffile}{Mart_fig11b}
\begin{fmfgraph*}(70,80)
\fmfstraight
\fmfleftn{i}{9}
\fmfrightn{o}{9}
\fmf{plain,tension=1.0}{i5,t1,t2,t3,o5}
\fmf{plain,width=3}{i1,b2,o1}
\fmf{photon,tension=0}{t2,b2}
\fmf{phantom}{i7,c1}
\fmf{phantom}{o7,c2}
\fmf{photon,tension=0,left=0.3}{t1,c1}
\fmf{photon,tension=0,left=0.3}{c2,t3}
\fmf{plain,left,tension=0.5}{c1,c2,c1}
\fmffreeze
\fmfv{label.angle=100,label.dist=10,label=$e$}{c1}
\fmfv{label=h}{i1}
\fmfv{label.angle=-150,label=$\mu$}{i5}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\caption{Item \#12, {\em eVP loop in SE} are radiative corrections with VP effects.
This is Fig.\,11(b) from a publication by the Martynenko group~\cite{Krutov:2014:JETP120_73} which
is the same as Fig.\,4 in Pachucki~\cite{Pachucki:1996:LSmup}.
It is Karshenboim's Fig.\,6(d) in Ref.~\cite{Korzinin:2013:PRD88_125019}.}
\label{fig:item_12}
\end{figure}
Item \#30 is the hadronic vacuum polarization (hVP) in $\mu$SE (see Fig.\,\ref{fig:item_30}). This item has only been calculated by the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,VIII e) which we adopt as \textit{our choice}.
\begin{figure
\begin{center}
\begin{minipage}{0.5\columnwidth}
\mbox{~}\vfill
\centering
\begin{fmffile}{item_30}
\begin{fmfgraph*}(70,80)
\fmfstraight
\fmfleftn{i}{9}
\fmfrightn{o}{9}
\fmf{plain,tension=1.0}{i5,t1,t2,t3,o5}
\fmf{plain,width=3}{i1,b2,o1}
\fmf{photon,tension=0}{t2,b2}
\fmf{phantom}{i7,c1}
\fmf{phantom}{o7,c2}
\fmf{photon,tension=0,left=0.3}{t1,c1}
\fmf{photon,tension=0,left=0.3}{c2,t3}
\fmf{dashes,left,tension=0.5}{c1,c2,c1}
\fmffreeze
\fmfv{label.angle=100,label.dist=10,label=$h$}{c1}
\fmfv{label=h}{i1}
\fmfv{label.angle=-150,label=$\mu$}{i5}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\caption{Item \#30, hadronic VP in SE contribution,
corresponds to Fig.~6(e) in Karshenboim {\it et\,al.}'s~\cite{Korzinin:2013:PRD88_125019}.}
\label{fig:item_30}
\end{figure}
Item \#13 is the mixed $e$VP + $\mu$VP (see Fig.\,\ref{fig:item_13}). The calculations from Borie \cite{Borie:2014:arxiv_v7} (p.\,4) and the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,3, Tab.\,1) roughly agree, whereas the value from the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,VIII b) is 0.002\,\ensuremath{\mathrm{meV}}\xspace larger. As \textit{our choice} we take the average.
\begin{figure
\begin{center}
\begin{minipage}{0.22\columnwidth}
(a)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{item_13a}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain,tension=1.0}{t1,t7}
\fmf{plain,width=3 }{b1,b7}
\fmf{phantom,tension=0.0001}{t4,c1,x1,c2,c4,x2,c5,b4}
\fmf{photon,tension=0}{t4,c1}
\fmf{photon,tension=0}{c2,c4}
\fmf{photon,tension=0}{c5,b4}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmf{plain,left,tension=0}{c4,c5,c4}
\fmffreeze
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=180,label=$e$}{l1}
\fmfpoly{phantom}{c4,l3,c5,l4}
\fmfv{label.angle=180,label=$\mu$}{l3}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{40px}
\begin{minipage}{0.35\columnwidth}
(b)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{item_13b}
\begin{fmfgraph*}(100,70)
\fmfstraight
\fmftopn{t}{8}
\fmfbottomn{b}{8}
\fmf{plain,tension=1.0}{t1,t8}
\fmf{plain,width=3 }{b1,b8}
\fmf{phantom,tension=0.0001}{t3,c1,c2,b3}
\fmf{photon,tension=0}{t3,c1}
\fmf{photon,tension=0}{c2,b3}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmf{phantom,tension=0.0001}{t6,c3,c4,b6}
\fmf{photon,tension=0}{t6,c3}
\fmf{photon,tension=0}{c4,b6}
\fmf{plain,left,tension=0}{c3,c4,c3}
\fmffreeze
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=110,label.dist=10,label=$e$}{l1}
\fmfpoly{phantom}{c3,l3,c4,l4}
\fmfv{label.angle=100,label.dist=10,label=$\mu$}{l3}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\vspace{-10px}
\caption{Item \#13, the mixed eVP-$\mu$VP contribution.}
\label{fig:item_13}
\end{figure}
Item \#31 is the mixed $e$VP + hVP (see Fig.\,\ref{fig:item_31}) which has only been calculated by the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,VIII c). We adopt their value as \textit{our choice}.
\begin{figure
\begin{center}
\begin{minipage}{0.22\columnwidth}
(a)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{item_31a}
\begin{fmfgraph*}(60,70)
\fmfstraight
\fmftopn{t}{7}
\fmfbottomn{b}{7}
\fmf{plain,tension=1.0}{t1,t7}
\fmf{plain,width=3 }{b1,b7}
\fmf{phantom,tension=0.0001}{t4,c1,x1,c2,c4,x2,c5,b4}
\fmf{photon,tension=0}{t4,c1}
\fmf{photon,tension=0}{c2,c4}
\fmf{photon,tension=0}{c5,b4}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmf{dashes,left,tension=0}{c4,c5,c4}
\fmffreeze
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=180,label=$e$}{l1}
\fmfpoly{phantom}{c4,l3,c5,l4}
\fmfv{label.angle=180,label=$h$}{l3}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{40px}
\begin{minipage}{0.35\columnwidth}
(b)\hfill\mbox{~}
\vspace{5px}\\
\begin{fmffile}{item_31b}
\begin{fmfgraph*}(100,70)
\fmfstraight
\fmftopn{t}{8}
\fmfbottomn{b}{8}
\fmf{plain,tension=1.0}{t1,t8}
\fmf{plain,width=3 }{b1,b8}
\fmf{phantom,tension=0.0001}{t3,c1,c2,b3}
\fmf{photon,tension=0}{t3,c1}
\fmf{photon,tension=0}{c2,b3}
\fmf{plain,left,tension=0}{c1,c2,c1}
\fmf{phantom,tension=0.0001}{t6,c3,c4,b6}
\fmf{photon,tension=0}{t6,c3}
\fmf{photon,tension=0}{c4,b6}
\fmf{dashes,left,tension=0}{c3,c4,c3}
\fmffreeze
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\fmfpoly{phantom}{c1,l1,c2,l2}
\fmfv{label.angle=110,label.dist=10,label=$e$}{l1}
\fmfpoly{phantom}{c3,l3,c4,l4}
\fmfv{label.angle=100,label.dist=10,label=$h$}{l3}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\vspace{-10px}
\caption{Item \#31, the mixed eVP- and hadronic VP contribution,
comes from the Uehling correction to the hadronic VP correction.
See Fig.~6(c) in Karshenboim {\it et\,al.}'s~\cite{Korzinin:2013:PRD88_125019}.}
\label{fig:item_31}
\end{figure}
Item \#32, the muon VP in SE correction shown in Fig.\,\ref{fig:item_32} is not included as a separate item in our Tab.\,\ref{tab:LS:QED}. It should already be automatically included in the QED contribution which has been rescaled from the QED of electronic $^3$He$^+$ by a simple mass replacement $m_e\rightarrow m_\mu$ \cite{Karshenboim:PC:2015}.
This is the case only for QED contributions where the particle in the loop is the same as the bound particle - like in this case, a muon VP correction in a muonic atom. The size of this item \#32 can be estimated from the relationship found by Borie \cite{Borie:1981:HVP}, that the ratio of hadronic to muonic VP is 0.66. With the Karshenboim group's value of item \#30 \cite{Korzinin:2013:PRD88_125019} one would obtain a value for item \#32 of $-0.0004/0.66\,\ensuremath{\mathrm{meV}}\xspace = -0.0006\,\ensuremath{\mathrm{meV}}\xspace$. This contribution is contained in our item \#21, together with the dominating item \#12 (see also p.\,10 of Ref.\,\cite{Borie:2014:arxiv_v7}).
\begin{figure
\begin{center}
\begin{fmffile}{item_32}
\begin{fmfgraph*}(70,70)
\fmfstraight
\fmfleftn{i}{9}
\fmfrightn{o}{9}
\fmf{plain,tension=1.0}{i5,t1,t2,t3,o5}
\fmf{plain,width=3}{i1,b2,o1}
\fmf{photon,tension=0}{t2,b2}
\fmf{phantom}{i7,c1}
\fmf{phantom}{o7,c2}
\fmf{photon,tension=0,left=0.3}{t1,c1}
\fmf{photon,tension=0,left=0.3}{c2,t3}
\fmf{plain,left,tension=0.5}{c1,c2,c1}
\fmffreeze
\fmfv{label.angle=100,label.dist=10,label=$\mu$}{c1}
\fmfv{label=h}{i1}
\fmfv{label.angle=-150,label=$\mu$}{i5}
\end{fmfgraph*}
\end{fmffile}
\end{center}
\caption{Item \#32, muon VP in SE contribution, is automatically included
in a rescaled electronic $^3$He$^+$ QED value of higher order SE contributions
(see text).}
\label{fig:item_32}
\end{figure}
Item \#21 is a higher-order correction to $\mu$SE and $\mu$VP of order $\alpha^2(Z\alpha)^4$ and $\alpha^2(Z\alpha)^6$. This item has only been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,2, Tab.\,6). On p.\,10 she points out that this contribution includes the ``fourth order electron loops'', which is our item \#12. It also contains our item \#32. We adopt her value as \textit{our choice}.
Item \#14 is the hadronic VP of order $\alpha(Z\alpha)^4$.
It has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,6) and the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,29, Tab.\,1).
Borie assigns a 5\% uncertainty to their value. However, in her Ref.\,\cite{Borie:2014:arxiv_v7} there are two different values of item \#14, the first on p.\,5 (0.219\,\ensuremath{\mathrm{meV}}\xspace) and the second in Tab.\,6 on p.\,16 (0.221\,\ensuremath{\mathrm{meV}}\xspace). Regarding the given uncertainty this difference is not of interest. In our Tab.\,\ref{tab:LS:QED}, we report the larger value which is further from that of the Martynenko group in order to conservatively reflect the scatter.
Martynenko {\it et\,al.}\ did not assign an uncertainty to their value. However, for \ensuremath{\mu}{\rm d}\xspace \cite{Krutov:2011:PRA84_052514} they estimated an uncertainty of 5\%. As \textit{our choice} we take the average of their values and adopt the uncertainty of 5\% (0.011\,\ensuremath{\mathrm{meV}}\xspace).
Item \#17 is the Barker-Glover correction \cite{Barker:1955}. It is a recoil correction of order $(Z\alpha)^4m_r^3/M^2$ and includes the nuclear Darwin-Foldy term that arises due to the Zitterbewegung of the nucleus. As already discussed in App.\,A of \cite{Krauth:2016:mud}, we follow the atomic physics convention \cite{Jentschura:2011:DF}, which is also adopted by CODATA in their report from 2010 \cite{Mohr:2012:CODATA10} and 2014 \cite{Mohr:2016:CODATA14}. This convention implies that item \#17 is considered as a recoil correction to the energy levels and not as a part of the rms charge radius.
This term has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,6), the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,21, Tab.\,1), and Jentschura \cite{Jentschura:2011:PRA84_012505} and \cite{Jentschura:2011:SemiAnalytic} (Eq.\,A.3). As \textit{our choice} we use the number given by Borie and Jentschura as they give one more digit.
Item \#18 is the term called ``recoil, finite size'' by Borie. It is of order $(Z\alpha)^5\langle r \rangle_{(2)}/M$ and is linear in the \textit{first} Zemach moment. It has first been calculated by Friar \cite{Friar:1978:Annals} (see Eq.\,F5 in App.\,F) for hydrogen and has later been given by Borie \cite{Borie:2014:arxiv_v7} for \ensuremath{\mu}{\rm d}\xspace, \ensuremath{\mu^4}{\rm He}\ensuremath{^+}\xspace, and \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace. We discard item \#18 because it is considered to be included in the elastic TPE \cite{Pachucki:PC:2015,Yerokhin:2016:RCFS}. It has also been discarded in \ensuremath{\mu}{\rm p}\xspace \cite{Antognini:2013:Annals}, \ensuremath{\mu}{\rm d}\xspace \cite{Krauth:2016:mud}, and \ensuremath{\mu^4}{\rm He}\ensuremath{^+}\xspace \cite{Diepold:2016:muHe4theo}. For the muonic helium-3 ion, item \#18 in \cite{Borie:2014:arxiv_v7} (Tab.\,6) amounts to 0.4040\,\ensuremath{\mathrm{meV}}\xspace, which is five times larger than the experimental uncertainty of about 0.08\,\ensuremath{\mathrm{meV}}\xspace (see Eq.~\ref{eq:uncertainty}), so it is important that the treatment of this contribution is well understood.
Item \#22 and \#23 are relativistic recoil corrections of order $(Z\alpha)^5$ and $(Z\alpha)^6$, respectively.
Item \#22 has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,6), the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,22, Tab.\,1), and Jentschura \cite{Jentschura:2011:SemiAnalytic} (Eq.\,32). They agree perfectly.
Item \#23 has only been calculated by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,23, Tab.\,1) whose value we adopt as \textit{our choice}.
Item \#24 are higher order radiative recoil corrections of order $\alpha(Z\alpha)^5$ and $(Z^2\alpha)(Z\alpha)^4$. This item has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,6) and the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,25, Tab.\,1). Their values differ by 0.015\,\ensuremath{\mathrm{meV}}\xspace. As \textit{our choice} we adopt the average.
Item \#28 is the radiative (only eVP) recoil of order $\alpha(Z\alpha)^5$. It consists of three terms which have been calculated by Jentschura and Wundt \cite{Jentschura:2011:SemiAnalytic} (Eq.\,46). We adopt their value as \textit{our choice}. Note that a second value (0.0072\,\ensuremath{\mathrm{meV}}\xspace) is found in \cite{Jentschura:2011:PRA84_012505}. However, this value is just one of the three terms, namely the seagull term, and is already included in \#28 (see \cite{Jentschura:2011:SemiAnalytic}, Eq.\,46).
The total sum of the QED contributions without explicit nuclear structure dependence is summarized in Tab.\,\ref{tab:LS:QED} and amounts to
\begin{equation}\label{eq:LS:QED}
\ensuremath{\Delta E_\mathrm{r\mathrm{-indep.}}^\mathrm{LS}} = \ensuremath{1644.3466}\pm\ensuremath{ 0.0146}\,\ensuremath{\mathrm{meV}}\xspace.
\end{equation}
Note that Borie, on p.\,15 in Ref.\,\cite{Borie:2014:arxiv_v7} attributes an uncertainty of 0.6\,\ensuremath{\mathrm{meV}}\xspace to her total sum.
The origin of this number remains unclear \cite{Borie:PC:2017}.
Its order of magnitude is neither congruent with the other uncertainties given in Ref.\,\cite{Borie:2014:arxiv_v7} nor with other uncertainties collected in our summary. Thus it will not be taken into account.
\subsection{Nuclear structure contributions}
\label{sec:LS:nuclstruc}
Terms that depend on the nuclear structure are separated into one-photon exchange (OPE) contributions and two-photon exchange (TPE) contributions.
The OPE terms (also called \textit{radius-dependent contributions}) represent the finite size effect which is by far the largest part of the nuclear structure contributions and are discussed in Sec.\,\ref{sec:LS:Radius}. They are parameterizable with a coefficient times the rms charge radius squared. These contributions are QED interactions with nuclear form factor insertions.
The TPE terms can be written as a sum of elastic and inelastic terms, where the latter describe the polarizability of the nucleus. These involve contributions from strong interaction and therefore are much more complicated to evaluate, which explains why the dominant uncertainty originates from the TPE part. The TPE contributions are discussed in more detail in Sec.\,\ref{sec:LS:Pol}.
The main nuclear structure corrections to the $n$S states have been given up to order $(Z\alpha)^6$ by Friar \cite{Friar:1978:Annals} (see Eq.\,(43a) therein)
\begin{widetext}
\begin{equation}\label{app:eq:friar}
\Delta E_{\rm fin. size} = \frac{2\pi Z\alpha}{3}|\Psi(0)|^2\left( \langle r^2\rangle - \frac{Z\alpha m_r}{2} \langle r^3\rangle_{(2)} + (Z\alpha)^2(F_{\rm REL} + m_r^2F_{\rm NREL}) \right),
\end{equation}
\end{widetext}
where $\Psi(0)$ is the muon wave function at the origin, $\langle r^2\rangle$ is the second moment of the charge distribution of the nucleus, i.e.\ the square of the rms charge radius, $\ensuremath{r_E} ^2$.
$\langle r^3\rangle_{(2)}$ is the Friar moment~\footnote{\label{footnote:friar}$\langle r^3\rangle_{(2)}$ has been called ``third Zemach moment'' in \cite{Friar:1978:Annals}. To avoid confusion with the Zemach radius \ensuremath{r_Z} in the 2S hyperfine structure we adopt the term ``Friar moment'', as recently suggested by Karshenboim {\it et\,al.}~\cite{Karshenboim:2015:PRD91_073003}.}, and $F_{\rm REL}$ and $F_{\rm NREL}$ contain various moments of the nuclear charge distribution (see Eq.\,(43b) and (43c) in Ref.\,\cite{Friar:1978:Annals}). Analytic expressions for some simple model charge distributions are listed in App.\,E of Ref.\,\cite{Friar:1978:Annals}.
As the Schr\"odinger wavefunction at the origin $\Psi(0)$ is nonzero only for S states, it is in leading order only the S states which are affected by the finite size. However, using the Dirac wavefunction a nonzero contribution appears for the \ensuremath{2\textrm{P}_{1/2}}\xspace level \cite{Ivanov:2001:LS}. This contribution affects the values for the Lamb shift and the fine structure and is taken into account in the section below.
The Friar moment $\langle r^3\rangle_{(2)}$ has not been included in \ensuremath{\mu}{\rm d}\xspace \cite{Krauth:2016:mud} because of a cancellation \cite{Friar:1997:PRA56_5173,Pachucki:2011:PRL106_193007,Friar:2013:PRC88_034004} with a part of the inelastic nuclear polarizability contributions. The TRIUMF-Hebrew group pointed out \cite{NevoDinur:2016:TPE,Hernandez:2016:POLupdate}, that in the case of \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace however, a smaller uncertainty might be achieved treating each term separately. This discussion is not finished yet and we will therefore continue with the more conservative treatment as before. See Sec.\,\ref{sec:LS:Pol}.
\begin{landscape}
\begin{table}
\begin{minipage}{\linewidth}
\renewcommand{\baselinestretch}{1.1}
\renewcommand{\arraystretch}{1.5}
\caption[Nuclear structure-independent contributions to the Lamb shift]{
All known {\bf nuclear structure-independent} contributions to the
Lamb shift in \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace. Values are in meV.
Item numbers ``\#'' in the 1st column follow the nomenclature of
Refs.~\cite{Antognini:2013:Annals,Krauth:2016:mud}, which in turn follow the supplement
of Ref.~\cite{Pohl:2010:Nature_mup1}.
Items ``\#`` with a dagger $^\dagger$ were labeled ``New'' in Ref.~\cite{Antognini:2013:Annals},
but we introduced numbers in Ref.\,\cite{Krauth:2016:mud} for definiteness.
For Borie~\cite{Borie:2014:arxiv_v7} we refer to the most recent
arXiv version-7 which contains several corrections
to the published paper~\cite{Borie:2012:LS_revisited_AoP}
(available online 6 Dec.\ 2011).
For the Martynenko group, numbers \#1 to \#29 refer to rows in Tab.~I of
Ref.~\cite{Krutov:2014:JETP120_73}.
Numbers in parentheses refer to equations in the respective paper.
}
\label{tab:LS:QED}
\setlength\tabcolsep{1mm}
\setlength{\extrarowheight}{0.2mm}
\centering
\fontsize{6pt}{6pt}\selectfont
\hspace*{-6mm
\begin{tabular}{ c | l |f{7} l | f{5} l | f{6} l | f{4}@{\qquad}l | f{4}@{ }f{5} c c}
\hline
\hline
\# & Contribution & \cntl{2}{Borie (B)}
& \cntl{2}{Martynenko group (M)}
& \cntl{2}{Jentschura (J)}
& \cntl{2}{Karshenboim group (K)}
& \cnt{3}{Our choice} \\
& & \cntl{2}{\cite{Borie:2014:arxiv_v7}}
& \cntl{2}{Krutov {\it et\,al.}~\cite{Krutov:2014:JETP120_73}}
& \cntl{2}{Jentschura, Wundt \cite{Jentschura:2011:SemiAnalytic}}
& \cntl{2}{Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509}}
& \cnt{2}{value} & source & Fig. \\
& &&
&&
& \cntl{2}{Jentschura \cite{Jentschura:2011:PRA84_012505}}
& \cntl{2}{Korzinin {\it et\,al.}~\cite{Korzinin:2013:PRD88_125019}}
& \\
\hline
1 & NR one-loop electron VP (eVP) & &
& 1641.8862 & \#
& 1641.885 & \cite{Jentschura:2011:PRA84_012505}
& &
& &
& & \\
2 & Rel.\ corr.\ (Breit-Pauli) & (0.50934)~\footnote{Does not contribute to the sum in Borie's approach.}
& Tab.\,1
& 0.509
& \#7+\#10
& 0.509344 & \cite{Jentschura:2011:SemiAnalytic}(17),
\cite{Jentschura:2011:PRA84_012505}
& (0.509340) & \cite{Karshenboim:2012:PRA85_032509} Tab.\,IV
& &
& & \\
3 & Rel.\ one-loop eVP & 1642.412 & Tab. p.\,4
& &
& &
& &
& &
& & \\
19 & Rel.\ RC to eVP, $\alpha(Z\alpha)^4$ & -0.0140 & Tab.\,1+6
& &
& &
& &
& & & & \\
& Sum of the above & 1642.3980 & 3+19
& 1642.3955 & 1+2
& 1642.3943 & 1+2
& 1642.3954 & \cite{Korzinin:2013:PRD88_125019} Tab.\,I
&1642.3962 &~ \pm~0.0018
& avg & \ref{fig:uehling} \\
\hline
4 & Two-loop eVP (K\"all$\acute{\mathrm{e}}$n-Sabry) & 11.4107 & Tab. p.\,4
& 11.4070 & \#2
& &
& &
& 11.4089 &\pm~0.0019 & avg. & \ref{fig:item_4} \\
5 & One-loop eVP in 2-Coulomb lines $\alpha^2(Z\alpha)^2$ & 1.674 & Tab.\,6
& 1.6773 & \#9
& 1.677290 & \cite{Jentschura:2011:SemiAnalytic}(13)
& &
& 1.6757&~\pm~0.0017
& avg. & \ref{fig:item_5} \\ %
& Sum of 4 and 5 & 13.0847 & 4+5
& 13.0843 & 4+5
& &
& 13.0843 & \cite{Korzinin:2013:PRD88_125019} Tab.\,I
& (13.0846)\footnote{Sum of \textit{our choice} of item \#4 and \#5, written down for comparison with the Karshenboim group.}
&
& & \\
\hline
6+7& Third order VP & 0.073(3) & p.\,4
& 0.0689 & \#4+\#12+\#11
& &
& 0.073(3) & \cite{Korzinin:2013:PRD88_125019} Tab.\,I
& 0.0710 & \pm~0.0036
& avg. \\
\hline
29 & Second-order eVP contribution $\alpha^2(Z\alpha)^4 m $ & &
& 0.0018 & \#8+\#13
& &
& 0.00558 & \cite{Korzinin:2013:PRD88_125019} Tab.\,VIII ``eVP2''
& 0.0037 & ~\pm~ 0.0019
& avg\\
\hline
9 & Light-by-light ``1:3'': Wichmann-Kroll & -0.01969 & p.\,4
& -0.0197 & \#5
& &
& &
& &
& & \ref{fig:lbl}a \\
10 & Virtual Delbr\"{u}ck, ``2:2'' LbL & &
& \multicolumn{1}{l}{\multirow{2}{1mm}{~\,$\left.\rule{0pt}{3ex}\right\}0.0064$}}
& \multirow{2}{0mm}{\#6}
& &
&
&
&
&
& \ref{fig:lbl}b\\
9a$^\dagger$ & ``3:1'' LbL & &
& &
& &
& &
& &
& & \ref{fig:lbl}c\\
& Sum: Total light-by-light scatt. & -0.0134(6) & p.5+Tab.6
& -0.0133 & 9+10+9a
& &
& -0.0134(6) & \cite{Korzinin:2013:PRD88_125019} Tab.\,I
& -0.0134 & \pm~0.0006
& K \\
\hline
20 & $\mu$SE and $\mu$VP & -10.827368 & Tab.\,2+6
& -10.8286 & \#24
& &
& &
& -10.8280 &\pm~ 0.0006
& avg. &\ref{fig:onephotonSE}\\
11 & Muon SE corr.\ to eVP $\alpha^2(Z\alpha)^4$ & (-0.1277)~\footnote{In App.\,C of \cite{Borie:2014:arxiv_v7}, incomplete. Does not contribute to the sum in Borie's approach, see text.}
& Tab.\,16
& -0.0627 & \#28
& -0.06269 & \cite{Jentschura:2011:SemiAnalytic}(29)
& -0.06269 & \cite{Korzinin:2013:PRD88_125019} Tab.\,VIII (a
& -0.06269 &
&J, K & \ref{fig:item_11}\\
\hline
12 & eVP loop in self-energy $\alpha^2(Z\alpha)^4$ \quad
& \cnt{1}{ incl. in 21} &
& -0.0299 & \#27
& &
& -0.02992 & \cite{Korzinin:2013:PRD88_125019} Tab.\,VIII (d)
& \cnt{1}{incl. in 21} &
& B & \ref{fig:item_12} \\
30 & Hadronic VP loop in self-energy $\alpha^2(Z\alpha)^4 m$
& &
& &
& &
& -0.00040(4) & \cite{Korzinin:2013:PRD88_125019} Tab.\,VIII (e)
& -0.00040 &\pm~0.00004
& K & \ref{fig:item_30} \\
13 & Mixed eVP + $\mu$VP & 0.00200 & p.\,4
& 0.0022 & \#3
& &
& 0.00383 & \cite{Korzinin:2013:PRD88_125019} Tab.\,VIII (b)
& 0.0029 & \pm ~0.0009
& avg & \ref{fig:item_13} \\
31 & Mixed eVP + hadronic VP & &
& &
& &
& 0.0024(2) & \cite{Korzinin:2013:PRD88_125019} Tab.\,VIII (c)
& 0.0024 &\pm ~0.0002
& K & \ref{fig:item_31} \\
21 & Higher-order corr.\ to $\mu$SE and $\mu$VP & -0.033749 & Tab.\,2+6
& &
& &
& &
& -0.033749 &
& B\\
& Sum of 12, 30, 13, 31, and 21 & -0.031749 & 13+21
& -0.0277 & 12+13
& &
& -0.0241(2) & 12+30+13+31
& -0.0288 &
& sum\\
\hline
14 & Hadronic VP
& 0.221(11) & Tab.\,6
& 0.2170 & \#29
& &
& &
& 0.219 & \pm ~ 0.011
& avg. \\
\hline
17 & Recoil corr.\ $(Z\alpha)^4m_r^3/M^2$ (Barker-Glover) & 0.12654 & Tab.\,6
& 0.1265 & \#21
& 0.12654 & \cite{Jentschura:2011:SemiAnalytic}(A.3) \cite{Jentschura:2011:PRA84_012505}(15)
& &
& 0.12654 &
& B, J \\
18 & Recoil, finite size & (0.4040(10))~\footnote{Is not included, because it is a part of the TPE, see text.} &
& &
& &
& &
& &
& \\
22 & Rel.\ RC $(Z\alpha)^5$ & -0.55811 & p.9+Tab.6
& -0.5581 & \#22
& -0.558107 & \cite{Jentschura:2011:SemiAnalytic}(32)
& &
& -0.558107 &
& J \\
23 & Rel.\ RC $(Z\alpha)^6$ & &
& 0.0051 & \#23
& &
& &
& 0.0051 &
& M \\
24 & Higher order radiative recoil corr. & -0.08102 & p.9+Tab.6
& -0.0656 & \#25
& &
& &
& -0.0733 & \pm ~0.0077
& avg.\\
28$^\dagger$ & Rad.\ (only eVP) RC $\alpha(Z\alpha)^5$ & &
& &
& 0.004941 &
& &
& 0.004941 &
&J \\
\hline
\hline
&&&&&&&&&&&&&\\[-3ex]
& \bf Sum & \cntl{2}{1644.3916~\footnote{Including item \#18 and \#r3' yields 1644.9169\,meV, which is Borie's value from Ref.~\cite{Borie:2014:arxiv_v7} page 15. On that page she attributes an uncertainty of 0.6\,\ensuremath{\mathrm{meV}}\xspace to that value. This number is far too large to be correct, so we ignore it.}
}
& \cntl{2}{$1644.3431$
}
& &
& &
& {\ensuremath{\bf 1644}}\bf .{\ensuremath{\bf 3466}} & \pm~{\ensuremath{\bf 0}}\bf .{\ensuremath{\bf 0146}}
& \\ [-3ex]
&&&&&&&&&&&&&\\
\hline
\hline
\end{tabular}
\end{minipage}
\end{table}
\end{landscape}
\subsubsection{One-photon exchange contributions (finite size effect)}
\label{sec:LS:Radius}
Finite size contributions have been calculated by Borie (\cite{Borie:2014:arxiv_v7} Tab.\,14), the Martynenko group (\cite{Krutov:2014:JETP120_73} Tab.\,1), and the Karshenboim group (\cite{Karshenboim:2012:PRA85_032509} Tab.\,III). All of these contributions are listed in Tab.\,\ref{tab:LS:Radius}, labeled with \#r$i$.
Most of the terms, given in Tab.\,\ref{tab:LS:Radius}, can be parameterized as $c \cdot \ensuremath{{r_E}^2} $
with coefficients $c$ in units of meV\,\,\ensuremath{\mathrm{fm}^{-2}}\xspace. Borie and Karshenboim {\it et\,al.}\ have provided the contributions in this parameterization, whereas Martynenko {\it et\,al.}\ provide the total value in units of energy. However, the value of their coefficients can be obtained by dividing their numbers by \ensuremath{{r_E}^2} . The value they used for the charge radius \ensuremath{r_E} is 1.9660\,fm~\footnote{This value has been introduced by Borie \cite{Borie:2014:arxiv_v7} as an average of several previous measurements \cite{Sick:2008:rad_scatt,Rooij:2011:HeSpectroscopy,CancioPastor:2012:PRL108}.} \cite{Martynenko:PC:2016}. In this way the numbers from Martynenko {\it et\,al.}\ can be compared with the ones from the other groups.
Item \#r1, the leading term of Eq.\,(\ref{app:eq:friar}), is the one-photon exchange with a helion form factor (FF) insertion (see Fig.\,\ref{fig:OPE}). Item \#r1 is of order $(Z\alpha)^4m_r^3$ and accounts for 99\% of the OPE contributions. Borie (\cite{Borie:2014:arxiv_v7} Tab.\,14, $b_a$), the Martynenko group (\cite{Krutov:2014:JETP120_73} No.\,14), and the Karshenboim group (\cite{Karshenboim:2012:PRA85_032509} Tab.\,III, $\Delta_{FNS}^{(0)}$) obtain the same result which we adopt as \textit{our choice}. This contribution is much larger than the following terms, but its absolute precision is worse, which we indicate by introducing an uncertainty. For that we take the value from Borie which is given with one more digit than the values of the other authors and attribute an uncertainty of 0.0005\,\ensuremath{\mathrm{meV}}\xspace, which may arise from rounding.
\begin{figure
\begin{center}
\begin{fmffile}{ope}
\begin{fmfgraph*}(70,70)
\fmftop{i1,o1}
\fmfbottom{i2,o2}
\fmf{plain,tension=1.0}{i1,v1,o1}
\fmf{plain,width=3}{i2,v2,o2}
\fmf{photon,tension=0}{v1,v2}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{v2}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{center}
\caption{
Item \#r1, the leading nuclear finite size correction
stems from a one-photon interaction with a helion form factor insertion,
indicated by the thick dot.}
\label{fig:OPE}
\end{figure}
Item \#r2 and \#r2' are the radiative correction of order $\alpha(Z\alpha)^5$. The equation used for the calculation of item \#r2 is given in Eq.\,(10) of \cite{Eides:1997:two-photon}. It has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,14, $b_b$) and the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,26, only Eq.\,(92)). Note that the value from the Martynenko group was published with a wrong sign.\\
Very recently the Martynenko group updated their calculation of higher-order finite size corrections \cite{Faustov:2017:rad_fin_size} using more realistic, measured nuclear form factors. The results contain a coefficient (in our work termed item \#r2) which agrees with the old value, and an additional, previously unkown term which cannot be parametrized with $\rh^2$ and therefore is given as a constant. This constant is found in our Tab.\,\ref{tab:LS:Radius} as item \#r2'. In Ref.\,\cite{Faustov:2017:rad_fin_size} the values are given for the 1S state but can easily be transferred to the 2S state via the $1/n^3$ scaling. For the 2S state this results in
\begin{equation}
\begin{split}
1/8\times(&-0.6109) \,\ensuremath{\mathrm{meV}}\xspace\\
=& ~1/8\times(-0.1946\,\rh^2 + 0.1412) \,\ensuremath{\mathrm{meV}}\xspace\\
=& ~-0.0243\,\ensuremath{\mathrm{meV}}\xspace/\,\ensuremath{\mathrm{fm}}\xspace^2 \rh^2 + 0.0177\,\ensuremath{\mathrm{meV}}\xspace.
\end{split}
\end{equation}
Borie and Martynenko get the same result for item \#r2, which we adopt as \textit{our choice}. Additionally we adopt the constant term from Martynenko as item \#r2'.
Item \#r3 and \#r3' are the finite size corrections of order $(Z\alpha)^6$. They have first been calculated in Ref.\,\cite{Friar:1978:Annals}. Item \#r3 and \#r3' consider third-order perturbation theory in the finite size potential correction and relativistic corrections of the Schr\"odinger wave functions. There are also corrections in the TPE of the same order $(Z\alpha)^6$, but these are of different origin.
Borie \cite{Borie:2014:arxiv_v7} (Tab.\,14, $b_c$ and Tab.\,6) and the Martynenko group \cite{Krutov:2014:JETP120_73} (Eq.\,(91)) follow the procedure in Ref.\,\cite{Friar:1978:Annals} and then separate their terms into a part with an explicit \ensuremath{{r_E}^2} dependence (item \#r3) and another one which is usually evaluated with an exponential charge distribution, since a model independent calculation of this term is prohibitively difficult \cite{Borie:2014:arxiv_v7}.
Differences in sorting the single terms have already been noticed in the \ensuremath{\mu}{\rm d}\xspace case \cite{Krauth:2016:mud}, where we mentioned that e.g.~the term $\ensuremath{\langle r^2 \rangle} \langle \ln(\mu r)\rangle$ in $F_{\rm REL}$ of Eq.\,\ref{app:eq:friar} is attributed to \#r3 and \#r3' by Martynenko {\it et\,al.}\ and Borie, respectively. The difference in this case amounts to 0.007\,\ensuremath{\mathrm{meV}}\xspace for \#r3'. Note that in Eq.\,(91) from the Martynenko group \cite{Krutov:2014:JETP120_73}, the charge radius has to be inserted in units of GeV$^{-1}$, with $\ensuremath{r_E} = 1.966\,\ensuremath{\mathrm{fm}}\xspace\,\widehat{=}\,9.963\,\mathrm{GeV}^{-1}$.
Item \#r4 is the one-loop $e$VP correction (\textit{Uehling}) of order $\alpha(Z\alpha)^4$. It has been calculated by all three groups, Borie \cite{Borie:2014:arxiv_v7} (Tab.\,14, $b_d$), Martynenko {\it et\,al.}~\cite{Krutov:2014:JETP120_73} (No.\,16, Eq.\,(69)), and Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509} (Tab.\,III, $\Delta E_{FNS}^{(2)}$).
On p.\,31 of \cite{Borie:2014:arxiv_v7}, Borie notes that she included the correction arising from the K\"all\'en-Sabry potential in her $b_d$. This means that her value already contains item \#r6, which is the two-loop $e$VP correction of order $\alpha^2(Z\alpha)^4$.
Item \#r6 has been given explicitly only by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,18, Eq.\,73). The sum of Martynenko {\it et\,al.}'s \#r4 and \#r6 differs by 0.016\,\ensuremath{\mathrm{meV}}\xspace/fm$^2$ from Borie's result. Using a charge radius of 1.9660\,fm this corresponds to roughly 0.06\,\ensuremath{\mathrm{meV}}\xspace and, hence, causes the largest uncertainty in the radius-dependent OPE part. The origin of this difference is not clear \cite{Borie:PC:2017,Martynenko:PC:2017}. A clarification of this difference is desired but does not limit the extraction of the charge radius.
As \textit{our choice} we take the average of the sum (\#r4+\#r6) of these two groups. The resulting average does also reflect the value for \#r4 provided by Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509}.
Item \#r5 is the one-loop $e$VP correction (\textit{Uehling}) in second order perturbation theory (SOPT) of order $\alpha(Z\alpha)^4$. It has been calculated by all three groups, Borie \cite{Borie:2014:arxiv_v7} (Tab.\,14, $b_e$), the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,17, Eq.\,70), and the Karshenboim group \cite{Karshenboim:2012:PRA85_032509} (Tab.\,III, $\Delta E_{FNS}^{(1)}$).
On p.\,31 of \cite{Borie:2014:arxiv_v7}, Borie notes that she included the two-loop corrections to $\epsilon_{VP2}$ in her $b_e$.
This means that her value already contains item \#r7, which is the two-loop $e$VP in SOPT of order $\alpha^2(Z\alpha)^4$.
Item \#r7 has only been given explicitly by the Martynenko group \cite{Krutov:2014:JETP120_73} (No.\,19). The sum of Martynenko {\it et\,al.}'s \#r5+\#r7 differs by 0.003\,\ensuremath{\mathrm{meV}}\xspace from Borie's result. As \textit{our choice} we take the average of the sum (\#r5+\#r7) of these two groups.
Again here, {\it our choice} reflects the value for \#r5 provided by Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509}, too.
Item \#r8 is the finite size correction to the \ensuremath{2\textrm{P}_{1/2}}\xspace level of order $(Z\alpha)^6$. It has only been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,14, $b(2p_{1/2}$). This correction is the smallest in this section and is the only term which affects the \ensuremath{2\textrm{P}_{1/2}}\xspace level. In consequence, the effect on the Lamb shift is inverse, i.e.~if the 2P level is lifted ``upwards'', the Lamb shift gets larger. Thus, in contrast to Borie, we include this correction with a positive sign. At the same time this term decreases the fine structure ($\ensuremath{2\textrm{P}_{3/2}}\xspace - \ensuremath{2\textrm{P}_{1/2}}\xspace$ energy difference) and is hence listed in Tab.\,\ref{tab:fs} as item \#f10 with a negative sign.
The total sum of the QED contributions with an explicit dependence of \ensuremath{{r_E}^2} is summarized in Tab.\,\ref{tab:LS:Radius} and amounts to
\begin{multline}\label{eq:LS:Radius}
\ensuremath{\Delta E_\mathrm{r\mathrm{-dep.}}^\mathrm{LS}} (\ensuremath{{r_E}^2} )\\ = -\RADVAL \ensuremath{(98)}\,\ensuremath{\mathrm{meV}}\xspace\,\ensuremath{\mathrm{fm}^{-2}}\xspace\,\ensuremath{{r_E}^2} + 0.1354(33) \,\ensuremath{\mathrm{meV}}\xspace.
\end{multline}
\subsubsection{Two-photon exchange contributions to the Lamb shift}
\label{sec:LS:Pol}
\begin{figure}
\begin{center}
\begin{minipage}{0.3\columnwidth}
(a)\hfill\mbox{~}
\vspace{0px}\\
\begin{fmffile}{tpe_elastic_1}
\begin{fmfgraph*}(70,60)
\fmftop{i1,o1}
\fmfbottom{i2,o2}
\fmf{plain}{i1,t1,txx,t2,o1}
\fmf{plain,width=3}{i2,b1,bb,b2,o2}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b1}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b2}
\fmf{photon,tension=0}{t1,b1}
\fmf{photon,tension=0}{b2,t2}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{30px}
\begin{minipage}{0.3\columnwidth}
(c)\hfill\mbox{~}
\vspace{0px}\\
\begin{fmffile}{tpe_inelastic_1}
\begin{fmfgraph*}(70,60)
\fmftop{i1,o1}
\fmfbottom{i2,o2}
\fmf{plain}{i1,t1,txx,t2,o1}
\fmf{plain,width=3}{i2,b1}
\fmf{plain,width=3}{b2,o2}
\fmfpoly{smooth,filled=30,pull=1.4,tension=0.2,background=white+blue}{b1,b10,b2,b11}
\fmffreeze
\fmfshift{14up}{b10}
\fmfshift{14down}{b11}
\fmf{photon,tension=0}{t1,b1}
\fmf{photon,tension=0}{b2,t2}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\vspace{6ex}\\
\begin{minipage}{0.3\columnwidth}
(b)\hfill\mbox{~}
\vspace{0px}\\
\begin{fmffile}{tpe_elastic_2}
\begin{fmfgraph*}(70,60)
\fmftop{i1,o1}
\fmfbottom{i2,o2}
\fmf{plain}{i1,t1,txx,t2,o1}
\fmf{plain,width=3}{i2,b1,bb,b2,o2}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b1}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b2}
\fmf{photon,tension=0}{t1,b2}
\fmf{photon,tension=0}{b1,t2}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{30px}
\begin{minipage}{0.3\columnwidth}
(d)\hfill\mbox{~}
\vspace{0px}\\
\begin{fmffile}{tpe_inelastic_2}
\begin{fmfgraph*}(70,60)
\fmftop{i1,o1}
\fmfbottom{i2,o2}
\fmf{plain}{i1,t1,txx,t2,o1}
\fmf{plain,width=3}{i2,b1}
\fmf{plain,width=3}{b2,o2}
\fmfpoly{smooth,filled=30,pull=1.4,tension=0.2,background=white+blue}{b1,b10,b2,b11}
\fmffreeze
\fmfshift{14up}{b10}
\fmfshift{14down}{b11}
\fmf{photon,tension=0}{t1,b2}
\fmf{photon,tension=0}{b1,t2}
\fmfv{label=h}{i2}
\fmfv{label.angle=-150,label=$\mu$}{i1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\caption{\label{fig:tpe}
(a)+(b) Elastic \ensuremath{\Delta E_\mathrm{Friar}^\mathrm{LS}}{}, and (c)+(d) inelastic \ensuremath{\Delta E_\mathrm{inelastic}^\mathrm{LS}}{}
two-photon exchange (TPE) contribution.
The thick dots in (a) indicate helion form factor insertions.
The blob in (c) and (d) represents all possible excitations of the nucleus.
}
\end{figure}
Historically, the two-photon exchange (TPE) contribution to the Lamb shift (LS) in muonic atoms has been considered the sum of the two parts displayed in Fig.\,\ref{fig:tpe}(a,b) and (c,d), respectively:
\begin{equation}
\label{eq:tpe}
\ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}} = \ensuremath{\Delta E_\mathrm{Friar}^\mathrm{LS}} + \ensuremath{\Delta E_\mathrm{inelastic}^\mathrm{LS}}
\end{equation}
with the elastic ``Friar moment'' contribution \ensuremath{\Delta E_\mathrm{Friar}^\mathrm{LS}}~\footnote{formerly known as ``third Zemach moment'', see footnote \textsuperscript{\ref{footnote:friar}} on p.\,\pageref{footnote:friar} for disambiguation.} and the inelastic part \ensuremath{\Delta E_\mathrm{inelastic}^\mathrm{LS}}, frequently termed ``polarizability''.
The elastic part, \ensuremath{\Delta E_\mathrm{Friar}^\mathrm{LS}} ~is shown in Fig.\,\ref{fig:tpe}(a,b). It is sensitive to the shape of the nuclear charge distribution, beyond the leading \ensuremath{\langle r^2 \rangle}~dependence discussed in Sec.\,\ref{sec:LS:Radius}. This part is traditionally parameterized as being proportional to the third power of the rms charge radius and it already appeared in Eq.\,(\ref{app:eq:friar}) as the second term proportional to $ \langle r^3\rangle_{(2)} $. The coefficient depends on the assumed radial charge distribution.
The inelastic part, \ensuremath{\Delta E_\mathrm{inelastic}^\mathrm{LS}} ~is shown in Fig.\,\ref{fig:tpe}(c,d). It stems from virtual excitations of the nucleus. The inelastic contributions are notoriously the least well-known theory contributions and limit the extraction of the charge radius from laser spectroscopy of the Lamb shift.
Eq.\,(\ref{eq:tpe}) is valid for the nuclear contributions as well as for the nucleon contributions. This means that elastic and inelastic parts have to be evaluated for both, respectively.
\begin{landscape}
\begin{table}
\begin{minipage}{\linewidth}
\footnotesize
\setlength\extrarowheight{3pt}
\centering
\caption[Nuclear structure-dependent contributions to the Lamb shift]{
Coefficients of the {\bf nuclear structure-dependent} one-photon exchange (OPE) contributions to the Lamb shift of \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace.
%
The values from the Martynenko group shown here are the published ones divided by $(1.9660\,\mathrm{fm})^2$, which is the radius they used. The numbers \#$i$ from the Martynenko group refer to rows in Tab.\,1 of Ref.\,\cite{Krutov:2014:JETP120_73} and numbers in parenthesis to Eqs.\ therein.
%
KS: K\"all\'en-Sabry, VP: vacuum polarization, SOPT: second-order perturbation theory.
%
Values are in meV/fm$^2$, except for {\#r2'} and {\#r3'}.
}
\label{tab:LS:Radius}
\fontsize{7pt}{7pt}\selectfont
\begin{tabular}{l|l |f{5}
c |f{5} c |f{4} c | f{7}@{ }f{5} c}
\hline
\hline
\#& Contribution & \cntl{2}{Borie (B)}
& \cntl{2}{Martynenko group (M)}
& \cntl{2}{Karshenboim group (K)}
& \cnt {3}{Our choice} \\
& & \cntl{2}{Borie \cite{Borie:2014:arxiv_v7} Tab.14}
& \cntl{2}{Krutov {\it et\,al.}~\cite{Krutov:2014:JETP120_73}}
& \cntl{2}{Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509}}
& & & \\
& & & & \cntl{2}{Faustov {\it et\,al.}~\cite{Faustov:2017:rad_fin_size}}
& &
& \cnt{2}{value}
& \cnt{1}{source} \\
\hline
r1 & Leading fin.\ size corr., $(Z\alpha)^4$
& -102.520 & $b_a$
& -102.52 & \#14, (61)
& -102.52 & $\Delta E_{FNS}^{(0)}$
& -102.520
& \pm~0.0010
& B,M,K \\
r2 & Radiative corr., $\alpha(Z\alpha)^5$
& -0.0243~\footnote{Borie uses Eq.\,(10) of \cite{Eides:1997:two-photon} to calculate
this term.
For further explanations, see text.}
& $b_b$
& -0.0243~\footnote{The value in Eq.\,92 of \cite{Krutov:2014:JETP120_73} was published with a wrong sign.}
& \#26, (92)
& &
& -0.0243 & & B,M \\
r3 & Finite size corr. order $(Z\alpha)^6$
&-0.1275 & $b_c$
& -0.1301
& \#26, (91)
& &
& -0.1288 & \pm0.0013
& avg. \\
\hline
r4 & Uehling corr.\ (+KS), $\alpha(Z\alpha)^4$
& &
& -0.3310 & \#16, (69)
& - 0.323 & $\Delta E_{FNS}^{(2)}$
& &
& \\
r6 &Two-loop VP corr., $\alpha^2(Z\alpha)^4$
& &
& -0.0026 &\#18, (73)
& &
& &
& \\
sum & r4+r6
&-0.3176 & $b_d$
& -0.3336 &
& &
& -0.3256 & \pm 0.0080
& avg. \\
\hline
r5 & One-loop VP in SOPT, $\alpha(Z\alpha)^4$
& &
& -0.5196 &\#17, (70)
& - 0.520 & $\Delta E_{FNS}^{(1)}$
& &
& \\
r7 & Two-loop VP in SOPT, $\alpha^2(Z\alpha)^4$
& &
& -0.0063 & \#19~\footnote{This term is represented by Fig.\,9(a,b,c,d) from the Martynenko group \cite{Krutov:2014:JETP120_73}. This figure includes equation (76) therein.}
& &
& &
& \\
sum& r5+r7
& -0.5217 & $b_e$
& -0.5259 &
& &
&-0.5238 &\pm 0.0021
& avg. \\
\hline
r8 & Corr.\ to the $2P_{1/2}$ level
& 0.00409 ~\footnote{The sign is explained in the text.}
& $b(2p_{1/2})$
& &
& &
& 0.00409 &
& B \\
\hline
& Sum of coefficients
&-103.507(5)~\footnote{The summed coefficient is given in Ref.~\cite{Borie:2014:arxiv_v7} on p.\,15, where Borie indicates the uncertainty of 0.005\,meV.}
&
& -103.5339 &
& - 103.37 & $\Delta E_{FNS}$
&-103.5184 & \pm 0.0098~\footnote{This uncertainty is the one obtained from averaging the above values (0.0084\,meV) and the one given by Borie in her sum of (0.005\,meV) added in quadrature.}
& \\
\hline
\lft{11}{~} \\
\hline
r2'& Rad.\ corr.\ $\alpha(Z\alpha)^5$\,[meV]~\footnote{Belongs to \#r2. Not parametrizable with $\rh^2$.}
& &
& 0.0177 & \cite{Faustov:2017:rad_fin_size}
& &
& 0.0177 &
& M \\
r3'& Remaining order $(Z\alpha)^6$\,[meV]~\footnote{Belongs to \#r3. Depends on the charge distribution in a non-trivial way, see text.}
& 0.121 & Tab.~6
& 0.11445 & (91)
& &
& 0.1177&
\pm0.0033&avg. \\
\hline
\hline
&&&&&&&&&& \\
& \bf Sum
& \multicolumn{2}{l|}{$-103.507~{\rh}^2$ + 0.121\,meV}
& \multicolumn{2}{l|}{$-103.5339~{\rh}^2$ + 0.1322\,meV}
& - 103.37~{\rh}^2 &
& \multicolumn{3}{l}{ \bf -103.5184(98)~$\boldsymbol{\rh^2}$ + 0.1354(33)\,meV} \\
&&&&&&&&&& \\
\hline
\hline
\end{tabular}
\end{minipage}
\end{table}
\end{landscape}
The nuclear parts of \ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}}\ are then given as $\delta E^A_{\rm Friar}$ and $\delta E^A_{\rm inelastic}$ for a nucleus with A nucleons, and the nucleon parts as $\delta E^N_{\rm Friar}$ and $\delta E^N_{\rm inelastic}$.
With that, the total (nuclear and nucleon) TPE is given as~\footnote{Compared to the notation of the TRIUMF-Hebrew group \cite{NevoDinur:2016:TPE}, the terms in Eq.\,(\ref{eq:tpe2}) correspond to $\delta^A_{\rm Zem}$, $\delta^N_{\rm Zem}$, $\delta^A_{\rm pol}$, and $\delta^N_{\rm pol}$, respectively.}
\begin{equation}
\label{eq:tpe2}
\ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}} = \delta E^A_{\rm Friar} + \delta E^N_{\rm Friar} + \delta E^A_{\rm inelastic} + \delta E^N_{\rm inelastic}.
\end{equation}
We refer here to two calculations of the TPE contributions.
The first stems from the TRIUMF-Hebrew group, who perform \textit{ab initio} calculations using two different nuclear potentials. They have published two papers on the TPE in muonic helium-3 ions: Detailed calculations are given in Nevo Dinur {\it et\,al.}~\cite{NevoDinur:2016:TPE}, and updated results are found in Hernandez {\it et\,al.}~\cite{Hernandez:2016:POLupdate}.
The second calculation has been performed by Carlson {\it et\,al.}~\cite{Carlson:2016:tpe}, who obtain the TPE from inelastic structure functions via dispersion relations.
The two calculations are very different, so that comparisons of any but the total value may be inexact \cite{Carlson:2016:tpe}. An attempt to compare the different approaches is given in Tab.\,II of Ref.\,\cite{Carlson:2016:tpe}. Here, we want to refer to this table only and later compare the total values as suggested. Note that we proceed differently to our previous compilation for \ensuremath{\mu}{\rm d}\xspace \cite{Krauth:2016:mud} (Tab.\,3), where we listed and compared 16 individual terms (labeled \#p1...16) which together yield the sum of the four terms of Eq.\,(\ref{eq:tpe2}).
\\\\
The nuclear Friar moment contribution is calculated by the TRIUMF-Hebrew group to be $\delta E^A_{\rm Friar}=10.49(24)\,\ensuremath{\mathrm{meV}}\xspace$ \cite{NevoDinur:2016:TPE,Hernandez:2016:POLupdate}.
Previous values have been given by Borie \cite{Borie:2014:arxiv_v7} (10.258(305)\,\ensuremath{\mathrm{meV}}\xspace) and Krutov {\it et\,al.}\,\cite{Krutov:2014:JETP120_73} (10.50(10)\,\ensuremath{\mathrm{meV}}\xspace)\footnote{Sum of 10.28(10)\,\ensuremath{\mathrm{meV}}\xspace and 0.2214(22)\,\ensuremath{\mathrm{meV}}\xspace, which correspond to line 15 and 20 from Tab.\,1 in Ref.\,\cite{Krutov:2014:JETP120_73}, respectively.} using a Gaussian charge distribution and assuming an rms radius of $1.966(10)$\,fm.
These uncertainties do not include the (rather large) dependence of the calculation on the charge distribution \cite{Krutov:2014:JETP120_73,Sick:2014:HeZemach}. This type of uncertainty is gauged within the ab-initio calculation of \cite{NevoDinur:2016:TPE} by using two different state-of-the-art nuclear potentials.
We therefore use the more recent value provided by the TRIUMF-Hebrew group. Their value also agrees with a value of 10.87(27)\,\ensuremath{\mathrm{meV}}\xspace which is obtained in \cite{NevoDinur:2016:TPE} from the third Zemach moment $\langle r^3\rangle_{(2)} = 28.15(70)\,\mathrm{fm}^3$ that was extracted from electrons scattering off $^3$He by Sick\,\cite{Sick:2014:HeZemach}.
\\
The nuclear polarizability contribution from the TRIUMF-Hebrew group is $\delta E^A_{\rm inelastic}=4.16(17)\,\ensuremath{\mathrm{meV}}\xspace$ \cite{NevoDinur:2016:TPE,Hernandez:2016:POLupdate}. The first calculation of the nuclear polarizability contribution in \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace has been published in 1961 \cite{Joachain:1961:pol}. The recent value from the TRIUMF-Hebrew group replaces a former one of $4.9\,\ensuremath{\mathrm{meV}}\xspace$ from Rinker \cite{Rinker:1976:he_pol} which has been used for more than 40 years now.
\\\\
As mentioned before, the total TPE contribution has a nuclear part and a nucleon part. The nucleon Friar moment contribution from the TRIUMF-Hebrew group amounts to $\delta E^N_{\rm Friar}=0.52(3)\,\ensuremath{\mathrm{meV}}\xspace$. They obtain this value using $\delta E^N_{\rm Friar}(\ensuremath{\mu}{\rm p}\xspace)=0.0247(13)\,\ensuremath{\mathrm{meV}}\xspace$ from \ensuremath{\mu}{\rm p}\xspace and scale it according to Eq.\,(17) in Ref.\,\cite{NevoDinur:2016:TPE}. This procedure has also been done in \cite{Krauth:2016:mud} for \ensuremath{\mu}{\rm d}\xspace~\footnote{
In Eq.\,(12) of Ref.\,\cite{Krauth:2016:mud}, we used a scaling of the nucleon TPE contribution by the reduced mass ratio to the third power, which is only correct for $\delta E^N_{\rm inelastic}$. $\delta E^N_{\rm Friar}$ should be scaled with the fourth power \cite{Friar:2013:PRC88_034004,NevoDinur:2016:TPE}.
This is due to an additional $m_r$ scaling factor compared to the proton polarizability term.
This mistake has no consequences for \ensuremath{\mu}{\rm d}\xspace yet, as the nuclear uncertainty is much larger, but the correct scaling is relevant for \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace and \ensuremath{\mu^4}{\rm He}\ensuremath{^+}\xspace.
}. $\delta E^N_{\rm Friar}(\ensuremath{\mu}{\rm p}\xspace)$ is a sum of the elastic term $(0.0295(13)\,\ensuremath{\mathrm{meV}}\xspace)$ and the non-pole term $(-0.0048\,\ensuremath{\mathrm{meV}}\xspace)$ which have been obtained by Carlson {\it et\,al.}\ in Ref.\,\cite{Carlson:2011:PRA84_020102}.\\
The nucleon polarizability contribution from the TRIUMF-Hebrew group amounts to $\delta E^N_{\rm inelastic}=0.28(12)\,\ensuremath{\mathrm{meV}}\xspace$. It is obtained using the proton polarizability contribution from \ensuremath{\mu}{\rm p}\xspace and scaling it with the number of protons and neutrons~\footnote{
Assuming isospin symmetry, the value of the neutron polarizability contribution used in \cite{NevoDinur:2016:TPE} is the same as the one of the proton, but an additional uncertainty of 20\% is added, motivated by studies of the nucleon polarizabilities \cite{Myers:2014:comptonScatt}.
}, as well as with the wavefunction overlap,
according to Eq.\,(19) of Ref.\,\cite{NevoDinur:2016:TPE}. Furthermore it is corrected for estimated medium effects and possible nucleon-nucleon interferences. The proton polarizability contribution used here amounts to $0.0093(11)\,\ensuremath{\mathrm{meV}}\xspace$ and is the sum of an inelastic term ($0.0135\,\ensuremath{\mathrm{meV}}\xspace$ \cite{Carlson:2014:PRA89_022504}) and the proton subtraction term $\delta^{p}_{\rm subtraction}=-0.0042(10)\,\ensuremath{\mathrm{meV}}\xspace$ which has been calculated for muonic hydrogen in Ref.\,\cite{BirseMcGovern:2012}.
\\\\
Summing up all nuclear and nucleon contributions evaluated by the TRIUMF-Hebrew group \cite{NevoDinur:2016:TPE,Hernandez:2016:POLupdate} yields a total value of the \ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}} ~of \cite{NevoDinur:2016:TPE,Hernandez:2016:POLupdate}
\begin{equation}
\begin{split}\label{eq:LS:TPEpotentials}
\ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}}(&{\rm nuclear~potentials}) \\
=& ~\delta E^A_{\rm Friar} + \delta E^N_{\rm Friar} + \delta E^A_{\rm inelastic} + \delta E^N_{\rm inelastic}\\
=& ~15.46(39)\,\ensuremath{\mathrm{meV}}\xspace.~\text{\footnotemark}
\end{split}
\end{equation}
\footnotetext{As explained in the introduction, we use a different sign convention, which explains the minus sign in Refs.\,\cite{NevoDinur:2016:TPE,Hernandez:2016:POLupdate}.}
Recently, Carlson {\it et\,al.}\ \cite{Carlson:2016:tpe} have also calculated the TPE in \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace. Their result of
\begin{equation}
\label{eq:LS:TPEdispersion}
\ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}}(\rm dispersion~relations) = 15.14(49)\,\ensuremath{\mathrm{meV}}\xspace
\end{equation}
is in agreement with the one from the TRIUMF-Hebrew group.
As \textit{our choice} we take the average of Eqs.\,(\ref{eq:LS:TPEpotentials}) and (\ref{eq:LS:TPEdispersion}) and remain with
\begin{equation}
\label{eq:LS:pol}
\ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}} = \POLVALRND \ensuremath{(52)}\,\ensuremath{\mathrm{meV}}\xspace.
\end{equation}
As conservative uncertainty we use the larger one (from Eq.\,(\ref{eq:LS:TPEdispersion})) and add in quadrature half the spread. A weighted average of the two values (Eq.\,(\ref{eq:LS:TPEpotentials}) and (\ref{eq:LS:TPEdispersion})) which would reduce the total uncertainty is not adequate as certain contributions are effectively fixed by the same data \cite{Gorchtein:PC:2016}.
\subsection{Total Lamb shift in \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace}
\label{sec:LS_total}
Collecting the
radius-independent (mostly) QED contributions
listed in Tab.~\ref{tab:LS:QED} and
summarized in Eq.~(\ref{eq:LS:QED}),
the radius-dependent contributions
listed in Tab.~\ref{tab:LS:Radius} and
summarized in Eq.~(\ref{eq:LS:Radius}),
and the complete TPE contribution \ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}}{}
from Eq.~(\ref{eq:LS:pol}),
we obtain for the $\mathrm{2S\rightarrow2P}$ energy difference in \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace
\begin{widetext}
\begin{equation}
\label{eq:LS:full}
\begin{aligned}
\Delta E(2S_{1/2}\rightarrow2P_{1/2}) ~ = & ~\, 1644.3466(~\,146)\,\mathrm{meV} \\
& + ~~\, 0.1354(~~~33)\,\mathrm{meV}
&-& ~ \RADVAL \ensuremath{(98)} ~ \rh^2 ~ \mathrm{meV/fm^2}\\
& + ~ \POLVALFINAL00(5200)\,\mathrm{meV}\\
= & ~ \, 1659.78(52)~\mathrm{meV}&-& ~ 103.518(10) ~ \rh^2 ~ \mathrm{meV/fm^2},
\end{aligned}
\end{equation}
\end{widetext}
where in the last step we have rounded the values to reasonable accuracies.
One should note that the uncertainty of \POLERRFINAL\,meV
from the nuclear structure corrections \ensuremath{\Delta E_\mathrm{TPE}^\mathrm{LS}}{},
Eq.~(\ref{eq:LS:pol}),
is about 30 times larger than the combined uncertainty of all
radius-independent terms summarized in Tab.~\ref{tab:LS:QED},
and 13 times larger than the uncertainty in the coefficient of the $\rh^2$-dependent term (which amounts to 0.038\,meV for $\rh=1.966$\,fm).
A further improvement of the two-photon exchange contributions in
light muonic atoms is therefore strongly desirable.
\section{2S hyperfine splitting}
\label{sec:HFS}
The 2S hyperfine splitting (HFS) in muonic helium-3 ions has been calculated by Borie \cite{Borie:2014:arxiv_v7} and Martynenko \cite{Martynenko:2008:muheHFS}.
(There is also the more recent paper \cite{Martynenko:2010:2SHFS_muHe} from Martynenko {\it et\,al.}, but it is less detailed and reproduces all numbers from \cite{Martynenko:2008:muheHFS}, with one exception to be discussed for \#h27.)
The values are summarized in Tab.~\ref{tab:hfshelium} and labeled with \#h$i$.
We also adapted the ordering according to increasing order/complexity of the terms and grouped them thematically as: Fermi energy with anomalous magnetic moment and relativistic corrections discussed in Sec.~\ref{sec:EFermi}, vacuum polarization and vertex corrections in Sec.~\ref{sec:VP}, nuclear structure contributions and corrections listed in Sec.~\ref{sec:nuclstruc}, and the weak interaction contribution in Sec.~\ref{sec:weak}.
\subsection{Fermi energy with muon anomalous magnetic moment and Breit corrections}
\label{sec:EFermi}
\subsubsection{h1 and h4 Fermi energy and muon AMM correction}\label{sec:EFermi2}
Item \#h1 is the Fermi energy $\Delta E_\mathrm{Fermi}$ which defines the main splitting of the 2$S$ hyperfine levels. Borie and the Martynenko group have both calculated the Fermi energy, however, their values disagree by 0.055\,\ensuremath{\mathrm{meV}}\xspace (see Tab.\,\ref{tab:hfshelium}). For the calculation Borie uses Eq.\,(13) in her Ref.\,\cite{Borie:2014:arxiv_v7}. Martynenko uses Eq.\,(6) in his Ref.\,\cite{Martynenko:2008:muheHFS}. The Fermi energy is calculated using fundamental constants only. Thus we repeated the calculation for both equations, the one from Borie and the one from Martynenko which resulted to be the same: Both equations yield the same result, as they should, which is
\begin{equation}\label{eq:Fermi}
\Delta E_{\rm Fermi} = \frac{8(\alpha^4 Z^3) m_r^3 }{3n^3m_{\mu} m_p}\mu_h=-171.3508\,\mathrm{meV},
\end{equation}
where $m_\mu$ is the muon mass, $m_p$ is the proton mass, $m_r$ is the reduced mass, and $\mu_h$ is the
helion magnetic moment to nuclear magneton ratio of $\mu_h = -2.127\,625\,308(25)$ \cite{Mohr:2016:CODATA14}. We use the value in Eq.\,(\ref{eq:Fermi}) as \textit{our choice}. This value agrees neither with Borie's value ($-171.3964\,\ensuremath{\mathrm{meV}}\xspace$) nor with the one from the Martynenko group ($-171.341\,\ensuremath{\mathrm{meV}}\xspace$).
The value for the Fermi energy corrected for the muon anomalous magnetic moment (AMM) $a_{\mu}$ is then also updated to
\begin{equation}\label{eq:FermiAMM}
\Delta E_\mathrm{Fermi,AMM}=\Delta E_\mathrm{Fermi}\cdot(1+a_{\mu})=-171.5506\,\mathrm{meV}
\end{equation}
with a correction of $-0.1998$\,meV.
All further corrections from Borie given as coefficients $\epsilon$, are applied to this value analogous to
\begin{equation}
\Delta E_\mathrm{Fermi,AMM}\cdot(1+\epsilon).
\end{equation}
Note, that in Tab.\,\ref{tab:hfshelium}, for the contributions given by Borie, we use her coefficients but apply them to our value of the Fermi Energy given in Eq.\,(\ref{eq:FermiAMM}). The value for the Fermi Energy in Eq.\,(\ref{eq:FermiAMM}) is evaluated to a precision of $0.0001\,\ensuremath{\mathrm{meV}}\xspace$. If the number of significant digits from Borie's coefficients is too small to yield this precision we attribute a corresponding uncertainty.
For example item \#h28* has the coefficient $\epsilon_{2\gamma}=0.0013$; here the coefficient is only precise up to a level of 0.00005, which we include as uncertainty. This uncertainty is propagated upon multiplication with the Fermi energy (Eq.\,(\ref{eq:FermiAMM})) and then yields 0.0086\,\ensuremath{\mathrm{meV}}\xspace.
\subsubsection{h2 Relativistic Breit correction}
Item \#h2 is the relativistic Breit correction of order $(Z\alpha)^6$. It is given congruently by both authors as $\Delta E_\mathrm{F,rel}^{\mathrm{B}}=-0.0775\,$meV and $\Delta E_\mathrm{F,rel}^{\mathrm{M}}=-0.078\,$meV, respectively.
We take the number from Borie as \textit{our choice}, which is given with one more digit and attribute an uncertainty of 0.0001\,\ensuremath{\mathrm{meV}}\xspace due to the precision in her coefficient.
\subsection{Vacuum polarization and vertex corrections}
\label{sec:VP}
\subsubsection{h8 and h9: Electron vacuum polarization in a one-photon one-loop interaction (h8) and in a one-photon two-loop interaction (h9)}
\label{sec:eVP}
The Feynman diagrams corresponding to \#h8 and \#h9 are analogous to those shown in Figs.~\ref{fig:uehling} and \ref{fig:item_4}, respectively, and constitute the analogs to the Uehling- and K\"all\'en-Sabry contributions in the Lamb shift. \#h8 is of order $\alpha(Z\alpha)^4$, \#h9 is of order $\alpha^2(Z\alpha)^4$.
Borie calculates the main electron VP contribution
("by modification of the magnetic interaction between muon and nucleus"),
which is a one-photon one-loop interaction.
It amounts to a correction $\epsilon_{VP1} = 0.00315$, which results in an energy shift of $-0.5405\,\ensuremath{\mathrm{meV}}\xspace$ (\#h8).
She also gives $\epsilon_{VP1}=2.511\cdot10^{-5}$ for one-photon two-loop interactions, resulting in $-0.0043\,\ensuremath{\mathrm{meV}}\xspace$ (\#h9).
These terms are evaluated on p.\,21 of her document \cite{Borie:2014:arxiv_v7}, using her Eq.\,(16).
Martynenko calculates these contributions to be $-0.540\,\ensuremath{\mathrm{meV}}\xspace$ and $-0.004\,\ensuremath{\mathrm{meV}}\xspace$, respectively.
These values are found in the table in Ref.\,\cite{Martynenko:2008:muheHFS}.
Martynenko mentions that his value for our item \#h9 consists of his Eqs.\,(15,16).
The numerical result from Eq.\,(15) corresponds to two separate loops (see our Fig.~\ref{fig:item_4}(a)) and is given as $-0.002\,$meV, whereas Eq.~(16) describes the two nested two-loop processes where an additional photon is exchanged within the electron VP loop (see our Fig.~\ref{fig:item_4}(b,c)).
One can conclude that its numerical value is also $-0.002\,$meV.
Both authors give congruent results within their precisions, as \textit{our choice} we write down the numbers by Borie which are given with one more digit. We attribute an uncertainty to item \#h8 due to the precision in Borie's coefficient.\\
\subsubsection{h5 and h7: Electron vacuum polarization in SOPT in one loop (h5) and two loops (h7)}
\label{sec:eVPSopt}
Items \#h5 and \#h7 are the SOPT contributions to items \#h8 and \#h9, respectively.
Borie's value for our item \#h5 is given by the coefficient $\epsilon_{VP2}=0.00506$ and her value for our item \#h7 by $\epsilon_{VP2}=3.928\cdot10^{-5}$. This results in energy shifts of $-0.8680(9)\,\ensuremath{\mathrm{meV}}\xspace$ and $-0.0067\,\ensuremath{\mathrm{meV}}\xspace$, respectively (those values are for point nuclei; the finite size correction is taken into account in our \#h25 and \#h26). The uncertainty in item \#h5 originates from the precision of $\epsilon_{VP2}$.
The corresponding values from Martynenko are $-0.869\,\ensuremath{\mathrm{meV}}\xspace$ (\#h5) and $-0.010\,\ensuremath{\mathrm{meV}}\xspace$ (\#h7).
Due to slight differences between the two authors, as \textit{our choice} we take the average of items \#h5 and \#h7, respectively. The uncertainty of item \#h5 is the above uncertainty and half the spread between both authors added in quadrature.
\subsubsection{h13 and h14: Vertex correction ($\hat{=}$ self energy happening at the muon-photon vertex)}
\label{sec:vertex}
Item \#h13 is the muon self-energy contribution of order $\alpha(Z\alpha)^5$
(it is the analogue to a part of item \#20 in the Lamb shift, see Fig.\,\ref{fig:onephotonSE}a).
It has only been calculated by Borie as
\begin{equation}
\epsilon_\mathrm{vertex}=\alpha(Z\alpha) \left( \ln{2}-\frac{5}{2} \right)=-0.9622\cdot10^{-4}\cdot Z.
\end{equation}
Its numerical value is thus $0.0330\,\ensuremath{\mathrm{meV}}\xspace$, however this includes a muon VP contribution of $-0.0069\,\ensuremath{\mathrm{meV}}\xspace$ (\#h12, see Sec.~\ref{sec:mVP}). For our item \#h13, we use the value from Borie as \textit{our choice}. We therefore should not include \#h12, which is discussed later.
Borie also cites a higher order correction of Brodsky and Erickson \cite{Brodsky:1966:radiative:hyperfine} which results in a correction of $-0.211\cdot10^{-4}\hat{=}-0.0036\,\ensuremath{\mathrm{meV}}\xspace$ (\#h14).
Very probably the sign of the energy shift is not correct because the coefficient is negative, but the Fermi energy of helium-3 also has a negative sign, thus the energy shift should be positive.
(The analogous contributions in muonic hydrogen and deuterium are negative, which is a further hint to a wrong sign since the helium-3 Fermi energy is negative, contrary to hydrogen and deuterium.)
\subsubsection{h12: Muon VP and muon VP SOPT}\label{sec:mVP}
Item \#h12 is the one-loop muon vacuum polarization. Borie on p.\,19 (below the equation of $\epsilon_\mathrm{vertex}$) of Ref.\,\cite{Borie:2014:arxiv_v7} gives the coefficient as $0.3994\cdot10^{-4}\cdot Z$. In combination with the Fermi energy this yields $-0.0069\,\ensuremath{\mathrm{meV}}\xspace$.
Martynenko obtains a value of $-0.007\,\ensuremath{\mathrm{meV}}\xspace$ which is congruent to Borie's value.
However, Borie's value of this contribution is already included in our item \#h13, which has been discussed in the previous section. Hence, we do not include it separately in `our choice'.
\subsubsection{h18 Hadronic vacuum polarization}
\label{sec:hVP}
Item \#h18 is the hadronic vacuum polarization. Borie gives this contribution as $ \epsilon_\mathrm{hVP}=0.2666\cdot10^{-4}\cdot Z$, which amounts to $-0.0091\,\ensuremath{\mathrm{meV}}\xspace$ on p.\,19 of her paper.
This contribution is analogous to our Fig.~\ref{fig:uehling}, but with a hadronic loop in the photon line.
Since Martynenko does not provide a value for hadronic VP in muonic helium-3 ions, we use Borie's value as `our choice'.
\subsection{Nuclear structure and finite size corrections}
\label{sec:nuclstruc}
Analogously to Sec.~\ref{sec:LS:nuclstruc}, we categorize the nuclear structure contributions to the 2S HFS as one-photon exchange (OPE) and two-photon exchange (TPE) processes, respectively.
We list first the by far dominant contribution to nuclear structure: the Zemach term, which is an elastic TPE process.
The following subsections describe the known elastic TPE corrections in the 2S HFS.
So far, to our knowledge there are yet no calculations with respect to the {\it inelastic} TPE contribution to the 2S HFS. Thus we only give a simplified estimate with a large uncertainty.
Later the section is concluded with the one-photon exchange (OPE) corrections to nuclear structure in the 2S HFS.
\subsubsection{h20 Zemach term and h23, h23b*, h28* nuclear recoil}\label{sec:Zemachterm}
Item \#h20 is the elastic TPE and the main finite size correction to the 2S HFS. This correction arises due to the extension of the magnetization density (Bohr-Weisskopf effect) and is also called the Zemach term \cite{Zemach:1956}. The Zemach term is usually parameterized as \cite{FriarSick:2004:Zemach}
\begin{equation}\label{eq:zemach_term}
\Delta E_{\rm Zemach}^{\rm HFS} = - \Delta E_{\rm Fermi, AMM}~ 2(Z\alpha)m_r~ \ensuremath{r_Z}
\end{equation}
with $m_r$ being the reduced mass and \ensuremath{r_Z} the Zemach radius of the nucleus \cite{Sick:2014:HeZemach}
\begin{equation}
\ensuremath{r_Z} = -\frac{4}{\pi}\int_0^\infty [G_E(q)G_M(q)-1]\,\frac{dq}{q^2}.
\end{equation}
Here, $G_E(q)$ and $G_M(q)$ are the electric and magnetic form factors of the nucleus, respectively.
The corresponding coefficient to the Fermi energy in Eq.\,(\ref{eq:zemach_term}) is given by Borie on p.\,23 of \cite{Borie:2014:arxiv_v7} as
\begin{equation}
\epsilon_{\rm Zem} = -2(Z\alpha)m_r~\ensuremath{r_Z} = -0.01506\mathrm{\,fm^{-1}}~\ensuremath{r_Z} .
\end{equation}
With our Fermi energy from Eq.\,(\ref{eq:FermiAMM}), item \#h20 is
\begin{equation}\label{eq:zemach_term2}
\Delta E_{\rm Zemach}^{\rm HFS} = 2.5836~\ensuremath{r_Z} \,\ensuremath{\mathrm{meV}}\xspace/\mathrm{fm} = 6.5312(413)\,\ensuremath{\mathrm{meV}}\xspace,
\end{equation}
where, in the second step, we inserted the most recent Zemach radius from Sick \cite{Sick:2014:HeZemach} ($\ensuremath{r_Z} = 2.528(16)\,$fm).
Note that Borie's published value of $\Delta E_{\rm Zemach}^{\rm HFS}$ differs from the one given here, because she uses a different Zemach radius of $\ensuremath{r_Z} = 2.562\,$fm, assuming a Gaussian charge distribution.
Martynenko, in his Ref.\,\cite{Martynenko:2008:muheHFS}, gives a value of $\Delta E_{\rm str}^{\rm HFS} = 6.047\,\ensuremath{\mathrm{meV}}\xspace$. This value contains a recoil contribution and is thus not directly comparable with our item \#h20. However, this value has been updated \cite{Martynenko:PC:2016} and is now available as two separate values of $\Delta E_{\rm str}^{\rm HFS} = 6.4435\,\ensuremath{\mathrm{meV}}\xspace = (6.4085 + 0.0350_{\rm recoil})\,\ensuremath{\mathrm{meV}}\xspace$.
The first can be compared to Eq.\,(\ref{eq:zemach_term2}).
The second is the recoil correction and listed in our table as item \#h23. Martynenko notes \cite{Martynenko:2008:muheHFS} that changing from a Gaussian to a dipole parameterization results in a change of the final number of 2\%.
Regarding our item \#h20, we do not consider the respective value from Martynenko because it is model-dependent and therefore carries a large uncertainty. This uncertainty can be avoided using the model-independent Zemach radius from Sick and the coefficient given by Borie as stated above.
A new contribution which hasn't been calculated for \ensuremath{\mu}{\rm p}\xspace and \ensuremath{\mu}{\rm d}\xspace is our item \#h23b*. It is an additional recoil contribution
which amounts to 0.038\,\ensuremath{\mathrm{meV}}\xspace. It has only been calculated by Martynenko and we adopt his value as \textit{our choice}. In order to account for the precision given by Martynenko, we write 0.0380(5)\,\ensuremath{\mathrm{meV}}\xspace.
Another contribution which has not been calculated for \ensuremath{\mu}{\rm p}\xspace and \ensuremath{\mu}{\rm d}\xspace is item \#h28*. It is a two-photon recoil correction, calculated by Borie in 1980 \cite{Borie:1980:mu3HeLS}, who followed the procedure of Grotch and Yennie \cite{Grotch:1969:EPM}. This contribution is not listed in Borie's recent Ref.\,\cite{Borie:2014:arxiv_v7}, but should be included \cite{Borie:PC:2014}. It is given by $\epsilon_{2\gamma}= 0.0013$ and therefore results in -0.2230(86)\,\ensuremath{\mathrm{meV}}\xspace, using our Fermi energy from Eq.\,(\ref{eq:FermiAMM}). The attributed uncertainty originates from the number of significant digits in $\epsilon_{2\gamma}$ (the value of the coefficient is considered to be accurate only to $\pm0.00005$). Regarding the contributions given by Martynenko, no overlap is found, which is why we list this item separately.
\subsubsection{h24 electron VP contribution to two-photon exchange}
\label{sec:h24}
Item \#h24, the electron VP contribution to the 2S HFS elastic two-photon exchange in muonic helium-3 ions is only calculated by Martynenko \cite{Martynenko:2008:muheHFS}.
The corresponding Feynman diagrams are shown in Fig.~4 of his helium 2S HFS paper \cite{Martynenko:2008:muheHFS}. These are analogous to our Fig.~\ref{fig:tpe}, but with a VP loop in one of the exchange photons.
A numerical value of the contribution is given in his Eq.\,(38) of 0.095\,meV and thus enters \textit{our choice}, where we write 0.0950(5)\,\ensuremath{\mathrm{meV}}\xspace and therefore account for the precision given by Martynenko.
\subsubsection{h15, h16, h17 radiative corrections to the elastic two-photon exchange}
\begin{figure
\begin{center}
\begin{minipage}{0.22\columnwidth}
(a)\hfill\mbox{~}
\vspace{7px}\\
\begin{fmffile}{item_h17ba}
\begin{fmfgraph*}(60,50)
\fmfstraight
\fmftopn{t}{10}
\fmfbottomn{b}{10}
\fmf{plain}{t1,t5,t10}
\fmf{plain,width=3}{b1,b5,b10}
\fmf{photon,tension=0.1,left=0.8}{t4,t7}
\fmf{photon}{t3,b3}
\fmf{photon}{t8,b8}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b3}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b8}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{25px}
\begin{minipage}{0.22\columnwidth}
(b)\hfill\mbox{~}
\vspace{7px}\\
\begin{fmffile}{item_h17bc}
\begin{fmfgraph*}(60,50)
\fmfstraight
\fmftopn{t}{11}
\fmfbottomn{b}{11}
\fmf{plain}{t1,t5,t11}
\fmf{plain,width=3}{b1,b5,b11}
\fmf{photon,tension=0.1,left=0.8}{t6,t10}
\fmf{photon}{t4,b4}
\fmf{photon}{t8,b8}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b4}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b8}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\hspace{25px}
\begin{minipage}{0.22\columnwidth}
(c)\hfill\mbox{~}
\vspace{7px}\\
\begin{fmffile}{item_h17bb}
\begin{fmfgraph*}(60,50)
\fmfstraight
\fmftopn{t}{9}
\fmfbottomn{b}{5}
\fmf{plain}{t1,t5,t9}
\fmf{plain,width=3}{b1,b3,b5}
\fmf{photon,tension=0.1,left=0.7}{t2,t8}
\fmf{photon}{t3,b2}
\fmf{photon}{t7,b4}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b2}
\fmfv{decor.shape=circle,decor.filled=full,decor.size=10}{b4}
\fmfv{label=h}{b1}
\fmfv{label.angle=-150,label=$\mu$}{t1}
\end{fmfgraph*}
\end{fmffile}
\end{minipage}
\end{center}
\caption{
(a) Item \#h15, $\mu$SE contribution to the elastic two-photon exchange;
(b) item \#h16 the vertex correction to the elastic two-photon exchange, which results in two terms (the vertex correction can take place either at one or the other photon); and
(c) item \#h17, spanning photon contribution to the elastic two-photon exchange, also referred to as jellyfish diagram.}
\label{fig:h15h16h17}
\end{figure}
Items \#h15, \#h16, and \#h17 are radiative corrections to the elastic two-photon exchange in the 2S hyperfine structure and represented in Fig.\,\ref{fig:h15h16h17}. They are partially given in Martynenko's Ref.\,\cite{Martynenko:2008:muheHFS}, but have been updated \cite{Martynenko:PC:2017} and result to be $-0.0101\,\ensuremath{\mathrm{meV}}\xspace$ (\#h15), $0.0333\,\ensuremath{\mathrm{meV}}\xspace$ (\#h16), and $0.0074\,\ensuremath{\mathrm{meV}}\xspace$ (\#h17). These numbers include recoil corrections and are based on Eqs.(24)-(27) from the Martynenko group \cite{Faustov:2014:radrec} and use a dipole parameterization of the helion form factor, as well as $\rh=1.966\,$fm.
For the moment, we will adapt these preliminary numbers including recoil considerations into \textit{our choice}.
\subsubsection{h22 inelastic two-photon exchange in the hyperfine structure}
In contrast to the Lamb shift, no calculations are available for the inelastic two-photon exchange (polarizability contribution) in the 2S HFS. We give an estimate of this value by calculating the ratio between the polarizability contribution and the Zemach term in the 1S ground state of (electronic) $\mathrm{^3He^+}$ and assume the ratio to be similar for the 2S state in \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace.
The 1S Zemach term for electronic $\mathrm{^3He^+}$ is found by using Eq.\,(\ref{eq:zemach_term}), but with the muon mass replaced by the electron mass and $n=1$.
Using the Zemach radius \ensuremath{r_Z} from Friar and Payne \cite{Friar:2005:PRC72} a value of 1717\,kHz is obtained. In order to obtain the total sum (polarizability + Zemach) of 1442\,kHz \cite{Friar:2005:PRC72}, a polarizability term of order $-300$\,kHz is missing. The ratio is then roughly $-1/6$. The Zemach term for muonic helium-3 ions (our item h20), obtained above, yields $\Delta E_{\rm Zem}\approx 6.5\,\ensuremath{\mathrm{meV}}\xspace$. The estimate for the polarizability contribution consequently follows with $\Delta E^{\rm HFS}_{\rm pol.}\approx-1.0\pm1.0\,\ensuremath{\mathrm{meV}}\xspace$, which includes a conservative 100\% uncertainty.
\subsubsection{h25 and h26 finite size correction to electron VP}
\label{sec:eVPfinitesize}
Borie gives the electron VP contributions \#h8 and \#h5 (eVP processes in OPE, see Sec.~\ref{sec:VP}) which are based on a point nucleus.
Additionally, she provides modified contributions which include the finite size effect on electron VP.
These are $\epsilon_{VP1}'=0.00295$ and $\epsilon_{VP2}'=0.00486$, respectively.
The difference between those values and \#h8 and \#h5 constitute finite size corrections.
Multiplied with the Fermi energy (including the AMM), these yield 0.0343(9)\,meV each and we attribute them to \#h25 and \#h26, analogous to the previous CREMA summaries. The uncertainty originates from the precision in Borie's coefficients.
Note that these are OPE processes.
\subsubsection{h27 and h27b nuclear structure correction in leading order and SOPT}
This correction is only given by Martynenko. The two terms are found in Fig.~5(a) and (b) of Ref.\,\cite{Martynenko:2008:muheHFS}, for leading and second order, respectively.
This correction is also an OPE process.
Care has to be taken here because
this contribution is given as 0.272\,meV in \cite{Martynenko:2008:muheHFS}, but as 0.245\,meV in a 2010 follow up paper \cite{Martynenko:2010:2SHFS_muHe} (however, this is the only term that changed between \cite{Martynenko:2008:muheHFS} and \cite{Martynenko:2010:2SHFS_muHe}).
As compared to muonic deuterium, Martynenko only gives the sum (h27 $+$ h27b) and not the single contributions.
In \cite{Martynenko:2008:muheHFS} the formulas he uses to calculate h27 and h27b are explicitly given as
\begin{equation}
\Delta E^{\mathrm{HFS}}_\mathrm{1\gamma,str}
=-\frac{4}{3}(Z\alpha)^2 m_r^2r_M^2 \cdot E_\mathrm{Fermi}\cdot\frac{1-n^2}{4n^2}
\label{eq:1gammaStr}
\end{equation}
\begin{multline}
\Delta E^{\mathrm{HFS}}_\mathrm{str,SOPT}(2S)\\
=\frac{4}{3}(Z\alpha)^2m_1^2r_E^2 \cdot E_\mathrm{Fermi}(2S)\cdot
(\ln(Z\alpha) - \ln 2), \label{eq:strSOPT}
\end{multline}
where $m_r$ is the reduced mass of the muon, $m_1$ is the muon mass, and $r_E$ and $r_M$ are the charge and magnetic radii, respectively.
Martynenko states to use $r_E\approx r_M=1.844\pm0.045$\,fm which is known to be outdated.
However, inserting Martynenko's Fermi energy, the radius he used, and fundamental constants into Eqs.\,(\ref{eq:1gammaStr}) and (\ref{eq:strSOPT}) yields a sum of 0.2251$\pm$0.0001\,meV which is neither congruent with \cite{Martynenko:2008:muheHFS} nor \cite{Martynenko:2010:2SHFS_muHe}.
Using Sick's 2014 values \cite{Sick:2014:HeZemach} for the charge and magnetic radii yields 0.2577$\pm$0.0001\,meV.
In the course of some private communications with Martynenko, he provided us his most current value of 0.2421\,meV for the sum of h27+h27b, and we use this preliminarily as \textit{our choice}.\vspace{20px}
\subsection{h19 weak interaction}
\label{sec:weak}
The contribution of the weak interaction to the 2S HFS of helium-3 is only given
by Borie. She cites Eides \cite{Eides:2012:Weak} and provides $\epsilon_\mathrm{weak}=1.5\cdot 10^{-5}\hat{=}-0.0026\,\ensuremath{\mathrm{meV}}\xspace$, which we adopt as \textit{our choice}.\vspace{20px}
\subsection{Total 2S HFS contribution}
In total, the 2S HFS contributions are given by
\begin{widetext}
\begin{equation}
\label{eq:hfs:total1}
\begin{aligned}
\Delta E^{\rm HFS}(\ensuremath{2\textrm{S}_{1/2}^{\textrm{F}=1}}\xspace - \ensuremath{2\textrm{S}_{1/2}^{\textrm{F}=0}}\xspace) =&~ -172.7457(89) \,\ensuremath{\mathrm{meV}}\xspace +2.5836\,\ensuremath{\mathrm{meV}}\xspace/\mathrm{fm}~r_Z &+&~ \Delta E_{\rm pol.}^{\rm HFS}\\
=&~ \hspace{50px}-166.2145(423) &-&~ 1.0(1.0)\,\ensuremath{\mathrm{meV}}\xspace\\
=&~ -167.2(1.0)\,\ensuremath{\mathrm{meV}}\xspace.
\end{aligned}
\end{equation}
\end{widetext}
Here, in the first line, we separated out the Zemach contribution and the estimate of the polarizability contribution. In the second line, the Zemach radius $r_Z = 2.528(16)\,$fm \cite{Sick:2014:HeZemach} is inserted and the estimated value of $\Delta E_{\rm pol.}^{\rm HFS}$ is shown. The polarizability is the dominant source of uncertainty in the hyperfine structure and prevents a precise determination of the helion Zemach radius from the measured transitions in the muonic helium-3 ion \cite{CREMA:mu3he}. A calculation of the polarizability contribution is therefore highly desirable.
Until then a precise measurement of the 1S or 2S HFS in muonic helium-3 ions can be used to experimentally determine a value of the polarizability contribution $\Delta E_{\rm pol.}^{\rm HFS}$. In essence, the measurement of the 2S HFS by the CREMA collaboration can be used to give the total TPE contribution to the HFS, $\Delta E^\mathrm{HFS}_\mathrm{TPE} = 2.5836\,\ensuremath{\mathrm{meV}}\xspace/\mathrm{fm}~r_Z + \Delta E_\mathrm{pol.}^{\rm HFS}$ with an expected uncertainty of 0.1\,\ensuremath{\mathrm{meV}}\xspace.
\begin{landscape}
\begin{table}
\begin{minipage}{\linewidth}
\footnotesize
\setlength\extrarowheight{2pt}
\centering
\caption{All contributions to the {\bf 2S hyperfine splitting (HFS)}.
%
The item numbers h$i$ in the first column follow
the entries in Tab.\,3 of Ref.\,\cite{Antognini:2013:Annals}.
However, the terms are now sorted by increasing complexity, analogous to their order in the text.
%
For Martynenko, numbers \#1 to \#13 refer to rows in
Tab.\,I of his Ref.\,\cite{Martynenko:2008:muheHFS},
whereas numbers in parentheses refer to equations therein.
%
Borie~\cite{Borie:2014:arxiv_v7} gives the values as coefficients $\epsilon$
to be multiplied with the sum of (h1+h4) of 'our choice' values. We list the resulting values
in meV.
%
AMM: anomalous magnetic moment,
PT: perturbation theory,
VP: vacuum polarization,
SOPT: second order perturbation theory,
TOPT: third order perturbation theory.
All values are in meV.
Values in brackets do not contribute to the total sum.
}
\fontsize{6pt}{6pt}\selectfont
\label{tab:hfshelium}
\begin{tabular}{l|l|f{3} l l| f{3} l l| f{2} l c}
\hline
\hline
& \cntl{1}{Contribution}
& \cntl{3}{Borie (B)}
& \cntl{3}{Martynenko group (M)}
& \cnt{3}{Our choice}\\
& \cntl{1}{~}
& \cntl{3}{Borie \cite{Borie:2014:arxiv_v7}}
& \cntl{3}{Martynenko \cite{Martynenko:2008:muheHFS}}
& \cnt{2}{value} & \cnt{1}{source}\\
\hline
h1& Fermi splitting, $(Z\alpha)^4$
& ( -171.3964) && p.\,19
& -171.341 &&\#1, (6)
& -171.3508~\footnote{calculated in this work and given in Eq.~(\ref{eq:Fermi}).}
&
& \\
h4& $\mu$AMM corr., $\alpha(Z\alpha)^4$
& ( -0.1999) &&
& -0.200 &&\#2, (7)
& -0.1998 & & \\
sum& (h1+h4)
& -171.5963 &&p.\,19
& (-171.541) &&
&
&
& \\
h2& Breit corr., $(Z\alpha)^6$
& -0.0775 &$\pm$ 0.0001&p.\,19
& -0.078&&\#3, (8)
& -0.0775 &$\pm$ 0.0001 & B\\
\hline
h8& One-loop eVP in OPE, $\alpha(Z\alpha)^4$ ($\epsilon_\mathrm{VP1}$)
& -0.5404 &$\pm$ 0.0009&p.\,21
&-0.540 &&\#4, (12)
& -0.5404 &$\pm$ 0.0009 & B\\
h9& Two-loop eVP in OPE, $\alpha^2(Z\alpha)^4$ ($\epsilon_\mathrm{VP1}$)
& -0.0043 &&p.\,21
& -0.004 &&\#5, (15,16)
&-0.0043 &
& B\\
h5& One-loop eVP in OPE, SOPT, $\alpha(Z\alpha)^4$ ($\epsilon_\mathrm{VP2}$)
& -0.8680 &$\pm$ 0.0009&p.\,21
&-0.869 &&\#7, (24)
& -0.8685 & $\pm$ 0.0010 & avg.\\
h7& Two-loop eVP in OPE, SOPT, $\alpha^2(Z\alpha)^4$ ($\epsilon_\mathrm{VP2}$)
& -0.0067 &&p.\,21
&-0.010& & \#8, (29,30)
& -0.0084 & $\pm$ 0.0017 & avg.\\
h13& Vertex, $\alpha(Z\alpha)^5$
&0.0330&&p.\,19
& &&
& 0.0330&
& B\\
h14& Higher order corr.\ of (h13), part with ln($\alpha$)
& 0.0036~\footnote{The sign from Borie is wrong and has been corrected here, see Sec.~\ref{sec:vertex}.} &&p.\,19
& &&
&0.0036 & & B\\
h12& one-loop $\mu$VP in 1$\gamma$ int., $\alpha^6$
& (-0.0069)& incl.\ in h13 &p.\,19 \& p.\,21
& -0.007 &&\#6, (12)
& \cnt{2}{incl.\ in h13} & B\\
h18& Hadronic VP, $\alpha^6$
& -0.0091 &&p.\,19
& &&
& -0.0091 & & B\\
\hline
h20& Fin.\ size (Zemach) corr.\ to $\Delta E_\mathrm{Fermi}$, $(Z\alpha)^5$
& 6.5312~\footnote{Calculated by combining Borie's coefficient with Sick's $r_Z$.}
& (=2.5836~$r_Z$/fm )
&p.\,23
& 6.4085
&($\pm$ 0.1)~\footnote{This uncertainty reflects the change in this contribution when moving from dipole parameterization to a Gaussian one.}
& priv.comm.
& 2.5836
& $r_Z$/fm
& B \\
h23& Recoil of order $(Z\alpha)(m_1/m_2$)ln($m_1/m_2)E_F$
&
&&
& 0.0350
&
&priv.comm.
& 0.0350 &
&M \\
h23b*&Recoil of order $(Z\alpha)^2(m_1/m_2)E_F$
&
&&
& 0.038
&
& \#13, (48)
& 0.0380 &$\pm$ 0.0005
& M\\
h28*& Two-photon recoil
& -0.2230
&$\pm 0.0086$& \cite{Borie:1980:mu3HeLS}
&
&
&
& -0.2230 &$\pm$ 0.0086
& B\\
h24& eVP in two-photon-exchange, $\alpha^6$
& &&
& 0.095 &
&\#10, (38)
& 0.0950 & $\pm$ 0.0005
& M\\
h15& muon self energy contribution in TPE, w/recoil
& &&
& -0.0101 &&priv.comm.
& -0.0101 & & M\\
h16& vertex correction contribution in TPE, w/recoil
& &&
& 0.0333 &&priv.comm.
& 0.0333 & & M\\
h17& jelly fish correction contribution in TPE, w/recoil
& &&
& 0.0074 &&priv.comm.
& 0.0074 & & M\\
\hdashline
h22a& Helion polarizability, $(Z\alpha)^5$
& &
& & &&
& &
& \\
h22b& Helion internal polarizability, $(Z\alpha)^5$
& &
&
& &&
& & & \\
sum& (h22a+h22b)
& &
&
& &
&
& (-1.0 &$\pm$ 1.0)~\footnote{Is a preliminary estimate, see text. It is therefore listed separately in the sum below.} & \\
\hdashline
h25& eVP corr.\ to fin.\ size in OPE (sim.\ to $\epsilon_\mathrm{VP2}$)
& 0.0343 &$\pm$ 0.0009& p.\,21
& &&
& 0.0343
~\footnote{Difference of two terms in Borie~\cite{Borie:2014:arxiv_v7}, see also Sec.~\ref{sec:eVPfinitesize}.}
&$\pm$ 0.0009 & B\\
h26& eVP corr.\ to fin.\ size in OPE (sim.\ to $\epsilon_\mathrm{VP1}$)
& 0.0343 &$\pm$ 0.0009& p.\,21
& &&
& 0.0343 &$\pm$ 0.0009 & B\\
h27+h27b& Nucl.\ struct.\ corr. in SOPT, $\alpha(Z\alpha)^5$
& &&
& 0.2421 &&priv.comm.
& 0.2421 & & M\\
\hline
h19& Weak interact.\ contr.
& -0.0026
&$\pm$ 0.0001&p.\,21
& &&
& -0.0026 &$\pm$ 0.0001 & B\\
\hline
\hline
&&&&&&&&&&\\
& \bf Sum
& -166.6988~\footnote{Borie's sum given in this table differs from her published one of -166.3745\,meV \cite{Borie:2014:arxiv_v7}. This is because we used an updated value of the Fermi energy (see Sec.\,\ref{sec:EFermi2}), a different value for the Zemach radius $r_Z$ (see Sec.\,\ref{sec:Zemachterm}), and included item \#h28* which has not been considered in Ref.\,\cite{Borie:2014:arxiv_v7}.} &
&
&-165.1998~\footnote{Martynenko's sum given in this table is different from the (superseded) published one of -166.615\,meV \cite{Martynenko:2008:muheHFS} because several terms have been changed and added upon private communication.} &
&
& \bf -172.\bf7457&$\pm$ \bf 0.0089 &\\
&
& &
&
& &
&
& \bf 2.\bf5836& \bf \,$\boldsymbol{r_Z/\mathrm{fm}}$ &\\
&
& &
&
& &
&
& \bf -1.\bf0&$\pm$ \bf 1.0 &\\
&&&&&&&&&&\\
\hline
\hline
\end{tabular}
\end{minipage}
\end{table}
\end{landscape}
\clearpage
\begin{landscape}
\begin{table}
\begin{minipage}{\linewidth}
\footnotesize
\setlength\extrarowheight{3pt}
\fontsize{8pt}{8pt}\selectfont
\centering
\caption{
Contributions to the {\bf 2P fine structure}. Items \# with an asterisk *
denote new contributions in this compilation.
%
The items \#f7a, \#f7d, and \#f7e
originate from the same graphs as the Lamb shift items \#11, \#12, and \#30,
respectively.
%
VP: vacuum polarization, AMM: anomalous magnetic moment, KS: K\"all\'en-Sabry.
%
All values are in meV.}
\label{tab:fs}
\begin{tabular}{l|l |f{5} |f{5} l |f{7} l |f{3} l l}
\hline
\hline
\# & Contribution & \cntl{1}{Borie (B)}
& \cntl{2}{Martynenko group (M)}
& \cntl{2}{Karshenboim group (K)}
& \cnt{3}{Our choice} \\
& & \cntl{1}{Borie \cite{Borie:2014:arxiv_v7} Tab.\,7}
& \cntl{2}{Elekina {\it et\,al.}~\cite{Elekina:2010:2Pmu3He} Tab.\,1}
& \cntl{2}{Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509}}
& &
& \\
& & \cntl{1}{~}
& \cntl{2}{~}
& \cntl{2}{Korzinin {\it et\,al.}~\cite{Korzinin:2013:PRD88_125019}}
& &
& \\
\hline
f1 & Dirac
& 144.4157 & & & && & & \\
f2 & Recoil
& -0.1898 & & & && & & \\
f3 & Contrib.\ of order $(Z\alpha)^4$
& & 144.18648 & l.\,1
& && & & \\
f4a & Contrib.\ of order $(Z\alpha)^6$
& & 0.01994 & l.\,3
& && & & \\
f4b & Contrib.\ of order $(Z\alpha)^6\,m^2/M$
& & -0.00060 & l.\,4
& && & & \\
sum & (f1+f2) or (f3+f4)
& 144.2259 & 144.20582 & & &
& 144.2159 & $\pm\,0.0100$ & avg. \\
\hline
f5a & eVP corr.\ (Uehling), $\alpha(Z\alpha)^4$
& & 0.12925 & l.\,5
& && & & \\
f5b & eVP corr.\ SOPT, $\alpha(Z\alpha)^4$
& & 0.14056 & l.\,7
& && & & \\
f13*& eVP corr.\ SOPT, $\alpha^2(Z\alpha)^4$
& & 0.00028 & l.\,9
& && & & \\
sum & f5+f13*
& 0.2696 & 0.27009 & & 0.26920 & \cite{Karshenboim:2012:PRA85_032509} Tab.IV
& 0.2696 & $\pm\,0.0004$ & avg. \\
\hline
f6a & two-loop $e$VP corr.\ (KS), $\alpha^2(Z\alpha)^4$
& & 0.00098 & l.\,10+11
& && 0.0010 & & M \\
f6b & two-loop $e$VP in SOPT, $\alpha^2(Z\alpha)^4$
& 0.0021 & 0.00234 & l.\,12+13
& 0.00242 & \cite{Korzinin:2013:PRD88_125019} Tab.IX ``eVP2''
& 0.0024 & & K \\
f7a & $\alpha^2(Z\alpha)^4 m$, like \#11
& & & & 0.000606 & \cite{Korzinin:2013:PRD88_125019} Tab.IX (a)
& 0.0006 & & K \\
f7d & $\alpha^2(Z\alpha)^4 m$, like \#12
& & & & 0.00164 & \cite{Korzinin:2013:PRD88_125019} Tab.IX (d)
& 0.0016 & & K \\
f7e & $\alpha^2(Z\alpha)^4 m$, like \#30$^*$
& & & & 0.000019(2) & \cite{Korzinin:2013:PRD88_125019} Tab.IX (e)
& 0.0000 & & K \\
f11*& $\alpha(Z\alpha)^6$
& & -0.00055 & l.\,8
& && -0.0006 & & M \\
f12*& one-loop $\mu$VP, $\alpha(Z\alpha)^4$
& & 0.00001 & l.\,6
& && 0.0000 & & M \\
\hline
f8 & AMM (second order)
& 0.3232 & & & && & & \\
f9 & AMM (higher orders)
& 0.0012 & & & && & & \\
sum & Total AMM (f8+f9)
& 0.3244 & 0.32446 & l.\,2
& && 0.3244
& &avg. \\
\hline
f10 & Finite size, $(Z\alpha)^6$~\footnote{This is item \#r8,
evaluated for a helion radius of 1.966(10)\,fm \cite{Borie:2014:arxiv_v7}, see text. The uncertainty is propagated from the charge radius, but is negligible.}
& -0.0158 & & & && -0.0158
& $\pm\,0.0002$ & B \\
\hline
\hline
&&&&&&\\
& \bf Sum & 144.8062 & 144.80315 & & &
& {\ensuremath{\bf 144}}\bf .{\ensuremath{\bf 7993}} & \bf $\pm$\,{\ensuremath{\bf 0}}\bf .{\ensuremath{\bf 0101}} & \\
&&&&&&\\
\hline
\hline
\end{tabular}
\end{minipage}
\end{table}
\end{landscape}
\section{2P levels}
\label{sec:2Plevels}
\subsection{2P fine structure}
\label{sec:fs}
Fine structure (FS) contributions have been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,7), the Martynenko group \cite{Elekina:2010:2Pmu3He} (Tab.\,1), and the Karshenboim group \cite{Karshenboim:2012:PRA85_032509} (Tab.\,4) and \cite{Korzinin:2013:PRD88_125019} (Tab.\,9). All of these contributions are listed in Tab.\,\ref{tab:fs} and labeled with \#f$i$.
The leading fine structure contribution of order $(Z\alpha)^4$ has been calculated by Borie using the Dirac wavefunctions (same as in Lamb shift). Her result (our item \#f1) has to be corrected by a recoil term (item \#f2) in order to be compared with the result from the Martynenko group. They use a nonrelativistic approach (our item \#f3) and then add relativistic corrections (our item \#f4a+b). Their total results differ by 0.02\,\ensuremath{\mathrm{meV}}\xspace. We take the average as \textit{our choice} and remain with an uncertainty of 0.01\,\ensuremath{\mathrm{meV}}\xspace. This is by far the dominant uncertainty in the 2P fine structure.
Item \#f5a and \#f5b are the one-loop $e$VP of order $\alpha(Z\alpha)^4$ in leading order and SOPT. Item \#f13* is the one-loop $e$VP contribution of order $\alpha^2(Z\alpha)^4$ in SOPT. All three items are given individually by the Martynenko group \cite{Elekina:2010:2Pmu3He} in lines 5, 7, and 9 of their Tab.\,1.
In Tab.\,7 of \cite{Borie:2014:arxiv_v7}, Borie's term ``Uehling(VP)'' presumably contains all these three items. Karshenboim {\it et\,al.}~\cite{Karshenboim:2012:PRA85_032509} (Tab.\,4) also calculate the sum of these items. All agree within 0.0009\,\ensuremath{\mathrm{meV}}\xspace and we take the average as \textit{our choice} which coincides with Borie's value.
Item \#f6a and \#f6b are the two-loop $e$VP (\textit{K\"all\'en-Sabry}) contribution of order $\alpha^2(Z\alpha)^4$ in leading order and SOPT. These terms have been calculated by Martynenko {\it et\,al.}~\cite{Elekina:2010:2Pmu3He} (Tab.\,1, line 10+11 and 12+13, respectively). Borie \cite{Borie:2014:arxiv_v7} and the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,IX) only calculated our item \#f6b. We therefore adopt the value provided by the Martynenko group for item \#f6a and the Karshenboim group's value of \#f6b as they included some higher order terms as well.
Items \#f7a, \#f7d, and \#f7e are of order $\alpha^2(Z\alpha)^4$ and have been calculated with high accuracy by the Karshenboim group \cite{Korzinin:2013:PRD88_125019} (Tab.\,IX). They correspond to the same Feynman diagrams as the Lamb shift items \#11, \#12, and \#30, shown in Figs.\,\ref{fig:item_11}, \ref{fig:item_12}, and \ref{fig:item_30}, respectively. We adopt the values from the Karshenboim group as \textit{our choice}.
Item \#f11* is a contribution of order $\alpha(Z\alpha)^6$ which has been calculated by Martynenko {\it et\,al.}~\cite{Elekina:2010:2Pmu3He} (Tab.\,1, line 8).
Item \#f12* is the one-loop $\mu$VP of order $\alpha(Z\alpha)^4$ which has been calculated by the Martynenko group as well \cite{Elekina:2010:2Pmu3He} (Tab.\,1, line 6). We adopt both of these values as \textit{our choice}.
The sum of items \#f8 and \#f9 is the muon anomalous magnetic moment (AMM) contribution of order $(Z\alpha)^4$. These items are labeled by Borie \cite{Borie:2014:arxiv_v7} as ``second order'' and ``higher orders'', respectively. Martynenko {\it et\,al.}~\cite{Elekina:2010:2Pmu3He} (Tab.\,1, line 2) provide the sum of these. Both groups agree very well. As \textit{our choice} we adopt the average.
Item \#f10 is the finite size correction to the \ensuremath{2\textrm{P}_{1/2}}\xspace level of order $(Z\alpha)^6$ which has only been calculated by Borie \cite{Borie:2014:arxiv_v7}. It is the same correction which appears in the radius dependent part of the Lamb shift as \#r8, with opposite sign and evaluated with a helion charge radius of 1.966(10)\,fm \cite{Borie:2014:arxiv_v7}. We adopt Borie's value as \textit{our choice} and add the uncertainty which we obtain from the given charge radius.
The total sum of the FS contributions is summarized in Tab.\,\ref{tab:fs} and amounts to
\begin{equation}\label{eq:fs}
\ensuremath{\Delta E_\mathrm{FS}} = \ensuremath{144.7993}\,\ensuremath{\mathrm{meV}}\xspace \pm \ensuremath{0.0101} \,\ensuremath{\mathrm{meV}}\xspace.
\end{equation}
It will enter the calculation of the 2P hyperfine structure in the following section. Note, that the uncertainty originates only from differences in the treatment of Dirac term (sum of items \#f1 to \#f4).
\subsection{2P hyperfine structure}
\label{sec:hfs}
The 2P hyperfine splitting is described by the Breit Hamiltonian. Off-diagonal terms appear in the matrix representation of this Hamiltonian in the basis of \ensuremath{2\textrm{P}_{1/2}^{\textrm{F}=1}}\xspace, \ensuremath{2\textrm{P}_{1/2}^{\textrm{F}=0}}\xspace, \ensuremath{2\textrm{P}_{3/2}^{\textrm{F}=2}}\xspace, and \ensuremath{2\textrm{P}_{3/2}^{\textrm{F}=1}}\xspace. These terms lead to a mixing of energy levels with same quantum number $F$ (see Fig.\,\ref{fig:energy_level}). This has first been calculated by Brodsky and Parsons \cite{Brodsky:1967:zeemanspectrum} for hydrogen and later has also been evaluated for muonic hydrogen by Pachucki \cite{Pachucki:1996:LSmup}.
In previous publications \cite{Antognini:2013:Annals,Krauth:2016:mud}, we also discussed the mixing of hyperfine states.
The traditional way \cite{Brodsky:1967:zeemanspectrum,Pachucki:1996:LSmup} is to calculate the FS (without perturbations from the HFS $F$ state mixing) and then include the so obtained FS in the evaluation of the Breit matrix. The centroids of the diagonal elements are now the virtual levels \ensuremath{2\textrm{P}_{1/2}}\xspace and \ensuremath{2\textrm{P}_{3/2}}\xspace. \textit{Afterwards} the mixing is included (via diagonalization) which means that the actual centroid is not at the position of the virtual levels anymore.
The 2P hyperfine structure has been calculated by Borie \cite{Borie:2014:arxiv_v7} (Tab.\,9) and Martynenko {\it et\,al.}~\cite{Elekina:2010:2Pmu3He} (Tab.\,2). We also calculated the splittings following Pachucki \cite{Pachucki:1996:LSmup}, who did the evaluation for \ensuremath{\mu}{\rm p}\xspace. The values which are listed in our Tab.\,\ref{tab:2Phfs} are not the published values, but the values which result when including our FS value from Sec.\,\ref{sec:fs}.
\begin{table*}
\setlength\extrarowheight{7pt}
\centering
\caption[2P levels from fine- and hyperfine splitting]{
{\bf 2P levels from fine- and hyperfine splitting}.
%
All values are in meV relative to the 2P$_{1/2}$ level.
%
The columns labeled with Borie and Martynenko include their HFS calculations,
but our value of the fine structure (2P$_{3/2} - $2P$_{1/2}$
energy splitting) $\ensuremath{\Delta E_\mathrm{FS}} =\FSVAL \ensuremath{(101)}\,$meV from Eq.\,(\ref{eq:fs}).
%
The column 'following \cite{Pachucki:1996:LSmup}' is calculated in this work following the
treatment of Pachucki for \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace, also including our value of the fine structure.
%
Uncertainties arise from differences between the published values and from the uncertainty in the fine structure value
\ensuremath{\Delta E_\mathrm{FS}}
}
\label{tab:2Phfs}
\begin{tabular}{l f{6} f{9} f{8} f{14}}
\hline
\hline
& \lft{1}{Borie \cite{Borie:2014:arxiv_v7}}
& \lft{1}{Martynenko \cite{Elekina:2010:2Pmu3He}}
& \lft{1}{following \cite{Pachucki:1996:LSmup}}
& \lft{1}{Our choice} \\[0.5ex]
\hline
\ensuremath{2\textrm{P}_{1/2}^{\textrm{F}=1}}\xspace & -14.7877 & -14.8080 & -14.7990 & -14.7979(102) \\
\ensuremath{2\textrm{P}_{1/2}^{\textrm{F}=0}}\xspace & 43.8458 & 43.9049 & 43.8797 & 43.8754(296) \\
\hline
\ensuremath{2\textrm{P}_{3/2}^{\textrm{F}=2}}\xspace& 135.7580 & 135.7552 & 135.7527 & 135.7554 (27)(101)_{\rm FS} \\% total uncertainty: (104)
\ensuremath{2\textrm{P}_{3/2}^{\textrm{F}=1}}\xspace& 160.0410 & 160.0459 & 160.0494 & 160.0452 (42)(101)_{\rm FS} \\% total uncertainty: (108)
\hline
\hline
\end{tabular}
\end{table*}
Borie in her Tab.\,9 lists the energies of the four 2P hyperfine levels relative to the \ensuremath{2\textrm{P}_{1/2}}\xspace fine structure state where she already included the $F$ state mixing. We reproduced her results using the Eqs.\ given in her Tab.\,9 and then inserted our \ensuremath{\Delta E_\mathrm{FS}} from our Eq.\,(\ref{eq:fs}). The result is listed in the second column of Tab.\,\ref{tab:2Phfs}. Borie mentions, she used the shielded helion magnetic moment, whereas the (unshielded) magnetic moment should be used. The change, however, appears only on the seventh digit and is therefore negligible.
In their Tab.\,2, Martynenko {\it et\,al.}\ provide the total splittings of the hyperfine structure levels, and at the end of their Sec.\,3, they list the term $\Delta = 0.173\,\ensuremath{\mathrm{meV}}\xspace$ originating from the mentioned $F$ state mixing. In order to include this term, the numbers in their Tab.\,2 first have to be divided according to the weight given by the number of $m_F$ states. $\Delta$ has then to be added to the two $F=1$ states. Furthermore, for the \ensuremath{2\textrm{P}_{3/2}}\xspace states, we add our \ensuremath{\Delta E_\mathrm{FS}} . The result is listed in the third column of our Tab.\,\ref{tab:2Phfs}.
Additionally, following Pachucki \cite{Pachucki:1996:LSmup}, we repeat his calculations in \ensuremath{\mu}{\rm p}\xspace for \ensuremath{\mu^3}{\rm He}\ensuremath{^+}\xspace. The off-diagonal elements are given by Eq.\,(85) of \cite{Pachucki:1996:LSmup}
\begin{multline}
\langle \, \ensuremath{2\textrm{P}_{1/2}^{\textrm{F}=1}}\xspace \, |\, V\, |\, \ensuremath{2\textrm{P}_{3/2}^{\textrm{F}=1}}\xspace \,\rangle\\
= \frac{1}{3}(Z\alpha)^4 \frac{m_r^3}{m_\mu m_h}(1+\kappa)
\left(1+\frac{m_\mu}{m_h}\frac{1+2\kappa}{1+\kappa}\right)\left(-\frac{\sqrt{2}}{48}\right),
\end{multline}
where we included the correct $Z$ scaling. $m_r$ is the reduced mass of the muonic helium-3 ion, $m_\mu$ ($m_h$) is the mass of the muon (helion), and $\kappa = -4.18415$~\footnote{The helion anomalous magnetic moment is obtained using the respective equation on p.\,17 of Borie's Ref.\,\cite{Borie:2014:arxiv_v7}, where this magnitude is denoted as $\kappa_2$.} is the
helion anomalous magnetic moment.
The diagonal terms are given by Eq.\,(86) therein
\begin{multline}
E_{\rm HFS}(\ensuremath{2\textrm{P}_{1/2}}\xspace)\\ = \frac{1}{3}(Z\alpha)^4 \frac{m_r^3}{m_\mu m_h}(1+\kappa)
\left(\frac{1}{3}+\frac{a_\mu}{6}+\frac{1}{12}\frac{m_\mu}{m_h}\frac{1+2\kappa}{1+\kappa}\right)
\label{eq:2Pone}
\end{multline}
\begin{multline}
E_{\rm HFS}(\ensuremath{2\textrm{P}_{3/2}}\xspace)\\ = \frac{1}{3}(Z\alpha)^4 \frac{m_r^3}{m_\mu m_h}(1+\kappa)
\left(\frac{2}{15}-\frac{a_\mu}{30}+\frac{1}{12}\frac{m_\mu}{m_h}\frac{1+2\kappa}{1+\kappa}\right)
\label{eq:2Pthree}
\end{multline}
with the anomalous magnetic moment of the muon $a_\mu = 1.165\,920\,89(63)\times10^{-3}$ \cite{Mohr:2016:CODATA14}.
Furthermore, Pachucki adds corrections due to vacuum polarization in his Eq.\,(89) and (90). With correct $Z$ scaling these are
\begin{align}
\delta E_{\rm HFS}(\ensuremath{2\textrm{P}_{1/2}}\xspace) =& \frac{1}{3}(Z\alpha)^4\frac{m_r^3}{m_\mu m_h}(1+\kappa) \cdot0.00022 \\
\delta E_{\rm HFS}(\ensuremath{2\textrm{P}_{3/2}}\xspace) =& \frac{1}{3}(Z\alpha)^4\frac{m_r^3}{m_\mu m_h}(1+\kappa) \cdot 0.00008.
\end{align}
They have to be added to Eqs.\,(\ref{eq:2Pone}) and (\ref{eq:2Pthree}), respectively.
Diagonalizing the matrix given in Eq.\,(91) of Ref.\,\cite{Pachucki:1996:LSmup} with entries determined by the above equations yields the values given as \textit{our choice} in Tab.\,\ref{tab:2Phfs}.
The diagonalization yields an $F$ mixing of $\Delta = 0.1724\,\ensuremath{\mathrm{meV}}\xspace$.
In the same manner as for the sections above, \textit{our choice} in Tab.\,\ref{tab:2Phfs} takes into account the spread of values from the different authors and additionally the uncertainty of our value of the fine structure which we obtained in Sec.\,\ref{sec:fs}. It is astonishing that the splitting of the \ensuremath{2\textrm{P}_{1/2}}\xspace states differs by as much as 0.04\,\ensuremath{\mathrm{meV}}\xspace between Borie and Martynenko. These states do not overlap with the nucleus, so it should be possible to determine them to much better precision. A precise calculation of these splittings is therefore highly welcome.
\section{Summary}
We have compiled all available contributions necessary to extract a charge radius of the helion from the Lamb shift measurement in muonic helium-3 ions, performed by the CREMA collaboration.
The total of the Lamb shift contributions are summarized in Eq.\,\ref{eq:LS:full}.\\
The nuclear structure-independent contributions of the Lamb shift, given in Tab.\,\ref{tab:LS:QED}, show good agreement within the four (groups of) authors. The uncertainty is dominated by the hadronic VP (\#14) and higher order radiative recoil corrections (\#24). The total uncertainty in Tab.\,\ref{tab:LS:QED}, however, is in the order of 0.01\,\ensuremath{\mathrm{meV}}\xspace and therefore sufficiently good (see also Eq.~\ref{eq:uncertainty}).\\
The nuclear structure-dependent part of the Lamb shift completely dominates the theoretical uncertainties. The one-photon exchange (finite size) contributions, where the coefficients are given in Tab.\,\ref{tab:LS:Radius}, have an uncertainty which corresponds to 0.04\,\ensuremath{\mathrm{meV}}\xspace, which already is above the ``ideal'' precision, mentioned in the introduction. This uncertainty is dominated by a disagreement in the terms \#r4 and \#r6.
The much larger uncertainty, however, stems from the two-photon exchange contributions (TPE), given in Eq.\,(\ref{eq:LS:pol}). Recently, two groups have published new calculations on the TPE with a precision of about 3\% ($\sim0.5\,\ensuremath{\mathrm{meV}}\xspace$). In terms of the helion charge radius this uncertainty corresponds to about
\begin{equation}
\sigma_{\rm theory}(\rh)\approx \pm 0.0013\,\ensuremath{\mathrm{fm}}\xspace.
\end{equation}
The expected experimental uncertainty will be about an order of magnitude smaller.
Thus, improving the theoretical uncertainty directly improves the extraction of the charge radius.
Isotope shift measurements generally benefit from cancellations of theory contributions that limit the absolute charge radii \cite{CancioPastor:2012:PRL108,Jentschura:2011:IsoShift}. For the present case of the muonic helium isotope shift it will be useful to exploit possible correlations between the nuclear and nucleon structure contributions, which dominate the total uncertainty of the muonic radii. The correlations could lead to a reduction of the uncertainty of the muonic isotope shift determination and shed light on the $4\,\sigma$ discrepancy in the electronic isotope shift measurements, see Fig.\,\ref{fig:iso_shift}. A further investigation of these correlations is therefore desired.
The total of the 2S HFS contributions are given in Tab.\,\ref{tab:hfshelium} and summarized in Eq.\,\ref{eq:hfs:total1}.
The uncertainty in the 2S HFS is completely dominated by the polarizability contribution, where no calculation exists. We have given a very rough estimate. The second largest uncertainty in the 2S HFS originates from the Zemach radius term (Bohr-Weisskopf effect).
The upcoming results of the CREMA experiment will be able to extract a value for the TPE in the 2S hyperfine splitting (sum of polarizability and Zemach radius contribution) from measured data. In this case the uncertainty will be limited by the experimental uncertainty.
For the 2P levels, we collect all fine structure terms from the various authors (Tab.\,\ref{tab:fs}) which are then used to calculate the hyperfine structure by means of the Breit matrix. The results are compared with two other groups (Tab.\,\ref{tab:2Phfs}). Here, the largest uncertainty originates from the leading order contributions (\#f1 to \#f4) in the fine structure (which is still sufficiently good) and from differing published values of the \ensuremath{2\textrm{P}_{3/2}}\xspace splitting. A clarification of this difference would be very welcome.
\\\\
Note added in proof: After this manuscript was accepted for publication, a paper by Karshenboim {\it et\,al.}\,\cite{Karshenboim:2016:mu3he} about the Lamb shift theory in muonic helium and tritium was published. They discuss the 2S-2P Lamb shift and the 2P fine- and hyperfine structure. The 2S hyperfine structure is not treated therein. The comparison of their values with ours has to be done carefully because Karshenboim {\it et\,al.}\ treat the mixing of the hyperfine levels (Brodsky Parsons contribution) differently. In their work the mixing is added as a perturbation to the fine structure. The traditional way, however, is to use the unperturbed fine structure and add the mixing as a perturbation to the hyperfine levels, which is what we do. Comparing the values one therefore has to subtract/add the Brodsky Parsons term printed in bold italic in \cite{Karshenboim:2016:mu3he}. Furthermore Karshenboim {\it et\,al.}\ neglect some known higher order terms and increase the uncertainty due to estimates of non-listed higher order contributions.
The comparison with the values in Ref. \cite{Karshenboim:2016:mu3he} yields the following (the numbers shown here are \textit{adapted} to the traditional treatment of the Brodsky Parsons contribution): For the radius-independent QED Lamb shift without TPE, Karshenboim {\it et\,al.}\ obtain a value of 1644.35(2)\,\ensuremath{\mathrm{meV}}\xspace which is in very good agreement with ours (Eq.\,\ref{eq:LS:QED}). In order to compare the radius-dependent (finite size) part we use a helion charge radius of 1.966\,\ensuremath{\mathrm{fm}}\xspace \cite{Borie:2014:arxiv_v7}. The value of Karshenboim {\it et\,al.}\ is then $-399.69(23)^\mathrm{theo}\,\ensuremath{\mathrm{meV}}\xspace$ which differs by 0.33(23)\,\ensuremath{\mathrm{meV}}\xspace ($1.4\sigma$) from our value of $-400.02(4)^\mathrm{theo}\,\ensuremath{\mathrm{meV}}\xspace$. This difference is the largest between our values and the ones from Karshenboim {\it et\,al.}. For the 2P fine structure, Karshenboim {\it et\,al.}\ obtain a value of $144.800(5)\,\ensuremath{\mathrm{meV}}\xspace-0.004\,\rh^2\,\ensuremath{\mathrm{meV}}\xspace/\,\ensuremath{\mathrm{fm}}\xspace^2$ which differs by 0.0142\,\ensuremath{\mathrm{meV}}\xspace ($1.3\sigma$) from ours. Regarding the 2P$_{1/2}$ hyperfine structure, the value from Karshenboim {\it et\,al.}\ of $-58.7150(7)\,\ensuremath{\mathrm{meV}}\xspace$ differs by 0.0417\,\ensuremath{\mathrm{meV}}\xspace ($1.3\sigma$) and has by far the smaller uncertainty. In our case the uncertainty arises from the huge difference between Borie and Martynenko. The 2P$_{3/2}$ splitting of $-24.2925(7)\,\ensuremath{\mathrm{meV}}\xspace$ agrees very well with our value.\\
However, all these differences are considerably smaller than the uncertainty of the two-photon contribution which we assumed to be 0.52\,\ensuremath{\mathrm{meV}}\xspace while Karshenboim {\it et\,al.}\ increase it to 0.86\,\ensuremath{\mathrm{meV}}\xspace. The final result for the charge radius will therefore not be changed significantly.
\section{Acknowledgments}
\label{sec:ack}
We are grateful to
E.~Borie and A.P.~Martynenko for insightful comments and for providing us with previously unpublished results.
We thank M.~Gorchtein and N.~Nevo Dinur for helpful discussions about the two-photon exchange in muonic helium-3 ions and the treatment of the Friar moment contribution.
We acknowledge valuable contributions in general from
S.~Bacca,
N.~Barnea,
M.~Birse,
E.~Borie,
C.E.~Carlson,
M.~Eides,
J.L.~Friar,
M.~Gorchtein,
F.~Hagelstein,
C.~Ji,
S.~Karshenboim,
A.P.~Martynenko,
J.~McGovern,
N.~Nevo Dinur,
K.~Pachucki, and
M.~Vanderhaeghen
and are thankful for
their valuable remarks and insightful discussions.
We proactively thank a future generation of motivated theorists for all future critical compilations of theory terms in light muonic atoms/ions.
The authors acknowledge support from the European Research Council
(ERC) through StG. \#279765 and CoG. \#725039, the Excellence Cluster PRISMA of the Unversity of Mainz, and the Swiss National Science
Foundation SNF, Projects 200021L\_138175 and 200021\_165854.
\section{Author contribution statement}
B.F.~and J.J.K.~set up the tables and wrote the manuscript. Both contributed equally to the paper. The paper was written under the supervision of and includes many comments and suggestions from A.A., F.K., and R.P., whereas M.D.~participated in the discussion.
All authors discussed the paper and participated in the review.
\bibliographystyle{mysty1}
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Энн Пэтчетт (, р. 2 декабря 1963) — американская писательница. Лауреат литературных премий «ПЕН/Фолкнер» и «Оранж» за роман «Бельканто», написанный в 2001 году. В 2012 журнал Time включил Пэтчетт в список 100 наиболее влиятельных людей в мире. На русском языке вышли ее романы «Бельканто» (2018), «Свои-чужие» (2018), «Прощальный фокус» (2019), «Предчувствие чуда» (2020), сборник эссе «Это история счастливого брака" (2020) и роман «Голландский дом» (2021), который на данный момент считается вершиной творчества Энн Пэтчетт. В 2020 году «Голландский дом» вошел в число финалистов Пулитцеровской премии.
Биография
Родилась Лос-Анджелесе в семье капитана полиции и медсестры. После развода родителей Пэтчетт в возрасте шести лет вместе с матерью и старшей сестрой переехала в Нэшвилл, где училась в частной католической школе для девочек — Академии Святого Бернарда. После окончания школы посещала в Нью-Йорке, курсы писательского мастерства в Университете Айовы, а также проходила обучение в Центре изящного искусства в Провинстауне, Массачусетс. Именно там Энн Пэтчетт приступила к работе над своей первой книгой - романом «Святой покровитель лжецов» («The Patron Saint of Liars»), который вышел в 1992 году.
Энн Пэтчетт сотрудничала со многими периодическими изданиями, писала для The Paris Review, The New Yorker, The New York Times Magazine, The Washington Post, O, The Oprah Magazine, ELLE, GQ, Gourmet, Vogue.
Известность пришла к ней после выхода четвертого романа — «Бельканто» («Bel Canto»). Книга принесла Пэтчетт несколько номинаций и побед в различных литературных премиях, а также вошла в ряд книжных рейтингов по итогам года. Роман был адаптирован для оперы, её премьера состоялась 7 декабря 2015 года. Три года спустя режиссёр Пол Вайц экранизировал книгу, главную роль в фильме «Бельканто» (2018) исполнила Джулианна Мур. В 2019 году роман "Бельканто", переизданный издательством "Синдбад", вошел в длинный список престижной литературной премии "Ясная поляна" в номинации "Иностранная литература".
В 2010 году Пэтчетт стала соосновательницей независимого книжного магазина Parnassus (открытие состоялось в ноябре 2011) в Нэшвилле, где из-за влияния корпораций не осталось ни одного книжного магазина:
Если отбросить условности, в список [100 наиболее влиятельных людей в мире] журнала TIME Энн Пэтчетт включили за противостояние компании Amazon.
Библиография
На английском языке
Романы
Репринт:
Нехудожественные произведения
Репринт:
На русском языке
Энн Пэтчетт. Бельканто. — Синдбад, 2018. — 448 с. Перевод Марины Карасевой. ISBN 978-5-00131-032-7, ISBN 978-5-00131-124-9
Энн Пэтчетт. Свои-чужие. — Синдбад, 2018. — 416 с. Перевод Александра Богдановского. ISBN 978-5-00131-011-2
Энн Пэтчетт. Прощальный фокус. — Синдбад, 2019. — 448 с. Перевод Елены Осеневой. ISBN 978-5-00131-121-8
Энн Пэтчетт. Предчувствие чуда. — Синдбад, 2020. — 448 с. Перевод Ирины Гиляровой. ISBN 978-5-00131-159-1
Энн Пэтчетт. Это история счастливого брака. — Синдбад, 2020. — 480 с. Перевод Сергея Кумыша. ISBN 978-5-00131-142-3
Энн Пэтчетт. Голландский дом. — Синдбад, 2021. — 320 с. Перевод Сергея Кумыша. ISBN 978-5-00131-231-4
Примечания
Ссылки | {
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Q: WebElement.findElement method is finding element under WebDriver scope I have the following HTML code on which I am trying to run my selenium test
<html>
<head></head>
<body>
<table id="Original">
<tr>
<td>Original-11</td>
<td>Original-12</td>
<td>Original-13</td>
</tr>
<tr>
<td>Original-21</td>
<td>Original-22</td>
<td>Original-23</td>
</tr>
<tr>
<td>Original-31</td>
<td>Original-32</td>
<td>Original-33</td>
</tr>
</table>
<br/><br/>
<table id="Duplicate">
<tr>
<td>Duplicate-11</td>
<td>Duplicate-12</td>
<td>Duplicate-13</td>
</tr>
<tr>
<td>Duplicate-21</td>
<td>Duplicate-22</td>
<td>Duplicate-23</td>
</tr>
<tr>
<td>Duplicate-31</td>
<td>Duplicate-32</td>
<td>Duplicate-33</td>
</tr>
</table>
</body>
</html>
The selenium java code looks like this:
public static void main(String[] args) throws Exception
{
System.setProperty("webdriver.chrome.driver", System.getProperty("user.dir") + "/src/test/drivers/chromedriver.exe");
WebDriver drv = new ChromeDriver();
drv.get("C:/Users/MYUserName/git/er_test/SeleniumTestHtml.html");
WebElement tableElement = drv.findElement(By.id("Duplicate"));
WebElement rowElement = tableElement.findElement(By.xpath("//tr[2]"));
WebElement cellElement = rowElement.findElement(By.xpath("//td[2]"));
System.out.println(rowElement.getText());
System.out.println(cellElement.getText());
drv.close();
drv.quit();
}
I am expecting a result as follows :
Duplicate-21 Duplicate-22 Duplicate-23
Duplicate-22
But I am getting this result :
Original-21 Original-22 Original-23
Original-12
Am I doing something wrong here ?
A: Your problem is that // makes you go back to search from top root element.
Which means
WebElement rowElement = tableElement.findElement(By.xpath("//tr[2]"));
Is as good as
WebElement rowElement = driver.findElement(By.xpath("//tr[2]"));
If you need to search only inside the current element then you should use
WebElement rowElement = tableElement.findElement(By.xpath(".//tr[2]"));
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PS4 Remote Play Now Available On PCs and Macs With 3.50 Firmware Update
Posted by Tim Verry | Apr 13, 2016 | General Tech | 13
game streaming10
Playstation 417
PSN5
remote play4
Source: Sony
Sony is rolling out a new firmware update for its PlayStation 4 gaming console. The 3.50 firmware update adds social networking features to schedule events and allow users to appear offline along with a major change that opens up Remote Play to allow game streaming from the PS4 to Macs and Windows PCs.
Users should start receiving the console update shortly. In order to stream to PCs, users will need to download the Remote Play utility for Windows or OS X. PC system requirements are modest requiring a minimum of a dual core (4 thread) Intel Core i5 560M (2.67 GHz) and 2GB of RAM when running Windows. Mac users can get by with an even lower end i5 520M (2.4 GHz). Users will need to be running the 32-bit or 64-bit versions of Windows (8.1 or 10) or Mac OS X 10.10 Yosemite or newer.
Sony recommends having a bare minimum of a 5Mbps symmetrical broadband internet connection in order to stream games to remote devices, and it recommends a connection with at least 12 Mbps download and upload speeds for the best results. Unfortunately, this rules out most DSL users, though they should still be able to play locally over their LAN. (It is not clear whether you can direct connect to the console to stream or if you have to go through a Sony server to stream, other remote play devices seem to be able to work only off of the LAN connection though so it should work.)
Sony makes it easy to play your games by supporting the DualShock 4 controller – users will simply need to plug it into the PC via USB cable and it will work as expected on PlayStation games. You will also need a Sony Entertainment Network account to pair devices and it is recommended to set the desired PS4 as your primary account. Specific setup instructions can be found here.
Streaming capabilities are currently limited as there is no support for streaming at 1080p resolution. Out of the box, Remote Play will stream at 540p and 30 FPS (frames per second). Users (preferably with wired devices including the PS4) can go into the settings and max it out at 720p and 60 FPS or dial it all the way down to 360p if you really need to play remotely over the internet with a small upload pipe.
Sony notes that not all games support Remote Play, but it seems like the majority of the console's catalog of games do.
There are several YouTube videos of users testing out Remote Play, and it does work. It seems to be a bit behind Xbox One streaming in the video quality and usability departments (e.g. no 1080p and you can't change resolution and frame rate on the fly). Hopefully Sony continues to flesh out the application and features.
Have you had a chance to try PS4 to PC game streaming? I'm now waiting for Microsoft to allow PC to Xbox One streaming hehe.
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StressedOutCat on April 13, 2016 at 9:56 am
Works great, I can play PS4
Works great, I can play PS4 games at work on my laptop with DS4
only downside is the colors seem like washed out a bit (even 720p).
it looks much better on my PSVita doing remote play, maybe is because that's a smaller oled screen.
but latency seems small, but then I got gigabit on both ends
so your experience may vary.
ajoy39 on April 13, 2016 at 11:37 am
The games lineup on PS4 is
The games lineup on PS4 is still kinda week but they continue to nail in home streaming. You can now stream to a Playstation TV (basically a vita in a box with no screen or inputs, works kinda the same way as the Steam Link but you can also play Vita games and use stuff like Netflix on it) PS Vita (Handheld) or any of my computers. I haven't tested streaming to computers yet, but if it works as well as streaming to the first two I have to say I'm really impressed with Sony on this. Good on them. I wish the Steam In Home Streaming ecosystem was this strong, the link is a good start but what I really want is a handheld and, at least the last time I used it (a couple months ago at this point) the Link streaming did not work as well as PS4 streaming to my Vita or Playstation TV
bidaum on April 13, 2016 at 1:35 pm
Lack of Win7 support seems
Lack of Win7 support seems bizarre… to me at least.
mAxius on April 13, 2016 at 3:35 pm
it is not really i am in the
it is not really i am in the same boat…
funandjam on April 13, 2016 at 2:43 pm
maybe i'm just too sensitive,
maybe i'm just too sensitive, but the added latency of streaming a game, even over a good wired network just kills the interest for me.
oh well, at least I only spent $50 on the steam link.
Gammett on April 13, 2016 at 3:55 pm
I have never understood the
I have never understood the need for steam link when at least for me I can just get a extremely long HDMI or display port cable and have not nearly the latency loss that a dedicated streaming box would bring. I do find interest in things like streaming over the internet and mobile devices [Nvidia's Shield Devices].
There isn't any getting
There isn't any getting around it, whether you stream games over your hardwired home network, wirelessly and/or over the internet, there is added latency. The least amount of latency will be over a hardwired home network, wireless would be in the middle and the most latency would be over the internet.
Best of luck to you with game streaming over the internet or wirelessly!
Anonymous on April 13, 2016 at 11:41 pm
Depending on the kinds of
Depending on the kinds of games you play it may not even matter. Obviously an FPS isn't going to be ideal. However, I know we've all seen the guy with like 300 ping trying to play the game for some reason.
Here's the problem with how
Here's the problem with how people just try to dismiss the latency as "oh, its just with ftp games" or "it's just with racing games" etc.:
It affects the desire to even want to stream games in the first place. I have to try and remember that when I stream games, to NOT try any of the games that would be affected and sometimes I forget and get upset once I realize it when the game has started. At that point I don't want to switch to a different game, i just turn off the Link and go back to hte PC where there is no added latency.
I'm at the point now where I view game streaming as equivalent to playing games on my phone, something to do when there is absolutely nothing else to do.
StressedOutCat on April 13, 2016 at 4:15 pm
the thing I am baffled by is
the thing I am baffled by is I can remote play with my vita over the internet to a playstation 4 (and now even with my PC)… yet microsoft RDP even over the same network has problems with running video let alone playing games.
I know its probably comparing apples to oranges.. but still.. you think functionality would have been added or improved since RDP been introduced.
Anonymous on April 13, 2016 at 5:46 pm
RDP is probably mostly
RDP is probably mostly uncompressed since video compression cam make a mess out of small text commonly used on desktop computer screens. With something like the PlayStation 4, it is made to work on a TV at 10 feet away, so it is not going to have small text at all. They can use a hardware video encoder to stream the screen images as if it is a video stream. If you want to stream video between PCs you generally have to set up a streaming media server specifically for that rather than trying to mirror the screen.
I actually haven't used windows in quite a while, so I don't know where they are as far as remote use. They have generally been way behind the UNIX world on such things. UNIX came out of the mainframe world in some respects, so remote use has always been present In some form. They had a network aware display system very early on. I don't know when such features became available, but even back in the early 90's you could very easily run an application on one machine and have it display on another since the display system (X server) has a network layer. These systems were not meant for playing media though. I don't think they ever supported playing sound.
Such systems will still have issues displaying video though, since the decoding would be done on the source machine and then the uncompressed frame would need to be sent over the network. Just like RDP or VNC type applications, compression must be kept to a minimum since most application computer applications would be unusable even with mild compression. If you have ever hooked a computer up to a TV, it becomes obvious why Steam big picture mode is required. It isn't worth it to implement video compression really. If they did in a desktop sharing type application, you would be decompressing the video for display and then re-compressing it on the same machine. It is much more efficient to just stream the compressed video with a streaming media server and decompress it on the display machine.
not sure if it all comes down
not sure if it all comes down to compression, cause RDP still feels jittery and laggy even when you scale the resolution back to 480p screen.. but guess it just comes to that the protocol is old and they are unable to update it really since allot of hardware based system still have to work with it. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,308 |
Charles Erskine may refer to:
Charles Erskine, Earl of Mar (1650–1689), Scottish nobleman
Charles Erskine, Lord Tinwald (1680–1763), Scottish judge, Lord Advocate, MP for Dumfriesshire 1722–41 and for Tain Burghs 1734–42
Charles Erskine (1716–1749), his son, Scottish lawyer, MP for Ayr Burghs
Charles Erskine (cardinal) (1739–1811), Italian-Scottish papal diplomat and cardinal
Sir Charles Erskine, 1st Baronet, of Alva (1643–1690), Scottish politician | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,384 |
/* File Description
* Original Works/Author: Thomas Slusny
* Other Contributors: None
* Author Website: http://indiearmory.com
* License: MIT
*/
using System;
using System.Linq;
namespace CookieLib.Graphics
{
/// <summary>
/// Enum of flags containing the effects that can be applied to a sprite when rendering it.
/// </summary>
[Flags]
public enum SpriteEffects : byte
{
/// <summary>
/// No effects specified.
/// </summary>
None = 0,
/// <summary>
/// Flips the sprite horizontally before rendering.
/// </summary>
FlipHorizontally = 1 << 0,
/// <summary>
/// Flips the sprite vertically before rendering.
/// </summary>
FlipVertically = 1 << 1,
/// <summary>
/// Flips the both vertically and horizontally before rendering.
/// </summary>
FlipVerticalHorizontal = FlipVertically | FlipHorizontally,
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,953 |
A totes (originalment en anglès, Playing for Keeps) és un telefilm canadenc dirigit per Gary Harvey, emès el 5 d'abril de 2009 a la xarxa CTV, i als Estats Units el 18 de gener de 2009 a Lifetime Movie Network. La pel·lícula va ser vista per aproximadament 1,6 milions d'espectadors quan es va emetre per primera vegada als Estats Units. El 2021 es va estrenar el doblatge en català a TV3.
Argument
A la Nicole, una jove blanca sense gaires aspiracions, li agrada sortir a la nit amb la seva amiga Maya. A la discoteca, miren de coincidir amb els jugadors de bàsquet professional de l'equip de la seva ciutat, Vancouver, del qual ella és una fervent seguidora. Una nit coneix Ty Rivers, una de les estrelles, afroamericà i casat, però propens a les aventures. La parella s'enamora però al cap de poc la Nicole s'adona que està embarassada perquè la pastilla anticonceptiva ha fallat. De sobte té un objectiu a la vida: ser mare, una determinació que contrasta amb els dubtes plantejats pel jugador, que d'entrada no reconeix la paternitat i en cap cas està disposat a abandonar la seva família. Ara bé, de mica en mica s'anirà interessant pel nen, mestís, cosa que donarà peu a una llarga batalla als tribunals.
Repartiment
Jennifer Finnigan: Nicole Alpern
Roger Cross: Ty Rivers
Doug Savant: Peter Marcheson
Brian Markinson: Daryl Alpern
Enuka Okuma: Beverly Rivers
Agam Darshi: Maya
Malcolm Stewart: Daniel Gibson
Chilton Crane: Bonnie Alpern
Sarah Edmondson: Amy Jansen
Sonja Bennett: Marsha Barclay
Ben Cotton: Peter
Glynis Davies: Juge Mitchell
Martin Brown: conductor de l'autobús
Referències
Enllaços externs
Fitxa de la producció a Force Four
Pel·lícules del 2009 doblades al català
Pel·lícules dramàtiques del Canadà
Pel·lícules del Canadà del 2009
Pel·lícules de drama romàntic de la dècada del 2000
Pel·lícules de drama biogràfic
Pel·lícules en anglès
Pel·lícules dramàtiques del 2009 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,720 |
To Yellowknife, With Love
By Brandy Saturley, Our CanadaUpdated: Jun. 18, 2019
The people who live in the Land of the Midnight Sun inspire this first-time Yellowknife visitor with their grit and creativity.
Photo: Brandy Saturley
Memories of the Midnight Sun: My 6 Days in Yellowknife
My journey through the Canadian landscape has taken me to many large cities and rural communities over the past couple of years in preparation for touring exhibitions of my paintings inspired by Canada. Back on Canada Day 2016, I had the opportunity to celebrate in Canada's North, on the edge of the Arctic Circle. I spent Canada Day week in Yellowknife on an epic journey in the land where the sun and the people never sleep.
I flew in over Great Slave Lake as the midnight sun chased us, throwing a gilded, glistening spotlight from rivers to lakes as we touched down. Global warming is having the most dramatic impact and causing visible shifts, say my hosts, who shared some pizza with me at a neighbourhood watering hole just a stroll down the gravel lane. That first night, I slept in an artist's shack moved to Old Town in 1980 from nearby Jolliffe Island; it seemed fitting I would rest my head inside this tiny piece of Canadian history.
Walking into downtown Yellowknife the following morning, I found myself at the Prince of Wales Northern Heritage Centre, named after the British prince himself. It is a thoughtful and impressive museum for its size, telling the story of the first peoples here, the Dene First Nations.
Did you know Yellowknife is also one of the best places in Canada for viewing the aurora borealis?
A Taste of Yellowknife Hospitality
My hosts in Yellowknife were a savvy media team: Kyle created YK Online, and Jen is the creative director for Tait Communications. Both are community connectors who made my trip both educational and entertaining.
We ventured into town and took in the Canada Day parade on a sweltering day in the "Arctic desert," as the region is often referred to locally. Our evening featured a cast of local characters and creators, and a feast of blueberries, whitefish, trout and a little homemade birch syrup delivered by Pike Mike. (Here are 10 more iconic Canadian dishes—and the best places in the country to find them.) Pike Mike is best known for his role on the Animal Planet series Ice Lake Rebels. A man with serious skills, he can weave a yarn, play the mandolin, help you catch record-sized trout and teach you about surviving off the land, all in one evening. After, we settled in for a patio roundtable of five creative women: a filmmaker, an engineer, a graphic artist, a printmaker and myself. It felt like a residency of sorts, all in one day in a small community in a land of extremes. At 1 a.m., it was still light out. Days are full here this time of year with the midnight sun, and everyone is trying to squeeze out every minute of this golden time.
Check out 10 more places in Canada every Canadian should visit.
Taking Roads Less Travelled
Our Saturday was spent exploring what this area of Canada does best: the wild and life on the edge. The day started with some off-road exploring by jeep and a hike to Cameron Falls with Kyle, Jen and Steve Schwarz, a geologist and Getty-selling photographer. (These are the most beautiful waterfalls in Canada.) The experience was vivid and informative; my brain was buzzing and firing on both halves. The rocky climbs were endless, and evidence of tectonic shifts and things bubbling up to the surface displayed blueprints left for geologists in this land so rich with minerals and precious metals.
In the afternoon, the lake beckoned and we hopped into a motorized canoe for a close-up tour of the colourful houseboats around Jolliffe Island. Parking the boat on an uninhabited island gave us another chance to explore lichen and moss-covered rock. Finds of the day included remnants of furry inhabitants and a claim stake from a prospector of the past. (This is what it's like taking a canoe trip through the Canadian Arctic.)
The Homes of Latham Island
Evening landed us at the WildCat Cafe, an Old Town log cabin turning out food since 1937. An evening walk took us to Latham Island through a neighbourhood of architecturally diverse homes with stunning vistas. (Check out this gorgeous gallery of Canadian architecture photography.) Many homes here—whether million-dollar or shack—display a nice rack of horns or a skull, and lots of Canadian flags. We ended the night with one more hike up and across rock to Pilot's Monument, which affords a 360-degree view of the town.
We set out the next day to circle Yellowknife by car, filling in the blank spots in my visit. The outskirts of the city are dotted with communities that blend expensive, contemporary properties with modular homes and funky workshop shacks. Everyone here seems to be a tinkerer, a creator, a craftsman or an artist. A seemingly inconspicuous shed can hide a meticulous and treasured workspace.
Don't miss these captivating roadside attractions across Canada.
Old Town Treasures
You can easily find a Timmy's or a $6 iced cappuccino here, which was well worth the bucks during the continuing sweltering heat. We took our custom coffees to the Lakeview Cemetery, with gravesites as meticulously crafted as the creations in the makers' sheds we'd seen earlier in the day. Some sites were encircled with white picket fences, and some had trees growing in the centre. Miners, children, Natives, hockey fans and even Elvis fanatics are present here, reflecting the lives I have seen in the area. Our day ended with a feast fit for a mineworker at the famous Bullocks Bistro in Old Town, a legendary shack brimming with diners' graffiti and things left stapled to the walls and ceiling. It's a sassy and humorous place serving up fish, bison and even caribou ribs. A thunderstorm and a rainbow marked our way home as we wrapped another full day on the edge. (Here's more breathtaking rainbow photography.)
On the Monday, I spent half a day visiting with two distinctive and well-known Arctic artists. Jen Walden is a painter, as well as a filmmaker, a hockey coach and founder of the Borderless Arts Movement in Yellowknife. Her distinctive, dimensional and textured style explores Canadian and northern life through people, wildlife and topography. I then went looking for Fran Hurcomb, a veteran Canadian photographer and photojournalist with more than 30 years of experience capturing Canada's North. Fran recently published a book about Yellowknife's Old Town, where she lives, depicting the area's vivid history and individuals from the past three decades.
Here are 12 more awesome attractions you didn't know were in Canada.
Bidding Adieu to Yellowknife
We then hit a molten-hot tarmac at Buffalo Airways and got a personal tour from Mikey McBryan, who's featured in the docu-series Ice Pilots. My evening included a visit with a husky from a sled dog team, seeing "YKEA" (the local dump is known as Yellowknife's IKEA; nothing gets thrown away here), scavenging and a late-night climb with a bottle of vino, some blueberries and stories as a red sun crested the horizon, not to set, but only to rest and rise again.
Before flying out, I had a chance to visit the talented folks over at the Aboriginal-owned Erasmus Apparel, which creates Aboriginal-inspired designs screen-printed on clothing, the perfect souvenir for this trip.
And then that was it for my six days on the edge of the Arctic Circle, where helping your neighbour really is the first order of business, and the only way to survive in this land of extreme weather and extreme living. This experience inspired many a painting when I returned home to my studio on Vancouver Island. These people have heart and grit and talent beyond whatever expectations I had going in. I love you, Yellowknife—see you for the freeze!
Check out 13 more reasons it's great living in Canada. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,301 |
//Copyright (c) Microsoft Corporation. All rights reserved.
using System;
using System.Collections.Generic;
using Microsoft.WindowsAPICodePack.Shell.Resources;
namespace Microsoft.WindowsAPICodePack.Shell
{
/// <summary>
/// The FolderTypes values represent a view template applied to a folder,
/// usually based on its intended use and contents.
/// </summary>
internal static class FolderTypes
{
/// <summary>
/// No particular content type has been detected or specified. This value is not supported in Windows 7 and later systems.
/// </summary>
internal static Guid NotSpecified = new Guid(
0x5c4f28b5, 0xf869, 0x4e84, 0x8e, 0x60, 0xf1, 0x1d, 0xb9, 0x7c, 0x5c, 0xc7);
/// <summary>
/// The folder is invalid. There are several things that can cause this judgement: hard disk errors, file system errors, and compression errors among them.
/// </summary>
internal static Guid Invalid = new Guid(
0x57807898, 0x8c4f, 0x4462, 0xbb, 0x63, 0x71, 0x04, 0x23, 0x80, 0xb1, 0x09);
/// <summary>
/// The folder contains document files. These can be of mixed format—.doc, .txt, and others.
/// </summary>
internal static Guid Documents = new Guid(
0x7d49d726, 0x3c21, 0x4f05, 0x99, 0xaa, 0xfd, 0xc2, 0xc9, 0x47, 0x46, 0x56);
/// <summary>
/// Image files, such as .jpg, .tif, or .png files.
/// </summary>
internal static Guid Pictures = new Guid(
0xb3690e58, 0xe961, 0x423b, 0xb6, 0x87, 0x38, 0x6e, 0xbf, 0xd8, 0x32, 0x39);
/// <summary>
/// Windows 7 and later. The folder contains audio files, such as .mp3 and .wma files.
/// </summary>
internal static Guid Music = new Guid(
0xaf9c03d6, 0x7db9, 0x4a15, 0x94, 0x64, 0x13, 0xbf, 0x9f, 0xb6, 0x9a, 0x2a);
/// <summary>
/// A list of music files displayed in Icons view. This value is not supported in Windows 7 and later systems.
/// </summary>
internal static Guid MusicIcons = new Guid(
0x0b7467fb, 0x84ba, 0x4aae, 0xa0, 0x9b, 0x15, 0xb7, 0x10, 0x97, 0xaf, 0x9e);
/// <summary>
/// The folder is the Games folder found in the Start menu.
/// </summary>
internal static Guid Games = new Guid(
0xb689b0d0, 0x76d3, 0x4cbb, 0x87, 0xf7, 0x58, 0x5d, 0x0e, 0x0c, 0xe0, 0x70);
/// <summary>
/// The Control Panel in category view. This is a virtual folder.
/// </summary>
internal static Guid ControlPanelCategory = new Guid(
0xde4f0660, 0xfa10, 0x4b8f, 0xa4, 0x94, 0x06, 0x8b, 0x20, 0xb2, 0x23, 0x07);
/// <summary>
/// The Control Panel in classic view. This is a virtual folder.
/// </summary>
internal static Guid ControlPanelClassic = new Guid(
0x0c3794f3, 0xb545, 0x43aa, 0xa3, 0x29, 0xc3, 0x74, 0x30, 0xc5, 0x8d, 0x2a);
/// <summary>
/// Printers that have been added to the system. This is a virtual folder.
/// </summary>
internal static Guid Printers = new Guid(
0x2c7bbec6, 0xc844, 0x4a0a, 0x91, 0xfa, 0xce, 0xf6, 0xf5, 0x9c, 0xfd, 0xa1);
/// <summary>
/// The Recycle Bin. This is a virtual folder.
/// </summary>
internal static Guid RecycleBin = new Guid(
0xd6d9e004, 0xcd87, 0x442b, 0x9d, 0x57, 0x5e, 0x0a, 0xeb, 0x4f, 0x6f, 0x72);
/// <summary>
/// The software explorer window used by the Add or Remove Programs control panel icon.
/// </summary>
internal static Guid SoftwareExplorer = new Guid(
0xd674391b, 0x52d9, 0x4e07, 0x83, 0x4e, 0x67, 0xc9, 0x86, 0x10, 0xf3, 0x9d);
/// <summary>
/// The folder is a compressed archive, such as a compressed file with a .zip file name extension.
/// </summary>
internal static Guid CompressedFolder = new Guid(
0x80213e82, 0xbcfd, 0x4c4f, 0x88, 0x17, 0xbb, 0x27, 0x60, 0x12, 0x67, 0xa9);
/// <summary>
/// An e-mail-related folder that contains contact information.
/// </summary>
internal static Guid Contacts = new Guid(
0xde2b70ec, 0x9bf7, 0x4a93, 0xbd, 0x3d, 0x24, 0x3f, 0x78, 0x81, 0xd4, 0x92);
/// <summary>
/// A default library view without a more specific template. This value is not supported in Windows 7 and later systems.
/// </summary>
internal static Guid Library = new Guid(
0x4badfc68, 0xc4ac, 0x4716, 0xa0, 0xa0, 0x4d, 0x5d, 0xaa, 0x6b, 0x0f, 0x3e);
/// <summary>
/// The Network Explorer folder.
/// </summary>
internal static Guid NetworkExplorer = new Guid(
0x25cc242b, 0x9a7c, 0x4f51, 0x80, 0xe0, 0x7a, 0x29, 0x28, 0xfe, 0xbe, 0x42);
/// <summary>
/// The folder is the FOLDERID_UsersFiles folder.
/// </summary>
internal static Guid UserFiles = new Guid(
0xcd0fc69b, 0x71e2, 0x46e5, 0x96, 0x90, 0x5b, 0xcd, 0x9f, 0x57, 0xaa, 0xb3);
/// <summary>
/// Windows 7 and later. The folder contains search results, but they are of mixed or no specific type.
/// </summary>
internal static Guid GenericSearchResults = new Guid(
0x7fde1a1e, 0x8b31, 0x49a5, 0x93, 0xb8, 0x6b, 0xe1, 0x4c, 0xfa, 0x49, 0x43);
/// <summary>
/// Windows 7 and later. The folder is a library, but of no specified type.
/// </summary>
internal static Guid GenericLibrary = new Guid(
0x5f4eab9a, 0x6833, 0x4f61, 0x89, 0x9d, 0x31, 0xcf, 0x46, 0x97, 0x9d, 0x49);
/// <summary>
/// Windows 7 and later. The folder contains video files. These can be of mixed format—.wmv, .mov, and others.
/// </summary>
internal static Guid Videos = new Guid(
0x5fa96407, 0x7e77, 0x483c, 0xac, 0x93, 0x69, 0x1d, 0x05, 0x85, 0x0d, 0xe8);
/// <summary>
/// Windows 7 and later. The view shown when the user clicks the Windows Explorer button on the taskbar.
/// </summary>
internal static Guid UsersLibraries = new Guid(
0xc4d98f09, 0x6124, 0x4fe0, 0x99, 0x42, 0x82, 0x64, 0x16, 0x8, 0x2d, 0xa9);
/// <summary>
/// Windows 7 and later. The homegroup view.
/// </summary>
internal static Guid OtherUsers = new Guid(
0xb337fd00, 0x9dd5, 0x4635, 0xa6, 0xd4, 0xda, 0x33, 0xfd, 0x10, 0x2b, 0x7a);
/// <summary>
/// Windows 7 and later. A folder that contains communication-related files such as e-mails, calendar information, and contact information.
/// </summary>
internal static Guid Communications = new Guid(
0x91475fe5, 0x586b, 0x4eba, 0x8d, 0x75, 0xd1, 0x74, 0x34, 0xb8, 0xcd, 0xf6);
/// <summary>
/// Windows 7 and later. The folder contains recorded television broadcasts.
/// </summary>
internal static Guid RecordedTV = new Guid(
0x5557a28f, 0x5da6, 0x4f83, 0x88, 0x09, 0xc2, 0xc9, 0x8a, 0x11, 0xa6, 0xfa);
/// <summary>
/// Windows 7 and later. The folder contains saved game states.
/// </summary>
internal static Guid SavedGames = new Guid(
0xd0363307, 0x28cb, 0x4106, 0x9f, 0x23, 0x29, 0x56, 0xe3, 0xe5, 0xe0, 0xe7);
/// <summary>
/// Windows 7 and later. The folder contains federated search OpenSearch results.
/// </summary>
internal static Guid OpenSearch = new Guid(
0x8faf9629, 0x1980, 0x46ff, 0x80, 0x23, 0x9d, 0xce, 0xab, 0x9c, 0x3e, 0xe3);
/// <summary>
/// Windows 7 and later. Before you search.
/// </summary>
internal static Guid SearchConnector = new Guid(
0x982725ee, 0x6f47, 0x479e, 0xb4, 0x47, 0x81, 0x2b, 0xfa, 0x7d, 0x2e, 0x8f);
/// <summary>
/// Windows 7 and later. A user's Searches folder, normally found at C:\Users\username\Searches.
/// </summary>
internal static Guid Searches = new Guid(
0x0b0ba2e3, 0x405f, 0x415e, 0xa6, 0xee, 0xca, 0xd6, 0x25, 0x20, 0x78, 0x53);
static Dictionary<Guid, string> types;
[System.Diagnostics.CodeAnalysis.SuppressMessage("Microsoft.Performance", "CA1810:InitializeReferenceTypeStaticFieldsInline")]
static FolderTypes()
{
types = new Dictionary<Guid, string>();
// Review: These Localized messages could probably be a reflected value of the field's name.
types.Add(NotSpecified, LocalizedMessages.FolderTypeNotSpecified);
types.Add(Invalid, LocalizedMessages.FolderTypeInvalid);
types.Add(Communications, LocalizedMessages.FolderTypeCommunications);
types.Add(CompressedFolder, LocalizedMessages.FolderTypeCompressedFolder);
types.Add(Contacts, LocalizedMessages.FolderTypeContacts);
types.Add(ControlPanelCategory, LocalizedMessages.FolderTypeCategory);
types.Add(ControlPanelClassic, LocalizedMessages.FolderTypeClassic);
types.Add(Documents, LocalizedMessages.FolderTypeDocuments);
types.Add(Games, LocalizedMessages.FolderTypeGames);
types.Add(GenericSearchResults, LocalizedMessages.FolderTypeSearchResults);
types.Add(GenericLibrary, LocalizedMessages.FolderTypeGenericLibrary);
types.Add(Library, LocalizedMessages.FolderTypeLibrary);
types.Add(Music, LocalizedMessages.FolderTypeMusic);
types.Add(MusicIcons, LocalizedMessages.FolderTypeMusicIcons);
types.Add(NetworkExplorer, LocalizedMessages.FolderTypeNetworkExplorer);
types.Add(OtherUsers, LocalizedMessages.FolderTypeOtherUsers);
types.Add(OpenSearch, LocalizedMessages.FolderTypeOpenSearch);
types.Add(Pictures, LocalizedMessages.FolderTypePictures);
types.Add(Printers, LocalizedMessages.FolderTypePrinters);
types.Add(RecycleBin, LocalizedMessages.FolderTypeRecycleBin);
types.Add(RecordedTV, LocalizedMessages.FolderTypeRecordedTV);
types.Add(SoftwareExplorer, LocalizedMessages.FolderTypeSoftwareExplorer);
types.Add(SavedGames, LocalizedMessages.FolderTypeSavedGames);
types.Add(SearchConnector, LocalizedMessages.FolderTypeSearchConnector);
types.Add(Searches, LocalizedMessages.FolderTypeSearches);
types.Add(UsersLibraries, LocalizedMessages.FolderTypeUserLibraries);
types.Add(UserFiles, LocalizedMessages.FolderTypeUserFiles);
types.Add(Videos, LocalizedMessages.FolderTypeVideos);
}
internal static string GetFolderType(Guid typeId)
{
string type;
return types.TryGetValue(typeId, out type) ? type : string.Empty;
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,029 |
El Teatro Civic es un gran teatro con capacidad para 2.378 personas situado en el centro de Auckland, Nueva Zelanda. Inaugurado el 20 de diciembre de 1929, fue reabierto en el año 2000 tras importantes esfuerzos de rehabilitación y conservación. Es un famoso ejemplo de teatro de estilo atmosférico, en el que las luces y el diseño se utilizaron para dar la impresión de estar sentado en un auditorio al aire libre por la noche, creando la ilusión de un cielo abierto completo con estrellas centelleantes.
Importancia
El Teatro Civic de Auckland es de importancia internacional por ser el cine atmosférico más antiguo que se conserva en Australasia (y también uno de los únicos siete de su estilo que queda en el mundo) y como el primer cine de este tipo construido a tal efecto en Nueva Zelanda. También es conocido por su vestíbulo de inspiración india, que incluye budas sentados, columnas salomónicas y techos abovedados. El auditorio principal fue diseñado en un estilo similar, imitando a un jardín morisco con torres, minaretes, torres y techos de tejas, así como varias estatuas de panteras abisinias. En su apertura podía albergar hasta 2.750 personas, e incluso con su actual aforo reducido sigue siendo el teatro más grande de Nueva Zelanda.
Historia
El Teatro Civic fue la creación de Thomas O'Brien, que construyó un imperio del cine en 1920 en la periferia de Auckland y llevó el cine atmosférico a Nueva Zelanda al abrir en 1928 el Teatro Empire De Luxe, de estilo morisco, en Dunedin.
Thomas O'Brien convenció a un grupo de acaudalados hombres de negocios de Auckland para construir un enorme cine atmosférico en Queen Street, logrando también obtener un préstamo de 180.000 dólares por parte del Bank of New Zealand.
El cine fue construido por Fletcher Construction. Sin embargo, el préstamo del BNZ y los altos costos de construcción llamaron la atención del Parlamento, a pesar de que el precio final se disparó a más de 200.000 dólares.
El Civic se abrió en medio de una gran fanfarria en diciembre de 1929, pero el inicio de la Gran Depresión contribuyó a asistencias decepcionantes, contribuyendo también la obstinada insistencia de O'Brien en mostrar películas británicas en lugar de películas estadounidenses, bastante más populares, con lo cual cayó finalmente en quiebra. Después de varias modificaciones durante las décadas siguientes, a finales de la década de 1990 el teatro fue finalmente restaurado aproximándolo a su diseño original.
El teatro también ha adquirido recientemente cierta fama por ser utilizado para representar a un teatro de Nueva York en escenas del remake de la película King Kong, dirigida por Peter Jackson.
Referencias
Enlaces externos
Civic Theater, sitio web oficial.
Civic
Auckland
Salas de cine de Nueva Zelanda | {
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} | 1,807 |
Q: can a struct be derived from a class in c#? Can a struct be derived from a class in c#?
If not, Why can primitive data types, such as int, be derived from the class object? Since the data type int is basically a struct type(value type).
Is this just an exception to the rule?
A: Integers and other value types (e.g. bool) are objects, because it allows them to leverage inheritance (i.e. they have access to the common .Equals(), .GetType(), .ToString() functions).
It's a design decision in the .NET framework. Rather than writing separate functions for all the value types under System.ValueType, they use a common code base.
Microsof's document on Types
A: All structs inherit System.ValueType, which in turn inherits Object.
You cannot change that.
A: When the run-time allocates a storage location for a type, or generates code to operate on one, it checks whether the type derives from System.ValueType but is not System.ValueType itself. Unless the storage location meets those criteria, it will hold a heap object reference, and any code to operate on its members (fields, methods, properties, etc.) will act upon the referenced object. Otherwise, the storage location will hold all the public and private fields of that type (which will be laid out identically in all storage locations of that type), and any code to operate on its members will operate on the storage location itself.
If an attempt is made to store a value type into a storage location of class ValueType, or a storage location which does not derive from ValueType, the system will generate a new heap object of the storage location's type, and then store a reference to that object in the appropriate storage location. Although storage locations of types deriving from System.ValueType, and code to access them, are treated specially by the run-time, heap object instances which inherit from System.ValueType (such as the newly-created one just mentioned) are simply heap objects that happen to derive from System.ValueType, and have inheritance behaviors which are essentially the same as other types, and may thus be passed around by code expecting to deal with heap references.
A statement like Object Foo = New System.Drawing.Point(3,4); actually involves three kinds of things:
*
*An unnamed temporary storage location of type `System.Drawing.Point(3,4)` which holds the private fields of that type (two integers), and is initialized to (3,4). Note that this storage location does not hold an `Object`; it holds two integers which the compiler knows represent its fields.
*A heap object of type `System.Drawing.Point`. Because this is a heap object, it inherits from `System.Object`, as do all heap objects.
*A storage location of type `Object`, which is represented by the variable name `Foo`.
The upshot of all this is that while value types may be defined as inheriting from ValueType which inherits from Object, and while heap objects whose types inherit from ValueType do inherit from Object, storage locations of value types do not hold things that inherit from Object.
A: The class hiearchy works as follows (simplified):
Object -> ValueType -> int
Object -> ValueType -> struct
Structs by definition of c# do not allow inheritance.
Here is a nice article describing the role of stucts within the C# language:
http://msdn.microsoft.com/en-us/library/aa288471(v=vs.71).aspx
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,330 |
Q: Return all fields, Group by And Aggregate functions in lambda I have such a table as shown below
I need to group each record by LifePluseCaseId column,
and each group must be selected by Min distance and Min Duration
And return All of the fields.
I have tried this:
var query = db.ApplicantCenterDistance.GroupBy(s => s.LifeplusCaseId)
.Select(s => new {
Id = s.Key,
MinDistance = s.Min(m => m.Distance),
Duration = s.Min(m => m.Duration)
}).ToList();
But i don't know how to get all fields in Select statement and what the role of key is.
Is s.Key equals to s.LifeplusCaseId?
A: From the above queries, each group will have multiple ApplicantCenterDistances record (in theory) because... well it's a group. If you really want to get all the items in each group as well, you can do like this (pseudo-code):
var productByCategory = await db.Products
.GroupBy(q => q.CategoryId)
.Select(q => new {
CategoryId = q.Key,
// Here q is also acting as a list of products with the same `CategoryId`
Products = q,
// Or if you only want some specific fields
ProductCustoms = q.Select(p => new {
Name = p.Name,
Color = p.Color,
// All fields you want
})
}).ToListAsync(); // Do whatever you want with the result
Now it makes much more sense right? The productByCategory is a list of groups, each group has the key (CategoryId), and has a list of products that has that matching CategoryId.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,000 |
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\section{Introduction}
\par This article deals with a linearized inverse scattering problem
considered by Nolan and Symes [23]. Acoustic waves are generated
at the surface of the earth, scatter off heterogeneities in the
subsurface and return to the surface. The full inverse problem
would use the pressure field at the surface to reconstruct an
image of the subsurface. We instead consider the linearized
operator $F$ which maps singular perturbations of a smooth
background sound speed in the subsurface, assumed known, to
perturbations of the resulting pressure field at the surface. The
goal is to left-invert $F$; standard techniques suggest studying
left invertibility of the normal operator $N=F^*F$. To start, we
make two assumptions: $(i)$ no single ray connects a source to a
receiver; and $(ii)$ no ray originating in the subsurface grazes
the surface. Under these assumptions, in the case of a {\it
single} source and receivers ranging over an open subset of the
surface, $\{x_3=0\}$, Rakesh\cite{ra} showed that $F$ is a Fourier integral
operator (FIO). Beylkin\cite{be} showed that if caustics do not occur
for the background soundspeed, $F^*F$ is a pseudodifferential
operator $ (\Psi DO)$.
\par For more general data acquisition geometries, the canonical
relation of $F$ depends on the sets of sources and receivers.
Nolan and Symes\cite{nosy} proved that, if both sources and receivers
vary over open and bounded subsets $\Sigma_r$ and $\Sigma_s $ of the
surface, then under the traveltime injectivity condition (TIC),
generalizing the no-caustic assumption, $F^*F$ is still a
$\Psi DO$. The same result was stated by ten Kroode, Smit and
Verdel\cite{ksv} and their proof was completed by Stolk\cite{st} who also
relaxed the TIC condition in low dimensions.
\par For applications in three spatial variables, an important problem is to
understand the nature of $F$ and $F^*F$ for the {\it marine} data acquisition
geometry \cite{nosy}, where measurements are made on the codimension
one submanifold
$\Sigma_{r,s}=\{ (r_1, r_2; s_1, s_2) \in \Sigma_r \times
\Sigma_s: s_2=r_2 \}$. This arises as follows: a seismic
vessel trails behind it both an acoustic source and recording
instruments. The point
source consists of an airgun which sends acoustic waves through the
ocean to the
subsurface. Reflections occur when the sound waves encounter
singularities in the material of the subsurface. The reflected
rays are received by a linear array of hydrophones towed behind the
vessel. The vessel then makes repeated passes along parallel lines
(say, parallel to $x_1$ axis).
\par The purpose of this paper is to consider the marine
geometry under the assumption that only the simplest, most prevalent type of
caustics, namely {\it fold caustics}, occur for the background soundspeed.
Fold caustics are initially defined as follows: A ray departing from
a source $s$ in
the direction $\alpha$
reaches at time $t$ a point denoted $x(t,\alpha)$ in the subsurface. If
there is a
source $s$, such that the spatial projection map $(t,\alpha ) \rightarrow x(t,
\alpha)$ has a
fold singularity and only singularities of this type, then we say that the
background soundspeed exhibits a fold caustic. By the
stability of folds, the maps $(t,\alpha ) \rightarrow x(t,
\alpha)$ also have at most fold singularities for all nearby
sources $s'$. However, it seems that the natural notion of a fold
caustic in the
context of the overdetermined marine data set considered here is the
requirement that
the analogous spatial projection be a {\it submersion with folds}, which is the
simplest singularity in the non-equidimensional setting. This will be
elaborated upon
in \S2 and \S4.
\par We now introduce the linearized scattering operator $F$
considered in \cite{nosy},\cite{ksv}. The model for the scattered
waves is given by
the wave equation:
\begin{eqnarray}\label{star}
\frac{1}{c^2(x)} \frac{\partial^2p}{\partial
t^2}(x,t)-\triangle p (x,t)&=&\delta(t)\delta(x-s)\\
p(x,t)&=&0,\ \ t < 0,\nonumber
\end{eqnarray}
where $x \in Y=\mathbb R^3_+= \{x \in \mathbb R^3, x_3 \geq 0 \}$
represents the Earth, $p(x,t)$
is the pressure field resulting from a pulse at the source $s$
and $c(x)$ is the velocity field. The linearization consists in
assuming $c$ to be of the form $c=c_0 + \delta c$, where $c_0 $ is
a smooth known background field. The associated pressure field
$p_0$ is also assumed known. The linearization of (\ref{star}) then
becomes
\begin{eqnarray}\label{starstar}
\frac{1}{c_0^2(x)}\frac{\partial^2 \delta
p}{\partial t^2}(x,t) -\triangle \delta p(x,t)&=& \frac{2\delta
c(x)}{c_0^3(x) } \frac{\partial^2 p_0}{\partial t^2}\\
\delta p &=&0,\ \ t < 0,\nonumber
\end{eqnarray}
where $p=p_0 + \delta p$. Now, for a given
data acquisition
submanifold $\Sigma_{r,s} \subset \partial Y \times \partial Y$
and appropriate time interval $(0,T)$, we define the linearized
scattering operator $F : \delta c \rightarrow \delta
p|_{\Sigma_{r,s} \times (0,T)}$. The assumption $(ii)$ ensures that
$F$ is an FIO (\cite{ha},\cite{ksv},\cite{ra},\cite{nosy}) and (i)
ensures that the composition $F^*F$ makes sense.
\par In the case
of the single source model, with only fold caustics appearing,
Nolan\cite{no} showed that $F$ is an FIO associated to a folding
canonical relation in the sense of \cite{meta} (also called a two-sided fold),
and stated that
the Schwartz kernel of the operator $F^*F$ belongs to a class of
distributions associated to two cleanly intersecting Lagrangians
in $(T^*Y \setminus 0) \times (T^*Y \setminus 0)$. This was fully
proved in \cite{fel}. The corresponding canonical relations are the
diagonal $\Delta$ and a folding canonical relation, different from
the original
one, which lies in $T^*X\times T^*Y$.
In this article we show that, for the
the marine geometry, the linearization
$F$ is an FIO associated to what we call a {\it folded cross cap }
canonical relation. We then prove that the Schwartz kernel of $F^*F$
belongs to a class of distributions with a microlocal structure
similar to that in the case of the single source geometry, but with
the order of the non-pseudodifferential part of $F^*F$ being
$\frac{1}{2}$ lower than in the case of a single source.
\emph{This means that artifacts arising in seismic imaging
from the presence
of fold caustics are $\frac{1}{2}$ derivative smoother for the marine
geometry than for the single source geometry.}
\par Composition of FIOs under other singular geometries arising in integral
geometry and inverse problems has been previously studied in, e.g.,
\cite{gu-book},\cite{gruh1},\cite{gruh3},\cite{gruh4},\linebreak\cite{no} and
\cite{fel}.
\par The article is organized as follows. In $\S 2$
we review some $C^\infty$ singularity theory and define the
submersion with folds
and cross cap singularities.
$\S 3$ is a review
of the distribution classes associated to two cleanly intersecting
Lagrangians, $I^{p,l}(\Lambda_0, \Lambda_1)$ and the operators which
have these as
their Schwartz kernels. In
$\S4$ we show that submersions with folds and cross caps appear
microlocally in the
marine geometry in the presence of the fold caustics, and we
formulate a general
class of canonical relations exhibiting these singularities.
$\S5$ is dedicated to analyzing a model folded cross cap canonical relation,
$C_0$, in $T^*\mathbb R^n \times T^*\mathbb R^{n-1}$ ; we establish the
composition calculus for $F^*F$, showing that $F^*F \in I^{p,l}
(\Delta, \tilde{C}_0)$ where $\tilde{C}_0$ is a folding canonical
relation. Finally, \S6 provides the extension of this to the general
class of folded
cross caps. We find a weak normal form for any folded cross cap
canonical relation $C \subset T^*X \times T^*Y$ which allows us to
show that $F^*F \in I^{p,l} (\Delta, \tilde{C})$, with
$\tilde{C}$ a folding canonical relation in $T^*Y \times T^*Y$.
\par We would like to thank Cliff Nolan for the helpful discussions, clarifying
\cite{no},
at the Institute for Mathematics and
its Applications, Minneapolis, in October, \nolinebreak2005.
\section{Fourier integral operators and singularity classes}
\par Let $X$ and $Y$ be manifolds and $I^m(X,Y;C)$ denote the class of
$m$-th order Fourier
integral operators (FIOs), $F: {\cal E}'(Y) \to {\cal D}'(X)$, associated
to a canonical relation $C \subset
(T^*X \setminus 0) \times (T^*Y \setminus 0)$. We will focus on
the composition calculus for two FIOs. Let $C_1 \subset (T^*X
\setminus 0) \times (T^*Y \setminus 0)$ and $C_2 \subset (T^*Y
\setminus 0) \times (T^*Z \setminus 0)$ be two canonical relations
and $F_1 \in I^{m_1}(X,Y;C_{1})$ and $F_2 \in I^{m_2}(Y,Z;C_{2})$.
If $ C_1 \times C_2 $ intersects $T^*X \times \Delta_{T^*Y}\times
T^*Z$ transversally, then H\"{o}rmander\cite{ho} proved that $F_1 \circ
F_2 \in I^{m_1+m_2}(X,Z;C_{1} \circ C_{2})$ where $C_{1} \circ
C_{2}$ is
the composition of $C_1$ and $C_2$ as relations in $T^*X \times T^*Y$
and $T^*Y \times T^*Z$. Duistermaat and Guillemin\cite{dugu} and
Weinstein\cite{we}
extended this calculus to the case of clean
intersection and showed that if $C_1 \times C_2$ and $T^*X \times
\Delta_{T^*Y} \times T^*Z$ intersect
cleanly with
excess $e$ then $A \circ B \in I^{m_1+m_2+e/2}(X,Z;C_1 \circ
C_2)$. In each of these cases, $C_{1} \circ C_{2}$
is again a smooth canonical relation.
However, in many interesting problems, these assumptions fail,
and it is important to analyze the composition and understand
the resulting operators. It turns out that the geometry of each
canonical relation and the structure of their projections play an
important role.
\par Let $ \pi _L$ and $ \pi_ R$ be the projections, to the left
and right, from $C$ to $T^*X$ and $T^*Y$, respectively. If either
one is a local diffeomorphism, so is the other one and then $C$,
is a local canonical graph. In the case of two canonical
relations, if at least one of $C_1$ and $C_2$ is a local canonical
graph, then $ C_1 \times C_2 $ intersects $T^*X \times
\Delta_{T^*Y}\times T^*Z$ transversally and the general
composition calculus applies.
\par Now consider the case when the projections are no longer local
diffeomorphisms.
When one of the projections
is singular, i.e., when the rank of its differential is
nonmaximal, then the other one is, too, and $C$ is called a {\it
singular canonical relation}. (Note: $C$ is still assumed to be smooth.)
\par Although corank($d \pi_L$)=corank($d \pi_R$) at all points, the
two projections,
$\pi_R$ and $\pi_L$, may have similar singularities or quite
different ones. The singularities considered in this article are
folds, submersion with folds and cross caps, which we now briefly
describe.\\
\par Let $f$ be a smooth function $f : V \to W,\ \ \textrm{ dim } V = \textrm{ dim } W=N$ and
$\mathcal{S}:={\mathcal S}(f) =\{ x \in V : \det (df (x))=0\}$.
\begin{definition} $f$ {\em has a } (Whitney) fold {\em
singularity along} $ {\cal S}$ {\em if} $d(det(df)) \neq 0$ {\em
on} ${\cal S}$, {\em so that} $ {\cal S}$ {\em is a smooth
hypersurface, } $df$ {\em drops rank by 1 there, and }
${\rm Ker} \ df(x)$ {\em intersects} $T_x{\cal S}$ {\em transversally for
every} $x \in {\cal S}$.
\end{definition}
\par Any map which has a fold singularity can be put into a local normal
form: $f(x_1,x_2, \dots, x_N)=(x_1,x_2, \dots, x_{N-1},x_N^2)$ with
respect to suitable local coordinates in the domain and codomain
\cite{gogu}.
\par Whitney folds are the singularities denoted by $S_{1,0}$ (in the
Thom theory
\cite{gogu}) and by $\Sigma_{1,0}$ (in the Boardman-Morin theory
\cite{mo1,mo2}) in the equidimensional case. The non-equidimensional
versions of
Whitney folds are submersions with folds and cross caps. We note the
difference in
notation: when
$\textrm{ dim } V\leq\textrm{ dim } W$, the singularity classes
$S_{r,0}=\Sigma_{r,0}$, while, if $\textrm{ dim } V > \textrm{ dim } W$, then
$S_{r,0}=\Sigma_{r+k,0}$, where
$k=\textrm{ dim } V-\textrm{ dim } W$.
\par Let $f$ be a smooth function $f : V \to W$, dim $V = N$, dim $W
= M$ , $ N > M.$
\begin{definition} {\em The map} $f$ {\em is a} submersion with
folds {\em if the only singularities of} $f$ {\em are of type}
$S_{1,0}$, {\em i.e.,
of type } $\Sigma_{N-M+1,0}$.
\end{definition}
\par One checks that $f$ is a submersion with folds as follows.
At points where
\noindent$\textrm{ rank } df\ge M-1$, by \cite{mo2}, we can choose suitable adapted
local coordinates on $V$ and $W$ such that $f$
has the form:
$f(x_1, x_2, \dots, x_{M-1}, x_M, \dots, x_N)=(x_1, x_2, \dots
x_{M-1}, f_1(x))$. The set $\mathcal{S}_1(f)$ where $f$ drops rank by $1$ is
described by $\mathcal{S}_1(f)= \{ x: \frac{\partial f_1}{\partial
x_i}=0, \ M \leq i
\leq N \}$. Then $f$ is a submersion with folds if $\mathcal{S}_1(f)$ is a
smooth submanifold, i.e., $\left\{ d\left(\frac{\partial f}{\partial
x_i}\right): \ \
M \leq i \leq N)\right\}$, is linearly
independent, and if the $(N-M+1)$-dimensional kernel of
$df$ is transversal to the tangent space to $\mathcal{S}_1(f)$ in $TV$.
These conditions can be combined \cite{mo1} into
\begin{equation}\label{swf}
\det\left[\frac{\partial^2f_1 }{\partial x_i
\partial x_j }\right]_{ M \leq i,j \leq N} \ne 0.
\end{equation}
and this is independent of the choice of adapted coordinates.
\par There are a finite number of local normal forms for a submersion
with folds, determined by the signature of the Hessian of $f$ \cite{gogu}:
\[f(x_1,x_2, \dots, x_N) =(x_1,x_2, \dots, x_{M-1}, x_M^2 \pm
x_{M+1}^2 \pm \cdots \pm x_N^2 ).
\]
In the case relevant here, $N=M+1$ and the last
entry is a quadratic form in two variables, which is either sign
definite or indefinite; we refer to these two possibilities as
${\it elliptic}$ and ${\it hyperbolic}$ respectively.
\par We now define the second singularity class of interest; like the class
of submersions with folds, it is stable under small $C^2$ perturbations. It is
now assumed that $\textrm{ dim } V=N, \ \ \textrm{ dim } W=M$ with $ N < M$, and
$g : V\to W$ is a smooth function.
\begin{definition} {\em We say that} $g$ {\em is a} cross cap {\em
if the only singularities of} $g$ {\em are of type} $ S_{1,0}$, {\em
i.e., of type}
$\Sigma_{1,0}$.
\end{definition}
\par To identify a cross cap, we use the description of \cite{mo1}.
At a point where $dg$ has rank $\ge N-1$, we can find suitable
adapted coordinates such that $g(x_1, x_2, \dots, x_{N-1}, x_N)=(x_1, x_2,
\dots, x_{N-1}, g_1, g_2, \dots g_q)$, where $q= M-N+\nolinebreak1$.
The set $\mathcal{S}_1(g)$ where $g$ drops rank by $1$
is given by \linebreak$\mathcal{S}_1(g)
=\{x: \frac{\partial g_i}{\partial x_N}=0, \quad 1 \leq i \leq q\}$.
Assume
that there is an $i_0$, such that $\frac{\partial ^2
g_{i_0}}{\partial x_N^2} (0) \neq 0$. Then, $g$ has a cross cap
singularity near 0 if the map $\chi : \mathbb R^N \rightarrow \mathbb R^q $ given by
$\chi(x_1, x_2, \dots x_N)=(\frac{\partial g_1}{\partial x_N},
\frac{\partial g_2}{\partial x_N}, \dots, \frac{\partial
g_q}{\partial x_N})$ satisfies $\textrm{ rank } d\chi(0) =q$. (Notice that this forces
$N \geq q$, i.e., $M \leq 2N-1$.) These conditions can be
reformualted as: $(i)$
$\mathcal{S}_1(g)$ is smooth and of codimension $q$; $(ii)$ the
$N\times N$ minors of
$dg$ generate the ideal of $\mathcal{S}_1(g)$; and $(iii)$ $\textrm { Ker } (dg)\cap
T\mathcal{S}_1(g)=(0)$.
\par As for folds, there is a local normal form for cross caps, due to
\linebreak Whitney\cite{wh} and Morin\cite{mo1}:
\begin{equation}\label{ccnf}
g(x_1,x_2, \dots, x_N) =(x_1,x_2, \dots, x_{N-1}, x_1 x_N, \dots
x_{M-N} x_N, x_N^2).
\end{equation}
\section{Distributions and operators associated \\ to two cleanly intersecting
Lagrangians}
\par Classes of distributions associated to two cleanly intersecting Lagrangian
manifolds were introduced by Melrose and Uhlmann
\cite{meuh} and Guillemin and Uhlmann\cite{guuh}. We briefly review their
definitions and properties.
\par First, one proves that any two pairs of cleanly intersecting
Lagrangian submanifolds are (micro)locally equivalent. Thus, one can
consider the
model pair
$(\tilde{\Lambda}_0,\tilde{\Lambda}_1)$ where
$\tilde{\Lambda}_0=T_0^*{\mathbb R}^n=\{(x,\xi): x=0\}$ and
$\tilde{\Lambda}_1= N^*\{x'' = 0 \}=\{ (x,\xi): x''=\xi'=0\}$ with
$x'=(x_1, x_2,\dots, x_k)$, and $x''=(x_{k+1}, x_{k+2},\dots,
x_n)$. One defines a class of
distributions
given by oscillatory integrals whose amplitudes are called {\it
product-type} symbols. Let $z=(x,s)$ be coordinates in ${\mathbb R}^m={\mathbb R}^n
\times {\mathbb R}^k$
and $(\xi, \sigma)$ the dual coordinates.
\begin{definition}$S^{p,l}(m,n,k)$ {\em is the set of all
functions} $a(z,\xi,\sigma) \in C^{\infty} ({\mathbb R}^m \times {\mathbb R}^n
\times {\mathbb R}^k )$ {\em such that for every}
$K \subset\subset {\mathbb R}^m$ {\em and
every} $\alpha \in {\mathbb Z}^n_+, \beta \in {\mathbb Z}^k_+, \gamma \in
{\mathbb Z}^m_+$ {\em there is a } $c_{\alpha\beta\gamma K}<\infty$
{\em such that}
\begin{equation}\label{prod-type}
|\partial_{\xi}^{\alpha}\partial_{\sigma}^{\beta}\partial_z^{\gamma}
a(z,\xi,\sigma)|
\le c_{\alpha\beta\gamma K}(1+ |\xi|)^{p- |\alpha|} (1+ |\sigma|)^{l-|
\beta|}, \forall (z,\xi,\tau) \in K \times {\mathbb R}^n \times
{\mathbb R}^k.
\end{equation}
\end{definition}
\begin{definition} {\em \cite{guuh} Let} $I^{p,l}({\mathbb R}^n;\tilde{\Lambda}_0,
\tilde{\Lambda}_1)$ {\em be the set of all distributions } $u$
{\em such that} $u=u_1 + u_2$ {\em with} $u_1 \in C^{\infty}_0$
{\em and}
$$u_2(x)=\int e^{i((x'-s)\cdot \xi'+x'' \cdot \xi''+ s
\cdot \sigma)} a((x,s),\xi,\sigma)d\xi d\sigma ds$$
{\em with} $a \in
S^{p',l'}(m,n,k)$ {\em where} $p'=p-\frac{n}{4}+\frac{k}{2}$ {\em and}
$l'=l-\frac{k}{2}$.
\end{definition}
\par At this point, if $X$ is a manifold of dimension $n$, we can define the
class $I^{p,l}(X;\Lambda_0,
\Lambda_1)$ for any pair of Lagrangians in $T^*X \setminus 0$ cleanly
intersecting in codimension $k$. The oscillatory integrals we use
are oscillatory integrals in sense of H\"{o}rmander[14, p.88].
\begin{definition} \emph{\cite{guuh}} $ \ u \in I^{p,l}(X;
\Lambda_0, \Lambda_1)$
{\em if} $ u=u_1 +
u_2 + \sum v_i$ {\em where} $u_1 \in I^{p+l}(\Lambda_0
\setminus \Lambda_1)$, $u_2 \in I^{p}(\Lambda_1 \setminus \Lambda_0)$,
{\em the sum} $\sum v_i$ {\em is locally finite and } $v_i=Fw_i$
{\em where} $F$ {\em is a zero order FIO associated to} $\chi ^{-1}$
{\em where } $\chi: T^*X\setminus 0 \to T^*{\mathbb R}^n \setminus 0$
{\em is a canonical
transformation such that} $\chi(\Lambda_j)\subseteq\tilde\Lambda_j, j=0,1$,
{\em microlocally, and }
$w_i
\in I^{p,l}({\mathbb R}^n;\tilde {\Lambda}_0, \tilde{\Lambda}_1)$.
\end{definition}
We say
that a distribution $u \in I^{r}(X;\Lambda_0 \setminus \Lambda_1)$
if, microlocally
away from $\Lambda_1$, $u \in I^r( X; \Lambda_0)$, the standard
H\"ormander class of
Fourier integral distributions on $X$ associated with $\Lambda_0$.
\begin{remark}\label{old-rem3.4} \emph{\cite{guuh}} $ \
{\rm If} \ u \in I^{p,l}(X;\Lambda_0, \Lambda_1) \ {\rm then} \
u \in I^{p+l}(X;\Lambda_0 \setminus \Lambda_1) \ {\text{\rm and also }}
\linebreak u
\in I^{p}(X;\Lambda_1 \setminus \Lambda_0)$.
\end{remark}
\par We will also use the notion of nondegenerate
phase functions which \\ parametrize two cleanly intersecting
Lagrangians, introduced by Mendoza [15]. Let $\lambda_0 \in
\Lambda_0 \cap \Lambda_1$ and $\Gamma \subset X \times {\mathbb R^k}
\times \left({\mathbb R}^N \setminus 0\right)$ an open, conic set.
\begin {definition} {\em \cite{men}\label{old-def3.6} A phase function}
$\phi(x,s,\theta)$ {\em defined
on} $\Gamma$ {\em is a } \linebreak parametrization {\em for the pair }
$(\Lambda_0,\Lambda_1)$ {\em if } \\
\indent i) $\phi_0(x,\theta):=\phi(x, 0, \theta)$ {\em where}
$\phi_0$ {\em is a nondegenerate phase function
parametrizing} $\Lambda_0$ {\em near} $\lambda_0$; {\em and} \\
\indent ii) $\phi_1(x,(\theta,\sigma)):=\phi(x,\frac{\sigma}{\mid \theta
\mid},\theta)$
{\em is a nondegenerate phase function parametrizing}
$\Lambda_1$ {\em near} $\lambda_0$.
{\em We also refer to } $\phi_1(x;\theta;\sigma)$ {\em as a } multi-phase
function {\em for }
$(\Lambda_0,\Lambda_1)$.
\end{definition}
\par For simplicity, we now focus on the case of codimension 1
intersection relevant
here, i.e., $k=1$. Let us consider the following example:
If $\tilde{\Lambda}_0=N^*\{ x'=0\}$ and
$\tilde{\Lambda}_1=N^*\{x=0\}$, with $x'=(x_2,x_3,\dots,x_n)$,
then $\phi(x,s,\theta')=x'\cdot \theta'\linebreak+x_1 s \mid \theta' \mid $
is a parametrization for $(\tilde{\Lambda}_0,\tilde{\Lambda}_1)$
since $\phi_0(x,\theta'):=\phi(x,0,\theta')=x'\cdot \theta'$, which is a
parametrization for $\tilde {\Lambda}_0$, and
$\phi_1(x,(\theta',\sigma)):=\phi(x,\frac{\sigma}{\mid \theta \mid},\theta)=
x'\cdot
\theta'+x_1 \sigma$ is a parametrization for $\tilde{\Lambda}_1$.
\begin{proposition}\label{old-prop3.7} {\em \cite{men} Let} $p_1$
{\em be a homogenous function of degree} $1$ {\em such
that} $p_1(\lambda_0)=0$ {\em and} $H_{p_1}$ {\em (the Hamiltonian
vector field associated to} $p_1)$ {\em is not tangent to}
$\Lambda_0$. {\em If } $\Lambda_1$ {\em is the flow out from}
$\Lambda_0 \cap \{p_1=0\}$ {\em by } $H_{p_1}$ {\em then there is
a parametrization} $\phi$ {\em for } $(\Lambda_0,\Lambda_1)$
{\em which can be chosen such that} $\frac{\partial
\phi}{\partial s}(x,s,\theta)=p_1(x,d_x \phi)$ {\em and}
$\phi(x,0,\theta)=\phi_0$ {\em with } $\phi_0$ {\em a specified
parametrization for} $\Lambda_0$.
\end{proposition}
\begin{remark}\label{old-rem3.8} {\em [15] If}
$\phi(x;\theta; \sigma)$ \ {\em is a multi-function for} $(\Lambda_0,
\Lambda_1)$ {\em near } $\lambda_0\in\Lambda_0\cap\Lambda_1$, {\em then, if }
$u\in\mathcal
D'(X)$ {\em has } $WF(u)$ {\em contained in a conic neighborhood of }
$\lambda_0$,
{\em then } $u\in I^{p,l}(X;\Lambda_0, \Lambda_1)$
{\em iff }
$$u(x)=\int e^{i
\phi(x;\theta;\sigma)} a(x;\theta;\sigma) d \sigma d \theta,$$
{\em where} $a \in S^{\tilde{p},\tilde{l}}(X\times (\mathbb R^N\setminus
0)\times\mathbb R)$,
{\em the space of } symbol-valued symbols of order
$\tilde{p},\tilde{l}$, {\em defined
by: for all }
$K\subset\subset X,\alpha\in\mathbb Z_+^N,\beta\in\mathbb Z_+,
\gamma\in\mathbb Z_+^n,$
\begin{equation}\label{ests-svs}|\partial^\alpha_{\theta}
\partial^\beta_{\sigma}\partial^\gamma_{x}
a(x;\theta;\sigma)| \leq
c_{\alpha\beta\gamma
K}(1+|\theta|+|\sigma|)^{\tilde{p}-|\alpha|}(1+|\sigma|)^{\tilde{l}-\beta},
\end{equation}
{\em with } $p=\tilde{p}+\tilde{l}+\frac{N+1}{2}-\frac{n}{4},
l=-\tilde{l}-\frac{1}{2}$.
\end{remark}
Finally, we define the classes of generalized (or {\it paired
Lagrangian}) Fourier
integral operators, to one of which we will show the normal operator
$F^*F$ belongs.
Recall that a {\em canonical relation} $C\subset(T^*X\setminus 0)\times
(T^*Y\setminus 0)$ is a smooth, conic submanifold such that
$C':=\left\{(x,y;\xi,\eta): (x,\xi;y,-\eta)\in C\right\}$ is a
Lagrangian submanifold
of $\left(T^*(X\times Y),\omega_{T^*(X\times Y)}\right)$, i.e, $C$
is a Lagrangian
with respect to $\omega_{T^*X}-\omega_{T^*Y}$.
\begin{definition}\label{old-def3.5} {\em If } $C_0,C_1\subset
(T^*X\setminus 0)\times
(T^*Y\setminus 0)$ {\em are smooth, conic canonical relations
intersecting cleanly,
then } $I^{p,l}(C_0,C_1):=I^{p,l}(X,Y;C_0,C_1)$ {\em denotes the set
operators }
$F:{\cal E}'(Y)\longrightarrow{ \cal D}'(X)$ {\em whose Schwartz
kernels are in }
$I^{p,l}(X\times Y; C_0',C_1')$.
\end{definition}
\section{Fold caustics in the marine geometry}
\par In three spatial dimensions, let $s$ be a fixed source on the
surface,\linebreak
$\{x_3=0\}$;
$H(x,\xi)=\frac{1}{2}(c_0(x)^{-2}-|\xi|^2)$ the Hamiltonian
associated to the smooth
background soundspeed $c_0(x)$ in (\ref{starstar}); and $\Lambda_s$
the image of
$T_s^*\mathbb R^3
\setminus 0$ under the bicharacteristic flow associated to $H$,
which is a Lagrangian submanifold of $T^*\mathbb R^3 \setminus 0$. The
assumption of a (point) fold caustic means that the only singularities of
the spatial projection $\pi_Y : \Lambda_s \rightarrow Y$ are folds. We make
use of the description of $\Lambda_s$ in Nolan\cite{no}. It can be
parametrized by $t_{inc}$, the time travelled by the incident
ray, and the takeoff direction $(p_1,p_2,p_3) \in \mathbb S ^2$. We
can change these coordinates to $(x_1, x_2, p_3)$ \cite{no}. Hence on
$\Lambda_s$, $x_3=f(x_1, x_2, p_3)$ and $(p_1,p_2)=(g_1(x_1, x_2,
p_3), g_2(x_1, x_2, p_3))$. In this new setting, det $d\pi_S= \frac
{\partial f}{\partial p_3}(x_1, x_2, p_3)$ and fold
caustics occur where $ \frac{\partial f}{\partial p_3} = 0$
and $\frac{\partial^2 f}{\partial p_3^2} \neq 0$.
\par In the marine geometry, the source $s=(s_1, s_2,0)$ is subject
to the restriction $s_2=r_2$, so we
consider just $s_1$ as an independent coordinate. Fix an $s_2\in\mathbb R$,
i.e., consider a single pass of the vessel, and let $\Lambda_{s_2}$
be the union of
the flowouts
$\left\{
\Lambda_{(s_1,s_2,0)}: s_1\in\mathbb R\right\}$, so that $\Lambda_{s_2}\subset
T^*\mathbb R^3\setminus 0$ is an involutive submanifold.
We say that fold
caustics (and no
worse) appear for the background sound speed if, considering
$s_1$ as a variable, the spatial projection
$\pi_Y:\Lambda_{s_2}\longrightarrow \mathbb R^3$ is a submersion with folds. By the
structural stability of submersions with folds, this condition will
then hold for all
$s_2'$ close to $s_2$. The presence of fold caustics may be
characterized as follows.
The variables
$x_3$ and
$(p_1, p_2)$ are functions of the other four:
$x_3=f(x_1, x_2, s_1, p_3)$, $(p_1,p_2)=(g_1(x_1, x_2,s_1, p_3),
g_2(x_1, x_2, s_1, p_3))$. The differential $d \pi_Y$ then becomes:
\[{d \pi_Y}= \left(\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 \\
\frac{\partial f} {\partial x_1} & \frac{\partial f}{\partial x_2}
& \frac{\partial f}{\partial s_1} & \frac{\partial f}{\partial
p_3}
\end{array} \right).\]
We have
\begin{eqnarray*}
{\rm rank} \ \ d \pi_Y = \left\{
\begin{array}{ccc}
2, & {\rm if}& \frac{\partial f}{\partial s_1}=\frac{\partial
f}{\partial p_3}=0\\
3, & {\rm if }& \frac{\partial f}{\partial s_1} \neq 0 \
{\rm or} \ \frac{\partial f}{\partial p_3} \neq 0.
\end{array} \right.
\end{eqnarray*}
\par Suppressing $s_2$, let $ {\mathcal S}_1^{\Lambda}:=
{\mathcal S}_1(\pi_Y)=\{ \frac{\partial f}{\partial p_3}=
\frac{\partial f}{\partial s_1}=0 \}$ be the critical set of $\pi_Y$, where
$\textrm{ rank } d\pi_Y$
drops by 1. At points of
${\mathcal S}_1^{\Lambda}$,
$\textrm { Ker } d\pi_Y$ is two-dimensional and spanned by
$\{ (0,0, \delta s_1, \delta p_3) \}$. The tangent space to ${\cal
S}_1^{\Lambda}$ is
\begin{equation}\label{tansone}
T {\cal S}_1^\Lambda= \textrm { Ker } (d_{x_1,x_2,p_3,s_1}(\frac{\partial
f}{\partial p_3})) \cap \textrm { Ker } (d_{x_1,x_2,p_3,s_1}(\frac{\partial
f}{\partial s_1})).
\end{equation}
where
\begin{equation}\label{dpthree}
d_{x_1,x_2,p_3,s_1}(\frac{\partial
f}{\partial p_3})= (\frac{\partial^2 f}{\partial x_1 \partial
p_3}, \frac{\partial^2 f}{\partial x_2
\partial p_3}, \frac{\partial^2 f}{
\partial p_3^2}, \frac{\partial^2 f}{\partial s_1 \partial p_3})
\end{equation}
and
\begin{equation}\label{dsone}
d_{x_1,x_2,p_3,s_1}(\frac{\partial
f}{\partial s_1})= (\frac{\partial^2 f}{\partial x_1 \partial
s_1}, \frac{\partial^2 f}{\partial x_2
\partial s_1}, \frac{\partial^2 f}{
\partial p_3 \partial s_1}, \frac{\partial^2 f}{\partial s_1 ^2}).
\end{equation}
Then, $\pi_Y$ is a submersion with folds if
\begin{eqnarray}\label{starstarstar}
{\cal S}_1^{\Lambda} \textrm{ is smooth, i.e., the
gradients in (\ref{dpthree}) and (\ref{dsone}) are}\ \ \ \ \nonumber\\
\textrm{linearly independent, and }T{\mathcal S}_1^{\Lambda}\textrm{
is transversal
to }\textrm { Ker } d\pi_S,\
\end{eqnarray}
i.e., if
\begin{equation}\label{detnz}
\left|\begin{array}{cc}
\frac{\partial^2 f}{\partial p_3^2}& \frac{\partial^2 f}{\partial s_1
\partial p_3}\\
\frac{\partial^2 f}{\partial p_3 \partial s_1}& \frac{\partial^2
f}{\partial s_1 ^2}
\end{array}\right|\ne 0.
\end{equation}
\par Next, we parametrize the canonical relation $C$ of $F$ in
terms of $s_1, x_1, x_2$ and
$p_3$ ; $(\alpha_1, \alpha_2, \sqrt {1-|\alpha|^2})$, the take off
direction of the reflected ray, writing $\alpha=
(\alpha_1,\alpha_2)$; and $\tau$, the variable dual to time.
Following \cite{no}, the canonical relation $C \subset T^*(\Sigma_{r,s} \times
(0,T)) \times T^*\mathbb R^3_+$ is parametrized as
\begin{eqnarray*}
C = \big\{ (&s_1&, r_1(\cdot), r_2(\cdot),
t_{inc}(\cdot) + t_{ref}(\cdot),
\sigma(\cdot),
\rho_1(\cdot), \rho_2(\cdot), \tau ; \\
&{ } x_1&, x_2, f(\cdot), -\tau(c_0^{-1}(\cdot)
\alpha_1 + g_1(\cdot)), -\tau (c_0^{-1}(\cdot)\alpha_2 + g_2(\cdot)),\\
&{ }&-\tau(c_0^{-1}(\cdot), \alpha)\sqrt{1-|\alpha|^2} + p_3) \big\}
\end{eqnarray*}
where
\begin{eqnarray*}
f(\cdot)&=&f(x_1,x_2, s_1, p_3); \\
r_j(\cdot)&=&r_j(x_1,x_2, f(x_1,x_2,
s_1, p_3), j=1,2;\\
t_{inc}(\cdot)&=&t_{inc}(x_1, x_2, p_3); \\
t_{ref}(\cdot)&=&t_{ref}(x_1,x_2, f(x_1,x_2, s_1, p_3);\\
\sigma(\cdot)&=&\sigma(x_1,x_2, f(x_1,x_2, s_1, p_3);\\
\rho_j(\cdot)&=&\rho_j(x_1,x_2, f(x_1,x_2, s_1, p_3), \alpha), j=1,2;\\
g_j(\cdot)&=&g_j(x_1, x_2, s_1, p_3), j=1,2 {\rm ; and }\\
c_0^{-1}(\cdot)&=&c_0^{-1}\left(x_1, x_2,f(\cdot)\right).
\end{eqnarray*}
It was proven in \cite{nosy} that $F:\mathcal E'\left(\mathbb
R_+^3\right)\longrightarrow\mathcal D'\left(\Sigma_{r.s}\times
(0,T)\right)$ is a
Fourier integral operator,
$F\in I^{\frac{3}{4}} (\Sigma_{r,s} \times (0,T), \mathbb R^3_+; C)$; we now
show that the presence of caustics of fold type (and no worse) imposes certain
conditions on $C$, namely that $\pi_R : C \rightarrow T^*\mathbb R^3
\setminus 0$ is a submersion with folds and \linebreak $\pi_L : C
\rightarrow T^*(\Sigma_{r,s} \times (0,T)) \setminus 0$ is a cross
cap. In fact, with respect with the coordinates above,
$\pi_R :
\mathbb R^7 \rightarrow \mathbb R^6$ is given by
\begin{eqnarray*}
\pi_R(x_1,x_2,p_3,s_1,\alpha_1,\alpha_2,\tau)= \big ( x_1, x_2,
f(\cdot); & -&\tau(c_0^{-1}(\cdot) \alpha_1 + g_1(\cdot)),\\
&-&\tau(c_0^{-1}(\cdot)
\alpha_2 + g_2(\cdot)),\\
&-&\tau(c_0^{-1}(\cdot) \sqrt{1-|\alpha|^2} + p_3)
\big ).
\end{eqnarray*}
Thus,
\[{d \pi_R}= \left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 0 & 0 & 0 & 0 & 0\\
\frac{\partial f}{\partial x_1} & \frac{\partial f}
{\partial x_2} & \frac{\partial f}{\partial p_3} & \frac{\partial
f}{\partial s_1} & 0 & 0 & 0\\
A_1 & A_2 & A_3 & A_4 & -\tau c_0^{-1} & 0 & -(c_0^{-1}\alpha_1+g_1)\\
B_1 & B_2 & B_3 & B_4 & 0 & -\tau c_0^{-1} & -(c_0^{-1}\alpha_2+g_2)\\
C_1 & C_2 & C_3 & C_4 &
\frac{-\tau c_0^{-1}\alpha_1}{\sqrt{1-|\alpha|^2}} &
\frac{-\tau c_0^{-1}\alpha_2}{\sqrt{1-|\alpha|^2} } &
-(c_0^{-1}\sqrt{1-|\alpha|^2}+p_3)
\end{array} \right)\]
\par Thus, rank $d \pi_R = \left\{ \begin{array}{ccc}
5, & {\rm if} & \ \ \frac{\partial f}{\partial p_3}= \frac{\partial
f}{\partial s_1}=0\\
6, & {\rm if } & \ \ \frac{\partial f}{\partial p_3} \neq 0 \
{\rm or} \ \frac{\partial f}{\partial s_1} \neq 0
\end{array} \right. $ because the matrix
\[ \left(\begin{array}{ccc}
-\tau c_0^{-1} & 0 & -(c_0^{-1}\alpha_1+g_1)\\
0 & -\tau c_0^{-1} & -(c_0^{-1}\alpha_2+g_2)\\
- c_0^{-1} \frac{\tau\alpha_1}{\sqrt{1-|\alpha|^2}} & -c_0^{-1}
\frac{\tau\alpha_2}{\sqrt{1-|\alpha|^2} } & -(c_0^{-1}\sqrt{1-|\alpha|^2}+p_3)
\end{array} \right)\]
is nonsingular \cite{no}. Hence, the critical set ${\mathcal S}_1^C:= {\mathcal
S}_1(\pi_R)$
is a smooth, codimension two submanifold.
(Recall that by general considerations
[4], this must equal ${\mathcal S}_1(\pi_L)$, and $d\pi_R$ and
$d\pi_L$ must drop rank
by the same amount at each point.) At these points,
$\textrm { Ker } d
\pi_R =
\{ (0, 0,
\delta p_3,
\delta s_1,
\delta \alpha_1,
\delta \alpha_2, \delta \tau) \}$ where $\delta \alpha_1, \delta
\alpha_2, \delta \tau$ depend on $\delta p_3, \delta s_1$.
The tangent space to ${\cal S}_1^C$ is
\[
T {\cal S}_1^C= \textrm { Ker }
\left(d_{x_1,x_2,p_3,s_1,\alpha_1,\alpha_2,\tau}(\frac{\partial
f}{\partial p_3})\right) \cap \textrm { Ker }
\left(d_{x_1,x_2,p_3,s_1,\alpha_1,\alpha_2,\tau}(\frac{\partial
f}{\partial s_1})\right),
\]
with
\begin{equation}\label{grad-one}
d_{x_1,x_2,p_3,s_1,\alpha_1,\alpha_2,\tau}(\frac{\partial
f}{\partial p_3})= (\frac{\partial^2 f}{\partial x_1 \partial
p_3}, \frac{\partial^2 f}{\partial x_2
\partial p_3}, \frac{\partial^2 f}{
\partial p_3^2}, \frac{\partial^2 f}{\partial s_1 \partial p_3}, 0, 0,0)
\end{equation}
and
\begin{equation}\label{grad-two}
d_{x_1,x_2,p_3,s_1,\alpha_1,\alpha_2,\tau}(\frac{\partial
f}{\partial s_1})= (\frac{\partial^2 f}{\partial x_1 \partial
s_1}, \frac{\partial^2 f}{\partial x_2
\partial s_1}, \frac{\partial^2 f}{
\partial p_3 \partial s_1}, \frac{\partial^2 f}{\partial s_1 ^2}, 0, 0,0),
\end{equation}
and so we see that $\textrm { Ker } d \pi_R$ is transversal to $T {\cal S}_1^C$
because of the condition (\ref{starstarstar}). Hence, $\pi_R$ is a submersion
with folds. Note that without further restrictions on $f$, the
projection $\pi_R$
can either an elliptic or hyperbolic submersion with folds.
\par Similarly, with respect to the above coordinates,
reordered for ease of display, $\pi_L: \mathbb R^7
\rightarrow \mathbb R^8$ is given by
\[\pi_L(s_1, x_1, x_2, \alpha_1,\alpha_2, p_3, \tau)= \big(s_1,
r_1(\cdot), r_2(\cdot),
t_{inc}(\cdot) + t_{ref}(\cdot); \sigma(\cdot),
\rho_1(\cdot), \rho_2(\cdot), \tau \big),
\]
and thus ${d \pi_L}= $
\[
\left(\begin{array}{ccccccc}
1 & 0 & 0 & 0 & 0 & 0 & 0\\
\frac{\partial r_1}{\partial s_1} & \frac{\partial r_1}{\partial
x_1} + \frac{\partial r_1}{\partial x_3}\frac{\partial
f}{\partial x_1} & \frac{\partial r_1}{\partial x_2} +
\frac{\partial r_1}{\partial x_3}\frac{\partial f}{\partial x_2} &
\frac{\partial r_1}{\partial \alpha_1} & \frac{\partial r_1}{\partial
\alpha_2} & \frac{\partial r_1}{\partial x_3} \frac{\partial f}{\partial
p_3} & 0\\
\frac{\partial r_2}{\partial s_1} & \frac{\partial r_2}{\partial
x_1} + \frac{\partial r_2}{\partial x_3}\frac{\partial f}{\partial
x_1} & \frac{\partial r_2}{\partial x_2} + \frac{\partial
r_2}{\partial x_3}\frac{\partial f}{\partial x_2}& \frac{\partial
r_2}{\partial \alpha_1}
& \frac{\partial r_2}{\partial \alpha_2} & \frac{\partial r_2}{\partial
x_3} \frac{\partial f}{\partial p_3} & 0\\
\frac{\partial t_{ref}}{\partial s_1} & \frac{\partial
t_{inc}}{\partial x_1} + \frac{\partial t_{ref}}{\partial
x_3}\frac{\partial f}{\partial x_1} & \frac{\partial
t_{inc}}{\partial x_2} + \frac{\partial t_{ref}}{\partial
x_3}\frac{\partial f}{\partial x_1} & \frac{\partial
t_{ref}}{\partial \alpha_1} &
\frac{\partial t_{ref}}{\partial \alpha_2} & \frac{\partial
t_{inc}}{\partial p_3} + \frac{\partial t_{ref}}{\partial x_3}
\frac{\partial f}{\partial p_3} & 0 \\
\frac{\partial \sigma}{\partial s_1} & \frac{\partial
\sigma}{\partial x_1} + \frac{\partial \sigma}{\partial
x_3}\frac{\partial f}{\partial x_1} & \frac{\partial
\sigma}{\partial x_2} + \frac{\partial \sigma}{\partial
x_3}\frac{\partial f}{\partial x_2}& \frac{\partial
\sigma}{\partial \alpha_1} & \frac{\partial \sigma}{\partial
\alpha_2} & \frac{\partial \sigma}{\partial x_3}
\frac{\partial f}{\partial p_3} & 0 \\
\frac{\partial \rho_1}{\partial s_1} & \frac{\partial
\rho_1}{\partial x_1} + \frac{\partial \rho_1}{\partial
x_3}\frac{\partial f}{\partial x_1}& \frac{\partial
\rho_1}{\partial x_2} + \frac{\partial \rho_1}{\partial
x_3}\frac{\partial f}{\partial x_2} & \frac{\partial
\rho_1}{\partial \alpha_1} & \frac{\partial \rho_1}{\partial
\alpha_2} &
\frac{\partial \rho_1}{\partial x_3} \frac{\partial \rho_1}{\partial p_3}
& 0\\
\frac{\partial \rho_2}{\partial s_1} & \frac{\partial
\rho_2}{\partial x_1} + \frac{\partial \rho_2}{\partial
x_3}\frac{\partial f}{\partial x_1}& \frac{\partial
\rho_2}{\partial x_2} + \frac{\partial \rho_2}{\partial
x_3}\frac{\partial f}{\partial x_2} & \frac{\partial
\rho_2}{\partial \alpha_1} & \frac{\partial \rho_2}{\partial
\alpha_2} &
\frac{\partial \rho_2}{\partial x_3} \frac{\partial \rho_2}{\partial p_3}
& 0\\
0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array} \right)\]
\par Since $t_{inc}$ and $p_3$ are independent coordinates, we have
$\frac{\partial t_{inc}}{\partial
p_3}=0$. Also, because of the choice of the coordinates $x_1, x_2,
x_3$, it follows that $\frac{\partial f}{\partial x_1}= \frac{\partial
f}{\partial x_2}=0$ at the caustic points \cite{no-per}.
\par From this, it follows that the rank of $d \pi_L = \left\{
\begin{array}{ccc}
6, & {\rm if} & \ \ \frac{\partial f}{\partial p_3}= \frac{\partial
f}{\partial s_1}=0\\
7, & {\rm if } & \ \ \frac{\partial f}{\partial p_3} \neq 0 \
{\rm or} \ \frac{\partial f}{\partial s_1} \neq 0,
\end{array} \right. $ \\ because the matrix
\[ \left(\begin{array}{cccc}
\frac{\partial r_1}{\partial x_1}& \frac{\partial r_1}{\partial x_2}
& \frac{\partial r_1}{\partial \alpha_1} & \frac{\partial r_1}{\partial
\alpha_2} \\
\frac{\partial r_2}{\partial x_1}& \frac{\partial r_2}{\partial x_2}
& \frac{\partial r_2}{\partial \alpha_1} & \frac{\partial r_2}{\partial
\alpha_2} \\
\frac{\partial \rho_1}{\partial x_1}& \frac{\partial \rho_1}{\partial
x_2} & \frac{\partial \rho_1}{\partial \alpha_1} & \frac{\partial
\rho_1}{\partial
\alpha_2}\\
\frac{\partial \rho_2}{\partial x_1}& \frac{\partial
\rho_2}{\partial x_2} & \frac{\partial \rho_2}{\partial
\alpha_1} & \frac{\partial \rho_2}{\partial \alpha_2}
\end{array} \right)\]
is nonsingular \cite{no}. Furthermore, $\mathcal{S}_1(\pi_L)=\mathcal{S}_1^C=
\{\frac{\partial f}{\partial p_3}=\frac{\partial f}{\partial
s_1}=0\}$ is smooth and
the $7\times7$ minors of $d\pi_L$ generate the ideal of $\mathcal{S}_1^C$.
Finally, it also
follows that
\[\textrm { Ker } d \pi_L = \{ (0, \delta x_1, \delta x_2, \delta \alpha_1,
\delta \alpha_2, \delta p_3, 0) \},
\]
where $\delta \alpha_1,
\delta \alpha_2, \delta x_1, \delta x_2$ depend on
$\delta p_3$, and thus $\textrm { Ker } d\pi_L\cap T {\cal S}_1^C=(0)$. Hence, $\pi_L$
is a cross cap.
\ \
\par This leads us to formulate a general class of canonical
relations with this structure.
\begin {definition}\label{def-fcc}
{\rm Let} $X$ { \rm and} $Y$ {\rm be manifolds of dimensions} $n$
{\rm and} $n-1,$ {\rm respectively, and let} $C$ {\rm be a
canonical relation in } $\left(T^*X \setminus 0\right) \times
\left(T^*Y \setminus 0\right)$. $C $ {\rm is a} {\it folded cross
cap} {\rm canonical
relation if:}
a) $\pi_R : C \rightarrow T^*Y \setminus 0$ {\rm is a submersion with
folds, with singular set} ${\cal S}_1,$
{\rm and } $\pi_R({\cal S}_1)$ {\rm is a nonradial hypersurface in}
$T^*Y \setminus 0$; {\rm and}\\
\indent b) $\pi_L : C \rightarrow T^*X \setminus 0$ {\rm has a cross cap
singularity along} ${\cal S}_1$ {\rm and} $\pi_L({\cal S}_1)$ {\rm
is a nonradial submanifold.}
{\rm We say that} $C$ {\rm is an
\emph{elliptic}, respectively,
\emph{hyperbolic}, folded cross cap if } $\pi_R$ {\rm is an elliptic,
respectively, hyperbolic,
submersion with folds}.
\end{definition}
\begin{remark} {\rm We will see in \S6 (see discussion following Prop.
6.2) that } $\pi_L({\cal S}_1)$ {\rm is necessarily } maximally
noninvolutive{\rm , defined
after (\ref{iii}) below.}
\end{remark}
\par Here, as usual, a conic submanifold $\Gamma \subset
T^*Y\setminus 0$ is {\it
nonradial} if for all $(y,\eta)\in\Gamma$, the canonical one-form
$\sum \eta_j dy_j$
does not vanish identically on $T_{(y,\eta)}\Gamma$. Since $\pi_R({\cal S}_1)$ is the
immersed image of ${\cal S}_1$, $\pi_R({\cal S}_1)$ is nonradial at $\pi_R(c)$ iff $\sum
(\pi_R^*\eta_j) d\pi_R^*(d y_j)\ne 0$ as an element of $T_{c}^*C$. We can
understand the significance of this for the canonical relation
arising in the marine
geometry as follows. Using the fact that $df=0$ at the caustics, the
expressions for
$\pi_R$ and $d\pi_R$ computed above imply that, at $c\in{\cal S}_1$,
\begin{equation}\label{pb}
\sum_{j=1}^3(\pi_R^*\eta_j) d\pi_R^*(d
y_j)=-\tau\left((c_0^{-1}\alpha_1+g_1) dx_1+
(c_0^{-1}\alpha_2+g_2) dx_2\right).
\end{equation}
In order to be 0 on $T_{c}{\cal S}_1$, this must be a linear combination of
(\ref{grad-one}) and (\ref{grad-two}). By (\ref{detnz}), the only possible
linear combination is the trivial one; however, by the expression for
$\pi_R$, this
forces $\eta=(0,0,\eta_3)$ for some $\eta_3\ne 0$. That is, the fold
caustic surface
in $Y$ must be horizontal at this point. While there certainly exist background
soundspeeds $c_0(\cdot)$ for which this happens, the most basic
examples of fold
caustics arising from refraction about a low velocity lens
(see Nolan and Symes \cite{nosy-fold}) have fold
surfaces which are not horizontal.
\par Similarly, since $\pi_L|_{{\cal S}_1}$ is an immersion,
$\pi_L({\cal S}_1)\subset T^*X$ is
nonradial iff
\linebreak$\sum (\pi_L^*\xi_j) d\pi_L^*(d x_j)\ne 0$ in $T^*_cC$. The
physical interpretation
of a radial point, i.e., a point $c\in{\cal S}_1$ where this fails, for the
marine geometry
is less clear and, to proceed, we will simply need to assume that
such points are
absent.
\vfil\eject
\section{Model case}
\par We showed in the previous section that the canonical relation
$C$ arising from the marine geometry in the presence of fold caustics is a
folded cross cap. To help understand the nature of the normal operator, we
first consider
a model folded cross cap canonical relation,
$C_0$, in
$(T^*\mathbb R^n \setminus 0) \times (T^*\mathbb R^{n-1} \setminus 0)$,
parametrized by the phase function
\[\phi_0(x,y,\theta'')=(x''-y'') \cdot \theta'' + \left((x_n^2-x_{n-1}^2)
y_{n-1}+x_ny_{n-1}^2\right)\theta_1,\]
\noindent where $(x, y, \theta'')
\in \mathbb R^n \times \mathbb R^{n-1} \times (\mathbb R^{n-2} \setminus 0)$, in the
region $\{ |\theta_1| \geq c|\theta''| \}$. Here and at various
points below we use
the notation $x=(x_1,\dots,x_n)=(x'', x_{n-1}, x_n)=(x_1,x''',x_n)$, $y=(y'',
y_{n-1})$ and
$\theta''= (\theta_1,\theta''')$.
\par For simplicity, in this section and the next one we will make
the choice
that $C$ is a hyperbolic folded cross cap. This corresponds to the
choice of the $(-)$ sign in the
$(x_{n-1}^2 - x_n^2)$ term of $\phi_0$. There are no significant changes
needed in the calculations for the elliptic case.
\par One easily calculates that
\begin{eqnarray*}
C_0=\big\{( x'', &x_{n-1}, x_{n}, \theta'', -2x_{n-1}y_{n-1}\theta_1,
y_{n-1}^2\theta_1+2x_ny_{n-1}\theta_1;\\
&y'', y_{n-1}, \theta'', -\left(2x_n
y_{n-1}+(x_n^2-x_{n-1}^2)\right)\theta_1)\\
&:
|\theta_1| \geq c|\theta''|, \ x_i=y_i, \ 2 \leq i \leq n-2,\\
&x_1-y_1
+(x_n^2-x_{n-1}^2)y_{n-1} +x_n y_{n-1}^2=0 \big\}
\end{eqnarray*}
\par We will verify that the projections of $C_0$ to the left and to
the right have the desired singularities. Note that $(x, y_{n-1},
\theta'')$ are coordinates on $C_0$. Hence, reordering the variables
for ease of display,
\begin{eqnarray*}
\pi_R (x_1,x''', y_{n-1},\theta_1, \theta''',x_{n-1},x_n)= \big(
x_1& +&(x_n^2-x_{n-1}^2)y_{n-1} +x_n
y_{n-1}^2, x''', y_{n-1}\\
&;& \theta'', -\left(2x_n
y_{n-1}+(x_n^2-x_{n-1}^2)\right)\theta_1\big)
\end{eqnarray*}
and
\[{d \pi_R}= \left(\begin{array}{ccccccc}
1 & 0 & A & 0 & 0 & B & D\\
0 & I_{n-3} & 0 & 0 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0 & 0 & 0\\
0 & 0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & 0 & I_{n-3} & 0 & 0\\
0 & 0 & E & F & 0 & 2x_{n-1}\theta_1 &
-2y_{n-1}\theta_1-2x_n\theta_1
\end{array} \right)\]
where $A= (x_n^2-x_{n-1}^2)+2x_ny_{n-1}, B=-2x_{n-1} y_{n-1},
D= 2x_ny_{n-1}+y_{n-1}^2, \\ E= -2x_n\theta_1$, and $F=
-2x_ny_{n-1}-(x_n^2-x_{n-1}^2)$. This is a $ (2n-2) \times (2n-1)$
matrix, with
$$ {\rm rank} \ \ d \pi_R = \left\{
\begin{array}{ccc}
2n-3, & {\rm if} & \ \ x_{n-1}=x_n+y_{n-1}=0\\
2n-2, & {\rm if } & \ \ x_{n-1} \neq 0 \ {\rm or} \ x_n+y_{n-1}
\neq 0.
\end{array} \right. $$
\par Let ${\cal S}_1^{C_0}:={\cal S}_1(\pi_R)=\{ (x,y_{n-1},\theta'') :
x_{n-1}=x_n+y_{n-1}=0
\} $ be the set where $ d\pi_R$ drops rank by $1$. Off of this
smooth, codimension
two submanifold, $\pi_R$ is a submersion. The kernel of $ \pi_R$ at
${\cal S}_1^{C_0}$ is spanned by $\{ \frac{\partial {\quad
}}{\partial x_{n-1}},
\frac{\partial { \quad}}{\partial x_n} \}$ and thus intersects the tangent
space $T {\cal S}_1^{C_0}$ transversally. We also note that the
Hessian, $(x_{n-1}, x_n) \rightarrow
-(x_n^2-x_{n-1}^2)\theta_1$, is sign-indefinite. We
thus conclude that $\pi_R$ is a hyperbolic submersion with folds.
\par As for $\pi_L$, we have
$$\pi_L(x'',x_{n-1},x_n, \theta_1, \theta''',y_{n-1})=\left(x;
\theta'', -2x_{n-1}y_{n-1}\theta_1,
\left(y_{n-1}^2+2x_ny_{n-1}\right)\theta_1\right),$$
so that
\[{d \pi_L}=
\left(\begin{array}{cccccc} I_{n-2} & 0 & 0 & 0 & 0 &
0\\
0 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & I_{n-3} & 0 \\
0 & A' & 0 & B' & 0 & -2x_{n-1}\theta_1 \\
0 & 0 & D' & E' & 0 & 2(x_n+y_{n-1})\theta_1
\end{array} \right)\]
where $A'= -2y_{n-1}\theta_1, B'=-2x_{n-1} y_{n-1},
D'= 2y_{n-1}\theta_1$, and $E'= 2x_ny_{n-1}+\nolinebreak y_{n-1}^2$.
This is a $ 2n \times (2n-1)$ matrix and
$$ {\rm rank } \ d \pi_L =
\left\{ \begin{array}{ccc}
2n-2, & {\rm if} & \ \ x_{n-1}=x_n+y_{n-1}=0\\
2n-1, & {\rm if } & \ \ x_{n-1} \neq 0 \ {\rm or} \ x_n+y_{n-1}
\neq 0.
\end{array} \right. $$
\par Thus, $d\pi_L$ drops rank by $1$ at ${\cal S}_1^{C_0}$, and off
of this set is an immersion. The kernel of $ \pi_L$ is spanned by $\{
\frac{\partial\quad}{\partial y_{n-1} }\}$ and intersects the tangent
space $T {\cal S}_1^{C_0}$ transversally. Also, the rank of the differential of
$(x,y_{n-1},\theta'') \rightarrow\linebreak
(-2x_{n-1}\theta_1, \left(2y_{n-1} +2 x_n\right)\theta_1)$ is $2$ at
${\cal S}_1^{C_0}$. We conclude that $\pi_L$ has a cross cap
singularity.
\par We note for future use the images
of $ {\cal S}_1^{C_0}$ under $\pi_L$ and $\pi_R$: since $\pi_R({\cal
S}_1^{C_0})=\{
(x'', -x_n, \theta'', x_n^2 \theta_1) \}$ and $ \pi_L({\cal
S}_1^{C_0})=\{ (x'', 0, x_n,\theta'', 0, -x_n^2 \theta_1) \}$, we have
\begin{equation}\label{ii}
\pi_R\left({\cal S}_1^{C_0}\right)= \left\{
\xi_{n-1}= x_n^2 \xi_1 \right\}
\end{equation}
and
\begin{equation}\label{iii}
\pi_L\left({\cal S}_1^{C_0}\right)= \left\{
x_{n-1}=\xi_{n-1}=
\xi_n+x_n^2\xi_1=0\right\}.
\end{equation}
Note that $\pi_R\left({\cal S}_1^{C_0}\right)$ is a nonradial hypersurface in
$T^*\mathbb R^{n-1}\setminus 0$, while
$\pi_L({\cal S}_1^{C_0})$ is a nonradial, codimension three submanifold
of $T^*\mathbb R^{n}\setminus 0$, given by
defining functions $p_1=\xi_{n-1}, p_2=\xi_1 x_{n-1},
p_3=\xi_n+x_n^2\xi_1$ with Poisson brackets $\{p_1, p_2 \}=1, \{p_1,p_3 \}=0,
\{p_2,p_3 \}=0$. This is \emph{maximally noninvolutive} in the sense that
$\omega_{T^*\mathbb R^n} |_{\pi_L({\cal S}_1)}$ has the maximal possible rank for a
codimension three submanifold of $T^*\mathbb R^{n}\setminus 0$, namely
$2n-4$.
\ \
\par Next, we calculate the composition $C_0^t \circ C_0=\{ (x,
\xi; y, \eta): \exists (z, \zeta) \ \ {\rm such
\ \ that}\\ \ \ (x, \xi; z, \zeta) \in C_0^t \ {\rm and } \
(z, \zeta; y, \eta) \in C_0 \}$. We have $(x, \xi; z, \zeta) \in C_0^t $
iff $(z,\zeta;x,\xi)\in\nolinebreak C_0$ iff
$$ z_i=x_i, \ \ 2 \leq i
\leq n-2;$$
$$ z_1-x_1 +(z_n^2-z_{n-1}^2)y_{n-1} +z_n
x_{n-1}^2=0; $$
$$ \zeta_i=\xi_i \ \ 1 \leq i \leq n-2; $$
$$ \zeta_n =2 z_n x_{n-1} \xi_1 + x_{n-1}^2\xi_{1}; $$
$$ \zeta_{n-1}= -2z_{n-1} x_{n-1} \xi_1 ;\textrm{ and } $$
$$ \xi_{n-1} = -(z_n^2-z_{n-1}^2)\xi_1-2z_n x_{n-1} \xi_1,$$
with $(z, \zeta; y, \eta) \in C_0 $ being determined by the same
equations, but where
$(y,\eta)$ replaces $(x,\xi)$. We thus obtain that $C_0^t\circ C_0$ consists of
$(x,\xi;y,\eta)$ such that, for some $(z_{n-1},z_n)\in\mathbb R^2$,
$$x_i=y_i, \ \ 2 \leq i \leq n-2; $$
$$ \xi_i=\eta_i, \ \ 1 \leq i \leq n-2 ; $$
$$ z_{n-1}(x_{n_1}-y_{n-1})\xi_1=0 ;$$
$$ (x_{n-1}-y_{n-1})(2 z_n +x_{n-1}+ y_{n-1}) =0 ;$$
$$ \xi_{n-1} = -\left((z_n^2-z_{n-1}^2)+2z_n x_{n-1}\right)
\xi_1;$$
$$ \eta_{n-1} = -\left((z_n^2-z_{n-1}^2)+2z_n y_{n-1}\right) \eta_1;\
\ \rm{ and } $$
$$y_1-z_1+(z_n^2-z_{n-1}^2)(x_{n-1}-y_{n-1})
+z_n(x_{n-1}^2-y_{n-1}^2)=0.$$
\par If $x_{n-1}=y_{n-1}$ , then $x_1=y_1$ and
$\xi_{n-1}=\eta_{n-1}$, so this contribution to
$C_0^t \circ C_0$ is contained in $ \Delta $, the diagonal canonical
relation in $T^*
\mathbb R^{n-1} \times T^*\mathbb R^{n-1}$.
\par If, on the other hand, $ x_{n-1} \neq y_{n-1}$ , then
$$x_i=y_i, \ \ 2 \leq i \leq n-2; $$
$$ \xi_i=\eta_i, \ \ 1 \leq i \leq n-2 ; $$
$$\xi_{n-1}= \frac{(x_{n-1}+y_{n-1})(3x_{n-1}-y_{n-1})}{4}
\xi_1 ;$$
$$\eta_{n-1}= \frac{(x_{n-1}+y_{n-1})(3y_{n-1}-x_{n-1})}{4}
\xi_1; \rm{ and }$$
$$x_1-y_1+ \frac{(x_{n-1}+y_{n-1})^2( x_{n-1}-y_{n-1})}{4}=0,$$
giving a contribution to $C_0^t \circ C_0$ contained in $ \tilde
{C}_0$, where $ \tilde{C}_0$ is the twisted conormal bundle,
\[\tilde {C}_0= N^* \left\{ x_1-y_1+
\frac{(x_{n-1}+y_{n-1})^2( x_{n-1}-y_{n-1})}{4}=0, x_i-y_i=0, \ 2
\leq i \leq n-2 \right\}'.\]
\par In conclusion, $ C_0^t \circ C_0 \subseteq \Delta \cup \tilde
{C}_0$, from which
it follows that $ \tilde{C}_0$ is symmetric, i.e.,
$ \tilde{C}_0^t = \tilde{C}_0$. It is
easy to see that both projections from $\tilde {C}_0$ have Whitney fold
singularities. Such canonical relations were introduced in \cite{meta}
and called {\it folding canonical relations}; they are
also referred to as {\it two-sided folds} \cite{grse} and we will use
this latter
terminology. One also sees that
$\tilde {C}_0$ intersects $\Delta$ cleanly in codimension $1$, and
furthermore, $\Delta \cap \tilde {C}_0$ is in fact the fold
surface of $\tilde {C}_0$. As described in $\S 3$, one has a well
defined class
of distributions associated to the two cleanly intersecting
Lagrangians $(\Delta', \tilde {C}_0')$, namely $I^{p,l} (\mathbb R^{n-1}\times
\mathbb R^{n-1};\Delta',\tilde {C}_0')$, for $p,l \in \mathbb R$. A distribution
is in this class iff it has an oscillatory representation,
\begin{eqnarray*}
u(x,y)&=&\int_{\mathbb R^{n-1}} e^{i
\{(x_1-y_1-\frac{(x_{n-1}-y_{n-1})(x_{n-1}+y_{n-1})^2}{4})\xi_1
+(x'''-y''')\cdot\xi'''+(x_{n-1}-y_{n-1})\xi_{n-1}\}} \nonumber\\
& &\hbox{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ }
a(x,y; \xi'';\xi_{n-1}) d\xi
\end{eqnarray*}
where $a$ is a symbol-valued
symbol, satisfying the estimates:
$$|\partial^\alpha_{\xi''}\partial^\beta_{\xi_{n-1}}\partial^\gamma_{x,y}
a(x,y,\xi)| \leq
c(1+|\xi|)^{\tilde{p}-|\alpha|}(1+|\xi_{n-1}|)^{\tilde{l}-\beta},$$
with $\tilde{p}=p+l+\frac12,\tilde{l}=-l-\frac12$.
\par Next we will show that if $F \in I^{m-\frac{1}{4}}(C_0)$ then
$F^*F \in I^{p,l} (\Delta, \tilde {C}_0)$, i.e., $K_{F^*F}\in I^{p,l}(\mathbb
R^{n-1}\times\mathbb R^{n-1};\Delta',\tilde C_0')$, for some
$p,l
\in \mathbb R$. We have:
$$Ff(x)=\int e^{
i\{(x''-y'')\cdot\theta''+((x_n^2-x_{n-1}^2)y_{n-1}+ x_n
y_{n-1}^2)\theta_1 \}} a(x,y, \theta) f(y) d\theta dy,$$
where $a\in S^{m+\frac12}$. Thus,
\[ K_{F^*F}(x,y)=\int e^{i\{\phi_0(z,y,\theta'')-\phi_0(z,x,\eta'')\}}
a(z,y,\theta'')\overline{a}(z,x,\eta'') dz d\theta'' d\eta''.
\]
\par After a stationary phase in the variables $z'',\eta''$, the phase function
becomes:
\begin{eqnarray*}
\phi(x,y, \theta'', z_{n-1}, z_{n})&=&
(x''-y'')\cdot\theta''+(z_n^2-z_{n-1}^2)(y_{n-1}-x_{n-1})\theta_1+
z_n(y_{n-1}^2-x_{n-1}^2)\theta_1\\
&=&(x''-y'')\cdot\theta''+
(y_{n-1}-x_{n-1})[(z_n^2-z_{n-1}^2)+
z_n(x_{n-1}+y_{n-1})]\theta_1
\end{eqnarray*}
and the amplitude becomes $\tilde{a}(x,y,z_{n-1},z_n;\theta'')\in S^{2m+1}$.
\par Following an idea from \cite{gruh4},\cite{fel}, we now make a
singular change
of variables,
$T: \mathbb R^n \rightarrow \mathbb R^{n-1},
\ \ T(\theta'', z_{n-1}, z_n)=(\xi'', \xi_{n-1})$, given by:
\begin{eqnarray*}
\xi_i&=&\theta_i, \ 1 \leq i \leq n-2,\\
\xi_{n-1}&=&-\left((z_n^2-z_{n-1}^2)+
z_n(x_{n-1}+y_{n-1})\right)\theta_1.
\end{eqnarray*}
The kernel of
$F^*F$ can then be rewritten as
$$K_{F^*F}(x,y)=\int_{\mathbb R^{n-1}} e^{ i \{ (x''-y'')\cdot\xi''
+(x_{n-1}-y_{n-1})\xi_{n-1}\}} b(x,y,\xi) d\xi,
$$
where, using the coarea formula [4, p.249],
\[ b(x,y;\xi) = \int_{ \{((z_n^2-z_{n-1}^2)\theta_1+
z_n(x_{n-1}+y_{n-1})\theta_1)= -\xi_{n-1}\}} \tilde{a}(x,y, z_{n-1},
z_n;\xi'')
\frac{d\nu}{J_{n-1}}
\]
with $J_{n-1}$ the $(n-1)$-Jacobian of $T$ and $d\nu$ the arc length measure
on\linebreak
$\left\{\left((z_n^2-z_{n-1}^2)+
z_n(x_{n-1}+y_{n-1})\right)\theta_1= -\xi_{n-1}\right\}$.
The Jacobian is given by
\begin{eqnarray*}
J_{n-1}&=&|\nabla_z
\xi_{n-1}|\\
&=&|( 2z_{n-1} \theta_1, -(2z_n+
x_{n-1}+y_{n-1})\theta_1)|\\
&=& 2
\left(z_{n-1}^2+(z_n+\frac{x_{n-1}+y_{n-1}}{2})^2\right)^{\frac{1}{2}}|\theta_1|,
\end{eqnarray*}
which vanishes to first order at
$z_{n-1}=z_n+\frac{x_{n-1}+y_{n-1}}{2}=0$. Note that, at these points,
$\xi_{n-1}=\frac{(x_{n-1}+y_{n-1})^2}{4}\xi_1$. Thus, $b$ is of order
$2m$ in $\xi''$
and has a conormal singularity of order $-1$ at
$\{\xi_{n-1}-\frac{(x_{n-1}+y_{n-1})^2}{4}\xi_1=0\}$; thus, it has an
oscillatory
representation
\[ b(x,y,\xi)=\int_{\mathbb R}
e^{i\{(\xi_{n-1}-\frac{(x_{n-1}+y_{n-1})^2}{4}\xi_1)\frac{\rho}{\xi_1})\}}
\tilde{b}(x,y; \xi; \rho) d \rho.\]
Finally, the kernel of $F^*F$ becomes
\begin{equation}\label{ker}
K_{F^*F}(x,y)=\int_{\mathbb R^n} e^{ i \{ (x''-y'')\cdot\xi''
+(x_{n-1}-y_{n-1})\xi_{n-1} +
(\xi_{n-1}-\frac{(x_{n-1}+y_{n-1})^2}{4}\xi_1)\frac{\rho}{\xi_1}
\}} \tilde{b}(x,y;\xi; \rho) d\xi d\rho
\end{equation}
where one can check that $\tilde{b}$ is a product-type
symbol, $\tilde{b} \in S^{2m, -1}(2n-\nolinebreak2,n-\nolinebreak1,1)$.
We now introduce a conic partition of unity in $(\xi,\rho)$, with supports in
$\{|\rho|\le 2|\xi|\}$ and $\{|\rho|\ge\frac12 |\xi|\}$, resulting in a
decomposition $K_{F^*F}=K^0+K^1$. Letting
$$\psi(x,y; \xi, \rho)= (x''-y'')\cdot\xi'' +(x_{n-1}-y_{n-1})\xi_{n-1}
+(\xi_{n-1}-\frac{(x_{n-1}+y_{n-1})^2}{4}\xi_1)\frac{\rho}{\xi_1},$$
one easily sees that on the region $\{|\rho|\le c |\xi|\}$, this is
a multi-phase
function for $(\Delta', \tilde{C}_0')$ in the sense of Def.
\ref{old-def3.6}, i.e., $\psi_0(x,y; \xi):=\psi_0(x,y, \xi,0)$
parametrizes the
diagonal Lagrangian $\Delta'$ and
$\psi(x,y;(\xi,\rho))$ parametrizes the Lagrangian $\tilde{C}_0'$.
Furthermore, on this region, the amplitude is a symbol-valued symbol,
belonging to
$S^{2m,-1}(\mathbb R^{2n-2}\times (\mathbb R^{n-1}\setminus 0)\times\mathbb R)$. In view of Remark
\ref{old-rem3.8}, $K^0\in I^{p,l}(\mathbb R^{n-1}\times\mathbb
R^{n-1};\Delta',
\tilde{C}_0')$, for some $p,l\in\mathbb R$.
The orders may also be found
by applying Remark \ref{old-rem3.8} to (\ref{ker}). We see that
\[
p=(2m-1) +\frac{(n-1)+1}2-\frac{2n-2}4=2m-\frac12
\]
and
\[
l=-(-1)-\frac12=\frac12.
\]
Hence, this contribution to $F^*F$ is in
$I^{2m-\frac{1}{2}, \frac{1}{2}}(\Delta,\tilde{C}_0)
:=I^{2m-\frac{1}{2}, \frac{1}{2}}(\mathbb R^{n-1},\mathbb R^{n-1};\Delta,\tilde{C}_0)$,
with $\tilde{C}_0$ a two-sided fold.
Next, we show that $K^1\in I^{2m-\frac{1}{2}}(\tilde {C}_0)$.
First, let $s=\xi_{n-1}/\xi_1$. Then we can express
$K^1(x,y)= \int L(x,y,s) ds$ ,
where $L$ has the phase function
$$\Psi(x,y,s;\rho;
\xi'')=(x''-y'')\cdot\xi'' +(x_{n-1}-y_{n-1})\xi_1 s
+(s-\frac{(x_{n-1}+y_{n-1})^2}{4})\rho$$
and amplitude $c\in S^{-1,2m+1}(\mathbb R^{2n-1}\times (\mathbb R\setminus
0)\times\mathbb R^{n-1})$.
Note that
$\Psi_0(x,y,s;\rho):=\Psi(x,y,s;\rho;0)$ parametrizes
\[
\Gamma_0:= N^*\{
s-\frac{(x_{n-1}+y_{n-1})^2}{4}=0 \}\subset T^*\mathbb R^{2n-1}\setminus 0
\]
and
$\Psi_1(x,y,s;(\rho,\xi'')):=\Psi(x,y,s;(\rho,\xi''))$ parametrizes
\[
\Gamma_1:= N^* \left\{ x_i-y_i=0, 2 \leq i\leq n-2,
\ x_1-y_1+s(x_{n-1}-y_{n-1})=
s-\frac{(x_{n-1}+y_{n-1})^2}{4}=0 \right\}.
\]
Thus, $\Psi$ is
a multi-phase function in the sense of Def $3.5$.
Introduce a cutoff
function, $\chi \in C_0^{\infty}(R^n), \chi=1 $
near $0$, set
$$L^0(x,y,s)=\int e^{i \Psi(x,y,s;\rho,
\xi'')}\chi(\frac{|\xi''|}{c|\rho|})
c(x,y;\rho;\xi'') d\xi'' d\rho,$$
and let $L^1= L- L^0$. We have $L^0 \in
I^{2m-\frac{1}{4},\frac{1}{2}} (\mathbb R^{2n-1};\Gamma_0, \Gamma_1)$, $L^1 \in
I^{2m-\frac{1}{4}} (\mathbb R^{2n-1};\Gamma_1)$ and $WF(L^1)$ is contained
in the complement
of a neighborhood of $\Gamma_0$.
\par Since $K^2$ is
simply $\pi_{*}L$, the pushforward of $L$ by $\pi$, the projection
$\pi(x,y,s)=(x,y)$, which is a submersion, we compute the pushforwards of the
Lagrangians $\Gamma_0,
\Gamma_1$. It follows from standard results about pushforwards \cite{ho}
that, for
$u\in\mathcal E'\left(\mathbb R^{2n-1}\right)$,
\begin{eqnarray*}
WF(\pi_{*}u) \subseteq \{ (x,y; \xi, \eta)&|&
\exists (\hat{x},
\hat{y}, s, \hat\xi, \hat\eta, \sigma) \in WF(u) \ \ s.t.\\
&{ }&(x,y) =
\pi(\hat{x},
\hat{y},s), (\hat\xi, \hat\eta, \sigma)= d \pi^t (\xi,
\eta) \}.
\end{eqnarray*}
Using $d \pi^t (\xi,\eta)=(\xi,\eta,0)$, it is then an easy
calculation that the
push forward of
$\Gamma_0$, and indeed any neighborhood of $\Gamma_0$ on which
$\sigma\ne0$, is empty,
so that
$\pi_*(L^0)\in C^\infty$, while
$\pi_*(\Gamma_1)\subseteq\tilde{C}_0'$.
In fact, $\pi_{*}$ is a FIO of order $- \frac{1}{4}$, and the application of
$\pi_*$ to $I^\cdot(\Gamma_1)$ is covered by the transverse intersection
calculus. Hence
$\pi_{*}: I^{r}(\Gamma_1)
\rightarrow
I^{r-\frac{1}{4}}(\tilde{C}_0')$, so that
\[
K^1=\pi_{*}(L^1) \mod C^\infty\in I^{2m-\frac{1}{2}} (\tilde{C}_0),
\]
and thus
$K_{F^*F}=K^0+K^1 \in I^{2m-\frac{1}{2},\frac12}(\Delta',\tilde{C}_0')$.
\vskip.2in
In conclusion, we have shown that $F^*F\in
I^{2m-\frac{1}{2},\frac12}(\Delta,\tilde{C}_0)$.
For the single source data acquisition
geometry, Nolan\cite{no} and Felea\cite{fel} showed that $F^*F \in
I^{2m,0} (\Delta, \tilde {C})$, with $\tilde{C}$ a two-sided fold. In
that situation the artifact, i.e., the part of $F^*F$ on
$\tilde{C} \setminus \Delta$,
has, by Remark \ref{old-rem3.4}, the same strength as on
$\Delta\setminus\tilde{C}$.
However, it follows from what we have shown here that for the
microlocal model $C_0$
of the marine geometry, the artifact is $\frac{1}{2}$ order smoother then the
pseudodifferential operator part. In the next section, we show
this in full generality for FIOs associated with folded cross caps.
\section{Composition calculus in the general case}
\par By a well known result of Melrose and
Taylor\cite{meta},
any two-sided fold can be conjugated, via canonical transformations
on the left and
right, to a normal form. Unfortunately, even if folded cross caps could be
conjugated to a normal form, such a result would presumably be difficult to
prove. However, we will be able to avoid this problem by finding a
weak normal form
for the canonical relation and thus for a phase function
parametrizing it, adapting a
method originally developed by Greenleaf and Uhlmann \cite{gruh3,gruh4} for
some canonical
relations arising in integral geometry, for which
$\pi_R$ belongs to a class containing the submersions with folds and
$\pi_L$ is maximally degenerate. This will be sufficient
for establishing the composition calculus for general folded cross
cap canonical
relations.
\vskip.2in
\begin{theorem}\label{main} {\em If} $F \in I^{m-\frac{1}{4}}(X,Y;C)$ {\em is
properly supported
and} $C$ {\em is a folded cross cap canonical relation, then}
$F^*F \in I^{2m- \frac{1}{2},\frac{1}{2}}(Y,Y;\Delta, \tilde{C})$.
{\em Furthermore, } $\tilde
{ C}$ {\em is a symmetric, two-sided fold in} $\left(T^*Y \setminus
0\right) \times
\left(T^*Y \setminus 0\right)$, {\rm and }
$\Delta\cap\tilde C$ {\rm equals the fold surface in } $\tilde C$.
\end{theorem}
\vskip.2in
\begin{remark}
It was shown in \cite{nosy} that, for the marine seismic data
geometry in three spatial dimensions, the linearized scattering map
$F$ belongs to $I^{1-\frac14}(\Sigma_{r,s}\times (0,T),\mathbb R^3; C)$.
Thus, in the presence of fold caustics and under the additional
nonradiality assumptions described below
Def. \ref{def-fcc}, it follows from Theorem \ref{main} that the
normal operator $F^*F$ lies in
$I^{\frac32.\frac12}(\mathbb R^3,\mathbb R^3;\Delta,\tilde{C})$, with $\tilde{C}$
as above.
\end{remark}
\vskip.2in
Before establishing a weak normal form for a general folded cross
cap, we first need
to find separate weak normal forms for each of the two projections,
$\pi_R$ and $\pi_L$.
\begin{proposition}\label{sympl-swf}
Let $f:V \longrightarrow W$ be a smooth, conic map, with $V$ a
smooth, conic manifold
of dimension $2n-1$ and $(W,\omega_W)$ a conic symplectic manifold of dimension
$2n-2$. Assume that $f$ is a submersion with folds at $v_0\in V$ and $f(V)$ is
nonradial in $W$. Then, there exist local homogeneous coordinates
$(s'',s_{n-1},\sigma'',\sigma''')\in\mathbb R^{n-2}\times\mathbb R\times(\mathbb R^{n-2}\setminus
0)\times\mathbb R^2$ on $V$ near $v_0$, and local canonical coordinates
$(y',\eta')\in T^*\mathbb R^{n-1}\setminus 0$ on $W$ near $w_0:=f(v_0)$, such that
$v_0=(0,0,e_1^*,0),\quad w_0=(0;e_1^*)$, and
\begin{equation}\label{swfnf}
f(v)=f(s'',s_{n-1},\sigma'',\sigma''')
=\left(s'',s_{n-1};\sigma'',A^{s,\sigma}(\sigma''',\sigma''')\right),
\end{equation}
where $A^\cdot\in C^\infty(\mathbb R^{2n-1},{S^2\mathbb R^2}^*)$ is homogeneous of
degree -1 in
$\cdot$ and takes values in the nonsingular quadratic forms of the
same signature as
Hess $f(v_0)$.
\end{proposition}
\noindent\emph{Proof.}
In the terminology of \cite{gruh4}, a submersion with folds has {\it
clean folds of multiplicity one}. Prop.\ref{sympl-swf} is then a
special case of
\cite[Lem. 3.A.1]{gruh4}, with slight change of notation, and $(N,m,n)$ in
\cite{gruh4} being $(n-2,1,2)$ here. $\square$
\vskip.2in
Applying (\ref{swfnf}) to $\pi_R:C\longrightarrow T^*Y\setminus 0$ and writing
$\sigma'''=(\sigma_{n-1},\sigma_n)$, we see that
${\cal S}_1^C={\cal S}_1(\pi_R)=\{\sigma_{n-1}=\sigma_n=0\}$ and $\pi_R({\cal
S}_1^C)=\{\eta_{n-1}=0\}$. Furthermore,
\begin{eqnarray}\label{omegc}
\omega_C &=& \pi_R^*(\sum_{j=1}^{n-1} d\eta_j\wedge dy_j)\nonumber\\
&=& \sum_{j=1}^{n-2} d\sigma_j\wedge ds_j + d\left(A^{s,\sigma)}
(\sigma''',\sigma''')\right)\wedge ds_{n-1}.
\end{eqnarray}
Hence,
\begin{equation}\label{ker-swf}
Ker \left(\omega_C|_{T{\cal S}_1}\right)=\mathbb
R\cdot\left(\frac{\partial}{\partial
s_{n-1}}+\dots\right).
\end{equation}
Also, as in the case of the model canonical relation, $C_0$, we see that
$\omega_C$ has rank $2n-2$ on $C\setminus {\cal S}_1^C$, and has rank
$2n-4$ both at
${\cal S}_1^C$ and on ${\cal S}_1^C$, i.e., restricted to $T{\cal S}_1^C$.
Thus, since $\omega_C=\pi_L^*\omega_{T^*X}$ as well, the image $\pi_L({\cal
S}_1)\subset T^*X\setminus 0$, which is smooth, conic, nonradial and
codimension 3,
must also be maximally noninvolutive. Finally, we note that, as a general fact
about canonical relations, for all $c_0=(x_0,\xi_0,y_0,\eta_0)\in C$,
the subspace
$d\pi_L(T_{c_0}C)\le T_{(x_0,\xi_0)}\left(T^*X\right)$ is involutive;
in particular,
for $c_0\in{\cal S}_1$, $d\pi_L(T_{c_0}C)$ is a codimension 2,
involutive subspace.
We
are thus led to establishing a (very) weak normal form for cross cap maps into
symplectic manifolds reflecting these extra conditions. Note that
this would apply to
a more general class of maps than cross caps, since at this point we
are only using
information concerning the first derivatives.
\vskip.2in
\begin{proposition}\label{sympl-cc}
Let $g:V\longrightarrow U$ be a smooth conic map, with $V$ a
smooth, conic manifold of dimension $2n-1$ and $(U,\omega_U)$ a conic
symplectic
manifold of dimension $2n$. Assume that $g$ has a cross cap singularity, with
critical set ${\cal S}_1$. Let $v_0\in{\cal S}_1, u_0=g(v_0)$. Assume that
$dg(T_{v_0}V)\le T_{u_0}(U)$ is involutive and $dg\left(T_{v_0}{\cal
S}_1\right)=T_{u_0}\left(g({\cal S}_1)\right)\le T_{u_0}U$ is maximally
noninvolutive for all $v_0\in{\cal S}_1$. Then, there exist local homogeneous
coordinates
\[
(t,\tau)=(t'',t_n,\tau'',\tau''')\in\mathbb R^{n-2}\times\mathbb R\times(\mathbb R^{n-2}
\setminus 0)\times\mathbb R^2
\]
on $V$ near $v_0$, and local canonical coordinates $(x,\xi)\in
T^*\mathbb R^n\setminus 0$ on
$U$ near $u_0:=g(v_0)$, such that $v_0=(0,0,e_1^*,0)$,
$u_0=(0;e_1^*)$, and, writing
$\tau'''=(\tau_{n-1},\tau_n)$,
\begin{equation}\label{ccnf}
g(t,\tau)=\left(t'',g_{n-1}(t,\tau),t_n;\tau'',
\gamma_{n-1}(t,\tau),\gamma_n(t,\tau)\right);
\end{equation}
\begin{equation}\label{gnonz}
\frac{\partial g_{n-1}}{\partial \tau_{n-1}}\ne 0;\quad {\text and}
\end{equation}
\begin{equation}\label{derivz}
\gamma_i|_{\tau'''=0}=\frac{\partial
\gamma_i}{\partial\tau_j}|_{\tau'''=0}=0,\quad
n-1\le i,j\le n.\end{equation}
\end{proposition}
\vskip.2in
Note that, with respect to these coordinates,
${\cal S}_1\subseteq\{\tau_{n-1}=\tau_n=0\}$,
$\Sigma^{2n-3}:=g({\cal S}_1)\subseteq\{x_{n-1}=\xi_{n-1}=\xi_n=0\}$ and
\begin{equation}\label{ker-cc}
Ker\left(\omega_C|_{T{\cal S}_1}\right)=\mathbb R\cdot\left(\frac{\partial}{\partial
\tau_{n-1}}+\dots\right).
\end{equation}
\vskip.2in
\noindent\emph{Proof.} By assumption, $\Sigma^{2n-3}\subset U$ is nonradial,
codimension 3 and maximally nonivolutive. By an application of
Darboux's Theorem,
there exist (\cite[Thm. 21.2.4]{ho-book}) local canonical coordinates
$(x,\xi)\in
T^*\mathbb R^n\setminus 0$ such that
$\Sigma^{2n-3}\subseteq\{x_{n-1}=\xi_{n-1}=\xi_n=0\}$
and
$u_0=(0,e_1^*)$. Letting $t_j=g^*(x_j),\quad \tau_j=g^*(\xi_j)$ for
$1\le j\le n-2$,
and $t_n=g^*(x_n)$, these functions have linearly independent
gradients at $v_0$. If
$\tau_{n-1},\tau_n$ are any two defining functions for ${\cal S}_1$,
homogeneous of degree
1, then $(t,\tau):=(t'',t_n,\tau'',\tau_{n-1},\tau_n)$ are local homogeneous
coordinates on $V$, with ${\cal S}_1\subseteq\{\tau_{n-1}=\tau_n=0\}$. With
respect to these
coordinates,
\[
g(t,\tau)=\left(t'',x_{n-1}(t,\tau),t_n;\tau'',\xi_{n-1}(t,\tau),\xi_n(t,\tau)\right)
\]
and $\textrm{ rank } dg(v) = (2n-3)+\textrm{ rank } \mathbf D(v)$, where
\[
\mathbf D(v)= \left(\begin{array}{cc}
\frac{\partial x_{n-1}}{\partial \tau_{n-1}} &
\frac{\partial x_{n-1}}{\partial \tau_n} \\
{ } & { } \\
\frac{\partial \xi_{n-1}}{\partial \tau_{n-1}} &
\frac{\partial \xi_{n-1}}{\partial \tau_n} \\
{ } & { } \\
\frac{\partial \xi_{n}}{\partial \tau_{n-1}} &
\frac{\partial \xi_{n}}{\partial \tau_n}
\end{array}\right) = \left( D_{n-1},D_n\right),\quad D_{n-1},D_n\in T_uU.
\]
Since $g$ is a cross cap, $\textrm{ rank } \mathbf D(v)=1,\forall v\in{\cal S}_1$, so that by
interchanging $\tau_{n-1}$ and $\tau_n$, if necessary, we can assume
that $D_{n-1}\ne
0$ and $D_n\in\mathbb R\cdot D_{n-1}$ for $v\in{\cal S}_1$ near $v_0$. Furthermore,
rotating in the
$x_{n-1},\xi_{n-1}$ plane, if necessary, we can likewise assume that
$\frac{\partial
x_{n-1}}{\partial \tau_{n-1}}(v_0)\ne 0$ and
$\frac{\partial \xi_{n-1}}{\partial \tau_{n-1}}(v_0)= 0$, so that
\[
\left|\frac{\partial x_{n-1}}{\partial \tau_{n-1}}(v)\right| >>
\left|\frac{\partial \xi_{n-1}}{\partial \tau_{n-1}}(v)\right|,\quad\forall v
{\textrm{ near }} v_0.
\]
Let
\[
\Pi(v)=dg(T_vV)=span\left\{
\{\frac{\partial}{\partial x_j}\}_{j=1}^{n-2},\frac{\partial}{\partial x_n},
\{\frac{\partial}{\partial \xi_j}\}_{j=1}^{n-2}, D_{n-1}
\right\};
\]
by assumption, $\mathbf\Pi(v)\le T_{g(v)}U$ is a codimension 2,
involutive subspace.
We have
\[
D_{n-1}(v)=a(v)\frac{\partial}{\partial x_{n-1}}
+ b(v) \frac{\partial}{\partial \xi_{n-1}}
+ c(v) \frac{\partial}{\partial \xi_{n}},
\]
with $|a|>>|b|$ near $v_0$. A simple calculation shows that a vector
\linebreak$\mathbf X=\sum_{j=1}^n \alpha_j \frac{\partial}{\partial x_{j}}
+ \beta_j \frac{\partial}{\partial \xi_{j}}\in T_{g(v)}U$ is in the symplectic
annihilator
$dg(T_vV)^\omega$ iff
\[
\alpha_j=\beta_j=0,\quad 1\le j\le n-2, \beta_n=0,\quad
a\beta_{n-1}-b\alpha_{n-1}-c\alpha_n=0.
\]
Thus, $\left\{b\frac{\partial}{\partial x_{n-1}} + a\frac{\partial}{\partial
\xi_{n-1}}, c\frac{\partial}{\partial x_{n-1}}+ a
\frac{\partial}{\partial \xi_n}
\right\}$ forms a basis for $dg(T_vV)^\omega$. Since $dg(T_vV)$ is involutive
iff $dg(T_vV)^\omega\le dg(T_vV)$, we must have $b(v)=c(v)=0$. Thus,
$\frac{\partial \xi_i}{\partial \tau_j}|_{\tau'''=0}=0,\quad n-1\le i,i\le n$.
Relabeling, this finishes the proof of Prop.\ref{sympl-cc}. $\square$
\vskip.2in
Now, let $C\subset (T^*X\setminus 0)\times (T^*Y\setminus 0)$ be a
folded cross cap
canonical relation, and apply both Prop.\ref{sympl-swf} to
$\pi_R:C\longrightarrow T^*Y\setminus 0$ and Prop.\ref{sympl-cc} to $\pi_L:C\longrightarrow
T^*X\setminus
0$ near $c_0\in C$. As in \cite{gruh3,gruh4}, we now attempt to
reconcile the two
resulting coordinate systems on $C$, $(s'',s_{n-1},\sigma'',\sigma''')$ and
$(t'',t_n,\tau'',\tau''')$. On $T{\cal S}_1$ (which is the same for both projections),
$\omega_C$ has rank $2n-4$. By (\ref{ker-swf}) and (\ref{ker-cc}), since the
hypersurface $\{\tau_{n-1}=0\}$ is transverse to $\textrm { Ker }(\omega_C)$, it
must be locally
expressible as $\{s_{n-1}=\tilde{F}(s'',\sigma'')\}$ for some smooth
$\tilde{F}$. Now
set
$F(s,\sigma)=-A^{s,\sigma}(\sigma''',\sigma''')\tilde{F}(s'',\sigma'')$;
then
$F=\pi_R^*f$, where $f(y,\eta)=-\eta_{n-1}\tilde{F}(y'',\eta'')$. Using
(\ref{omegc}), we can find a vector field $\mathbf Y$ on $C$ which satisfies
\[
i(\mathbf Y)\omega_C= dF =
-\tilde{F}(s'',\sigma'')\frac{\partial}{\partial s_{n-1}}
+ O(|\sigma'''|^2)\cdot (ds'',d\sigma'').
\]
Thus, $exp(H_f)$ is a canonical transformation of $T^*Y\setminus 0$, while
$exp(\mathbf Y)$ is an $\omega_C$-morphism of $C$, mapping $\{\tau_{n-1}=0,
\tau'''=0\}$ into $\{s_{n-1}=0,\sigma'''=0\}$. Applying these simultaneously on
$T^*Y$ and $C$, respectively, we see that one can assume that
$L:=\{\tau_{n-1}=0,\tau'''=0\}=\{s_{n-1}=0,\sigma'''=0\}$. Restricted to this
$(2n-4)$-dimensional submanifold, $\omega_C$ is symplectic, so by Darboux there
exists a canonical transformation
$\Phi(y'',\eta'')=(\Phi_{y''},\Phi_{\eta''})$ of
$T^*\mathbb R^{n-2}\setminus 0$ such that
\[
\Phi^*( s''|_L)= t''|_L\quad\textrm{ and } \Phi^*(\sigma''|_L)=\tau''|_L.
\]
Extend $\Phi$ to $\tilde{\Phi}:T^*Y\setminus 0\longrightarrow T^*Y\setminus 0$ by
\[
\tilde{\Phi}(y'',y_{n-1};\eta'',\eta_{n-1})=(\Phi_{y''}(y'',\eta''),
y_{n-1};\Phi_{\eta''}(y'',\eta''),\eta_{n-1})
\]
and compose $C$ on the right with the graph of $\tilde{\Phi}$. Then,
\[
\pi_L^*(x'')=t''=s'', \pi_L^*(x_{n-1}),
\pi_L^*(x_n)=t_{n},\pi_R^*(y_{n-1})=s_{n-1}\textrm{ and
}\pi_R^*(\eta'')=\sigma''
\]
form coordinates on $C$ near $c_0$.
\par Thus, we have so far shown that if $C$ is a folded cross cap,
we may assume that
$(x, y_{n-1},
\eta'')$ form (micro)local coordinates on
$C$. Therefore there is a generating function
$S(x, y_{n-1},\theta'')$ for
$C$, where
$S$ is $C^{\infty}$ and homogeneous of degree $1$ in $\theta''$
(\cite[Thm.21.2.18]{ho-book}). Hence, $C$ can be parametrized as
\begin{equation}\label{i}
C=\left\{ (x, d_x S; d_{\theta''} S,
y_{n-1}, -\theta'', -d_{y_{n-1}} S) \right\}
\end{equation}
and the phase function
$\chi(x,y,\theta'')=S(x,y_{n-1},\theta'')-y'' \cdot \theta''$ is a
parametrization for the Lagrangian $C'$. We now show that the
properties of $\pi_L$
and $\pi_R$ impose several conditions on $S$ and its derivatives,
forcing the phase
function to be very similar to the model phase $\phi_0$ considered in \S5.
\par To prepare the canonical relation $C$, we first replicate (\ref{ii}) and
(\ref{iii}).
Since $\pi_R({\cal S}_1^C)$ is a nonradial hypersurface in $T^*Y \setminus
0$, by Darboux's theorem, microlocally there
is a canonical transformation from $T^*Y \setminus 0$ to $T^*\mathbb R^{n-1}
\setminus 0$ taking $\pi_R({\cal S}_1^C)$ to
$\pi_R({\cal S}_1^{C_0})$, given by (\ref{ii}). Similarly,
$\pi_L({\cal S}_1^C)$ is a codimension three submanifold of $T^*X\setminus 0$,
which is maximally noninvolutive in the sense that $\omega_{T^*X}
|_{\pi_L({\cal
S}_1^C)}$ has rank $2n-4$. Thus, there exist defining functions $p_1,
p_2, p_3$ for $\pi_L({\cal S}_1^C)$ with $\{p_1, p_2 \}=1, \{p_1,p_3 \}=0,
\{p_2,p_3 \}=0$. By Darboux's theorem, we can find a
canonical transformation from $T^*X \setminus 0$ to $T^*\mathbb R^n
\setminus 0$ mapping $\pi_L({\cal S}_1^C)$ to $\pi_L({\cal S}_1^{C_0})$
given by (\ref{iii}).
\par Using (\ref{i}), $\pi_R(x, y_{n-1}, \theta'')=(d_\theta'' S, y_{n-1},
-\theta'', -d_{y_{n-1}}
S)$, so (\ref{ii}) implies
\begin{equation}\label{old-one}
-d_{y_{n-1}} S|_{\{x_{n-1}=0=x_n+y_{n-1}\}}=x_n^2\theta_1
\end{equation}
\par From (\ref{i}), $\pi_L (x, y_{n-1}, \theta'')=(x, d_x S)$ and
(\ref{iii}) implies
\begin{equation}\label{old-two}
d_{x_{n-1}} S|_{\{x_{n-1}=0=x_n+y_{n-1}\}} =0
\end{equation}
and
\begin{equation}\label{old-three}
d_{x_n} S |_{\{x_{n-1}=0=x_n+y_{n-1}\}}=-x_n^2 d_{x_1} S |_{\{
x_{n-1}=0=x_n+y_{n-1} \}}
\end{equation}
\par Relation (\ref{old-two}) means that
\begin{equation}\label{old-four}
S(x, y_{n-1}, \theta'')= S_0(x'',\theta'')+x_{n-1}^2
S_1(x, y_{n-1}, \theta'')+ (x_n+ y_{n-1}) S_2(x, y_{n-1}, \theta'').
\end{equation}
\par From (\ref{old-one}) we obtain
\begin{equation}\label{old-five}
S_2(x, y_{n-1}, \theta'')|_{ \{x_{n-1}=0=x_n+y_{n-1}
\} }=-x_n^2\theta_1 ,
\end{equation}
and hence
\begin{equation}\label{old-three}
S_2(x, y_{n-1}, \theta'')= - x_n^2\theta_1 + x_{n-1}
S_3(x, y_{n-1}, \theta'') + (x_n + y_{n-1}) S_4(x, y_{n-1}, \theta'').
\end{equation}
\par Identity (\ref{old-three}) implies that
\begin{equation}\label{old-six}
S_2(x, y_{n-1},
\theta'')|_{\{x_{n-1}=0=x_n+y_{n-1}\}}=- x_n^2 d_{x_1}
S|_{\{x_{n-1}=0=x_n+y_{n-1} \}}.
\end{equation}
\par Now, (\ref{old-five}) and (\ref{old-six}) imply $ d_{x_1}
S|_{\{x_{n-1}=0=x_n+y_{n-1}\}}= \theta_1$, and from (\ref{old-four}) we have
that $d_{x_1} S_0 =\theta_1$, so $S_0=x_1 \theta_1 +
\tilde{S}_0(x''', \theta''')$. At this point, our analysis shows
that the generating
function has the form
\begin{eqnarray}\nonumber
S(x, y_{n-1}, \theta'')= x_1\theta_1&+&\tilde{S}_0 (x''', \theta''')+
x_{n-1}^2 S_1(x, y_{n-1}, \theta'') \\
&+& (x_n+y_{n-1})[-x_n^2\theta_1
+x_{n-1} S_3(x, y_{n-1}, \theta'')\label{old-seven}\\
& { } & { } +( x_n+ y_{n-1}) S_4(x, y_{n-1},
\theta'') ] .\nonumber
\end{eqnarray}
\par Next we consider the differentials $d\pi_R$ and $d\pi_L$. One computes
\[
{d \pi_R}= \left(\begin{array}{ccccc}
d_{x''\theta''}S & d_{x_{n-1}\theta''}S & d_{x_n \theta''}S &
d_{y_{n-1}\theta''}S & d_{\theta''\theta''}S\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & -I_{n-2}\\
-d_{x''y_{n-1}}S & -d_{x_{n-1}y_{n-1}}S & -d_{x_n y_{n-1}}S &
-d_{y_{n-1}y_{n-1}}S & -d_{\theta''y_{n-1}}S
\end{array} \right)
\]
\par Evaluating at ${\cal S}_1^C$, by (\ref{old-seven}), this becomes \\
\[{d \pi_R|_{{\cal S}_1^C}}= \left(\begin{array}{ccccc}
d_{x''\theta''}S_0 & 0 & 0 & 0 & d_{\theta''\theta''}S_0\\
0 & 0 & 0 & 1 & 0\\
0 & 0 & 0 & 0 & -I_{n-2}\\
0 & S_3 & -2x_n\theta_1 + 2S_4 & 2S_4 & 0
\end{array} \right)\]
\par By assumption, $d\pi_R$ has rank $2n-3$ at ${\cal S}_1^C$.
Thus, $S_0(x'',\theta'')$ is
nondegenerate, $S_3|_{ \{x_{n-1}=0=x_n+y_{n-1} \}}$=0 and
$S_4|_{ \{ x_{n-1}=0=x_n+y_{n-1} \}}=x_n \theta_1$. We thus have
\begin{equation}\label{old-eight}
S_3(x, y_{n-1}, \theta'') =x_{n-1}S_5(x, y_{n-1},
\theta'')+(x_n+y_{n-1})S_6(x, y_{n-1}, \theta''),
\end{equation}
and
\begin{equation}\label{old-nine}
S_4(x, y_{n-1}, \theta'')= x_n \theta_1 +
x_{n-1}S_7(x, y_{n-1}, \theta'')+(x_n+y_{n-1})S_8(x, y_{n-1}, \theta'').
\end{equation}
\par Similarly,
\[{d \pi_L|_{{\cal S}_1^C}}= \left(\begin{array}{ccccc}
I_{n-2} & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
d_{x''x''}S & d_{x_{n_1}x''}S & d_{x_nx''}S & d_{y_{n-1}x''}S &
d_{\theta''x''}S \\
d_{x''x_{n-1}}S & d_{x_{n-1}x_{n-1}}S & d_{x_nx_{n-1}}S &
d_{y_{n-1}x_{n-1}}S & d_{\theta''x_{n-1}}S\\
d_{x''x_n}S & d_{x_{n-1}x_n}S & d_{x_nx_n}S & d_{y_{n-1}x_n}S &
d_{\theta''x_n}S
\end{array} \right).\]
\par Using relation (\ref{old-seven}) again, we have
\[{d \pi_L|_{{\cal S}_1^C}}= \left(\begin{array}{ccccc}
I_{n-2} & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 \\
d_{x''x''}S_0& 0 & 0 & 0 & d_{\theta''x''}S_0 \\
0 & 2S_1 & N & S_3 & 0\\
0 & N & 2S_4-2x_n \theta_1 & 2S_4-2x_n\theta_1 & 0
\end{array} \right),\]
where $ N= S_3+\partial_{x_{n-1}}S_4$.
\par By the cross cap condition, the rank of
$d \pi_L|_{\{x_{n-1}=0=x_n+y_{n-1} \}}$ is
$2n-2$. Thus, $S_3|_{\{x_{n-1}=0=x_n+y_{n-1} \}}=0$ and
$S_4|_{\{x_{n-1}=0=x_n+y_{n-1} \}}= x_n \theta_1$, and we
see that relations (\ref{old-eight}) and (\ref{old-nine}) follow
from the analysis of $\pi_L$ as well as
$\pi_R$.
Putting together all
the previous relations, the generating function becomes:
\begin{eqnarray}\label{old-ten}
S(x, y_{n-1}, \theta'')=
x_1\theta_1&+&\tilde{S}_0 (x''', \theta''')+ x_{n-1}^2 S_1(x,
y_{n-1}, \theta'')
\nonumber\\
& +& (x_n+y_{n-1})x_ny_{n-1}\theta_1\nonumber\\
&+&(x_n+y_{n-1})[ x_{n-1}^2 S_5(x, y_{n-1}, \theta'')\\
&{ }& \ \ \ \ \ \ \ \ \ \ +x_{n-1}(x_n+ y_{n-1}) S_6(x, y_{n-1},
\theta'')\nonumber\\
&{ }& \ \ \ \ \ \ \ \ \ \ +(x_n+y_{n-1})^2 S_7(x,y_{n-1}, \theta'')].\nonumber
\end{eqnarray}
\par Since $S_0$ is nondegenerate, $C \cap \{x_{n-1}=x_n=y_{n-1}=0
\}$ is the graph of a canonical transformation on
$T^*R^{n-2}$; we may thus assume that $C \cap
\{x_{n-1}=x_n=y_{n-1}=0 \}=\left\{ (x'', 0, \theta'', 0; x'', 0,
\theta'',0) \right\}$, so that $S_0(x'', \theta'') =x'' \cdot \theta''$,
and (\ref{old-ten}) becomes:
\begin{eqnarray*}
S(x, y_{n-1}, \theta'')=
x''\cdot\theta''&+& x_{n-1}^2 S_1(x, y_{n-1}, \theta'')+
(x_n+y_{n-1})x_ny_{n-1}\theta_1 \\
&+&(x_n+y_{n-1})[ x_{n-1}^2
S_5(x, y_{n-1}, \theta'')\\
&{ }& \ \ \ \ \ \ \ \ \ \ x_{n-1}(x_n+ y_{n-1}) S_6(x,
y_{n-1}, \theta'')\\
&{ }& \ \ \ \ \ \ \ \ \ \ (x_n+y_{n-1})^2 S_7(x,y_{n-1}, \theta'')]
\end{eqnarray*}
\par Letting
\begin{eqnarray*}
M(x,y_{n-1},\theta'')=x_{n-1}^2S_5(x,y_{n-1},\theta'')
+x_{n-1}(x_n+y_{n-1})S_6(x,y_{n-1},\theta'')\\
+(x_n
+ y_{n-1})^2S_7(x,y_{n-1},\theta''),
\end{eqnarray*}
we now see that the Lagrangian $C'$ is parametrized by the phase function
\begin{eqnarray}\label{old-eleven}
\chi(x,y,\theta'')&=&S(x,y_{n-1},\theta'')-y''\cdot\theta''\nonumber\\
&=& (x''-y'')\cdot\theta''+ \left(x_n^2 y_{n-1}
+x_ny_{n-1}^2\right)\theta_1\\
& { } & \ \ \ \ + x_{n-1}^2 S_1(x,y_{n-1}, \theta'' )
+(x_n+y_{n-1})M(x,y_{n-1},\theta'').\nonumber
\end{eqnarray}
\par The right projection becomes:
$$
\pi_R\big(x'', y_{n-1}, \theta'', x_{n-1}, x_n)=(x''+ x_{n-1}^2
\partial_{\theta''}S_1 + (x_n+y_{n-1})\partial_{\theta''}M,
y_{n-1}, \theta'',$$ $$ (x_n^2+2x_n y_{n-1})\theta_1+x_{n-1}^2
\partial_{y_{n-1}}S_1 + M + (x_n+y_{n-1})\partial_{y_{n-1}}M \big)
$$
\par Since $\pi_R$ is a submersion with folds,
$Hess(d\pi_R):Ker(d\pi_R)\longrightarrow Coker(d\pi_R)$,
its Hessian
at ${\cal S}_1^C$, is
nonsingular. At ${\cal S}_1^C$, we have $Ker(d\pi_R)=span\{\frac{\partial}{\partial
x_{n-1}},\frac{\partial}{\partial x_n}\}$ and
$$ Hess(\pi_R)= \left(\begin{array}{cc}
2 \partial_{y_{n-1}}S_1|_{S_1^C} + 2 S_5|_{S_1^C} & 2S_6|_{S_1^C} \\
2S_6|_{S_1^C} & 2 \theta_1 + 6 S_7|_{S_1^C}
\end{array} \right)$$
\par At a fixed point $c_0\in{\cal S}_1^C$, by a rotation in $x_{n-1},x_n$,
we can diagonalize this to obtain $S_6=0$ at $c_0$. Similarly,
$2\theta_1+6S_7$ represents the
$d\xi_n$ component of $Hess(\pi_L)$; by a canonical transformation,
we can assume that this equals $2\theta_1$ at $c_0$, so that $S_7$
vanishes at $c_0$. Thus,
$S_6$ and $S_7$ are small near $c_0$; note
also that, by the nondegeneracy of $Hess(\pi_R)$,
$\partial_{y_{n-1}}S_1 + 2 S_5 \neq 0$ near $c_0$.
\medskip
We are now ready to prove Theorem \ref{main}. Composing $F$ on the
left and right with elliptic FIOs of order $0$ associated with all
of the canonical transformations of $T^*X \setminus 0$ and $T^*Y
\setminus 0$ used above, we can assume that the Schwartz kernel of
$F$ is represented by an oscillatory integral with the phase
function given by (\ref{old-eleven}) and an amplitude $a\in S^{m+\frac12}$.
\par Let $\tilde {\chi}$ be the phase function of $F^*F$:
\begin{eqnarray*}
\tilde {\chi}&=&\chi(z,y,\eta'')-\chi(z,x,\xi'')\\
&=&(z''-y'')\cdot\eta'' +z_n^2y_{n-1}\eta_1+ z_ny_{n-1}^2\eta_1
+ z_{n-1}^2 S_1(z, y_{n-1}, \eta'') \\
& { } & \ \ \ \ +(z_n+y_{n-1})M(z,y_{n-1},\eta'')-(z''-x'')\xi''
-z_n^2x_{n-1}\xi_1 \\
& { } & \ \ \ \ -z_nx_{n-1}^2\xi_1
- z_{n-1} ^2S_1(z, x_{n-1}, \xi'') -(z_n+x_{n-1})M(z,x_{n-1},\xi'').
\end{eqnarray*}
\par We will use stationary phase in $z''$ and $ \eta''$: set
$d_{z''} \tilde{\chi}=0$ and $ d_{\eta''}\tilde{\chi}=0$, where
\begin{eqnarray*}
d_{z''} \tilde{\chi}&=& \eta''-\xi''+(z_n+y_{n-1})\partial_{z''}
M(z, y_{n-1},
\eta'')-(z_n+x_{n-1})\partial_{z''} M(z, x_{n-1}, \xi'')\\
& { } & \ \ \ \ + z_{n-1}^2 (\partial_{z''}S_1(z, y_{n-1}, \eta'')-
\partial_{z''}S_1(z, x_{n-1}, \xi''))\\
d_{\eta_1} \tilde{\chi} &=& z_1-y_1 + z_n^2y_{n-1} + z_n y_{n-1}^2
+(z_n+y_{n-1})\partial_{\eta_1}M(z, y_{n-1},
\eta'') \\
& { } & \ \ \ \ + z_{n-1}^2 \partial_{\eta_1} S_1(z, y_{n-1}, \eta'')\\
d_{\eta_i} \tilde{\chi} &=& z_i-y_i +(z_n+y_{n-1})
\partial_{\eta_i}M(z, y_{n-1}, \eta'')+ z_{n-1}^2
\partial_{\eta_i}S_1(z, y_{n-1},
\eta''), \ \ \ 2 \leq i \leq n-2.
\end{eqnarray*}
\par Notice that $d^2_{z'' \eta''} \tilde{\chi} $ is
nondegenerate. We may solve these equations implicitly for $z''$ and
$ \eta''$ in
terms of the other
variables:
\begin{eqnarray*}
\eta''&=& \xi'' + (z_n+x_{n-1}) \partial_{z''} M(z, x_{n-1},
\xi'')-(z_n+y_{n-1})\partial_{z''}M(z, y_{n-1},\eta'')\\
& { } & \ \ \ \ - z_{n-1}^2 (\partial_{z''}S_1(z, y_{n-1}, \eta'')-
\partial_{z''}S_1(z, x_{n-1},
\xi''))\\
z_i&=&y_i-(z_n+y_{n-1})\partial_{\eta_i}M(z, y_{n-1}, \eta'') -
z_{n-1}^2\partial_{\eta_i}S_1(z, y_{n-1}, \eta'') ,
\ \ \ 2 \leq i \leq n-2;\\
z_1&=&y_1-z_n^2y_{n-1} -z_ny_{n-1}^2-(z_n+y_{n-1})
\partial_{z_1}M(z, y_{n-1},
\eta'')- z_{n-1}^2 \partial_{\eta_1}S_1(z, y_{n-1}, \eta'').
\end{eqnarray*}
\par We have that $\partial_{z''}S_1(z, y_{n-1},
\eta'')- \partial_{z''}S_1(z, x_{n-1}, \xi'')$ vanishes at
$y_{n-1}= x_{n-1}$ and $\eta''=\xi''$, so we can write
\begin{eqnarray*}
\partial_{z''}S_1(z, y_{n-1}, \eta'')- \partial_{z''}S_1(z,
x_{n-1}, \xi'')= (y_{n-1}- x_{n-1}) \partial_{z'' y_{n-1}}S_1(z,
y_{n-1}, \eta'') \\ + (\eta''-\xi'') \partial_{z''
\eta''}S_1(z,y_{n-1}, \eta'')
\end{eqnarray*}
\par In a similar way,
\begin{eqnarray*}
(z_n&+&y_{n-1}) \partial_{z''} M(z, y_{n-1},
\eta'')-(z_n+x_{n-1})\partial_{z''}M(z, x_{n-1},\xi'') \\& = &
(y_{n-1}-x_{n-1})[ \partial_{z''}M(z, y_{n-1}, \eta'') +
(z_n+y_{n-1})\partial_{z'' y_{n-1}}M(z, y_{n-1}, \eta'')] \\ & &+
(\eta''- \xi'')(z_n+ y_{n-1})
\partial_{z'' \eta''}M(z,y_{n-1}, \eta )
\end{eqnarray*}
\par Thus, $\eta''- \xi''$ becomes
\begin{eqnarray*}
\eta''-\xi''&=&-(y_{n-1}-x_{n-1})[
\partial_{z''}M(z, y_{n-1}, \eta'') \\
& &+ (z_n+y_{n-1})\partial_{z''
y_{n-1}}M(z, y_{n-1}, \eta'') + z_{n-1}^2 \partial_{z''
y_{n-1}}S_1(z, y_{n-1}, \eta'')] \\
&\times&[I + (z_n+y_{n-1})
\partial_{z'' \eta''}M(z, y_{n-1}, \eta'')+ z_{n-1}^2 \partial_{z''
y_{n-1}}S_1(z, y_{n-1}, \eta'')]^{-1}
\end{eqnarray*}
\par The phase $\tilde {\chi}$ then becomes:
\begin{eqnarray*}
\tilde{\tilde
{\chi}}&=&(x''-y'')\cdot\xi''+z_n^2(y_{n-1}-x_{n-1})\xi_1+
z_n(y_{n-1}^2-x_{n-1}^2)\xi_1\\
& & +(y_{n-1} -x_{n-1})[z_{n-1}^2
\partial_{y_{n-1}}S_1(\cdot) + M(\cdot)
+ (z_n + y_{n-1}) \partial_{y_{n-1}}M(\cdot)]
\\
&=&(x''-y'')\cdot\xi'' + (y_{n-1}-x_{n-1}) \{ \left(z_n^2 + z_n
(y_{n-1}+x_{n-1})\right)\xi_1\\
& &+ z_{n-1}^2 (\partial_{y_{n-1}}S_1(\cdot) + S_5(\cdot))\\
& & + (z_n +
y_{n-1})[z_{n-1}S_6(\cdot) + (z_n + y_{n-1})S_7(\cdot) +
\partial_{y_{n-1}}M(\cdot) ] \}
\end{eqnarray*}
where $(\cdot)= (y'', z_{n-1}, z_n, y_{n-1}, \xi'')$ and the
amplitude becomes $\tilde{a}\in S^{2m+1}$.\\
\par Let
\begin{eqnarray*}
\ N (\cdot)&=&z_{n-1}^2 (\partial_{y_{n-1}}S_1(\cdot) + S_5(\cdot))\\
\ \ & &+ (z_n + y_{n-1})[z_{n-1}S_6(\cdot) + (z_n + y_{n-1})S_7(\cdot)
+\partial_{y_{n-1}}M(\cdot)]
\end{eqnarray*}
and
\begin{equation*}
\ Q(\cdot)=z_{n-1}S_6(\cdot) + (z_n + y_{n-1})S_7(\cdot) +
\partial_{y_{n-1}}M(\cdot)
\end{equation*}
with
\begin{eqnarray}\label{old-40}
\partial_{y_{n-1}}M&=&z^2_{n-1}\partial_{y_{n-1}}S_5+ z_{n-1}S_6+
z_{n-1}(z_n+y_{n-1})\partial_{y_{n-1}}S_6\nonumber\\
& &+
2(z_n+y_{n-1})S_7+(z_n+y_{n-1})^2\partial_{y_{n-1}}S_7.
\end{eqnarray}
\par Repeating the argument from \S5,
we make a singular change of variables,
\begin{eqnarray*}
\theta_i&=&\xi_i, \ \ \ 1 \leq i \leq n-2,\\
\theta_{n-1}&=& -(z_n^2\xi_1 + z_n (y_{n-1}+x_{n-1})\xi_1 +
N(\cdot)).
\end{eqnarray*}
\par We have
\begin{eqnarray*}
\nabla_{z_{n-1}, z_n} \theta_{n-1}&=&-\big(2z_{n-1} [
\partial_{y_{n-1}}S_1(\cdot)
+ S_5(\cdot)]+ (z_n + y_{n-1}) \partial_{z_{n-1}}Q(\cdot),\\
&{ }&\ \ \ \ 2z_n \theta_1 +(x_{n-1}+y_{n-1})\theta_1+
\partial_{z_n}N(\cdot) \big),
\end{eqnarray*}
so that $|\nabla_z \theta_{n-1}|=0 $ \ \ {\rm iff }
\begin{equation*}
z_{n-1}=- \frac{(z_n+y_{n-1})\partial_{z_{n-1}}Q(\cdot)}{2 (
\partial_{y_{n-1}}S_1(\cdot)
+ S_5(\cdot))}
\end{equation*}
and
\begin{equation*}
z_n=-\frac{x_{n-1}+y_{n-1}}{2}-\frac{\partial_{z_n}N(\cdot)}{2
\theta_1}.
\end{equation*}
At these points,
\begin{eqnarray*}
\theta_{n-1}&=& \frac{(x_{n-1}+y_{n-1})^2}{4} \theta_1- \frac{1}{4
\theta_1 }(\partial_{z_n}N(\cdot))^2\\
& { } & - (z_n+y_{n-1})^2\frac{(\partial_{z_{n-1}}Q)^2}{4
(\partial_{y_{n-1}}S_1(\cdot) +
S_5(\cdot))} - (z_n+y_{n-1}) Q(\cdot)\\
&:=& \frac{(x_{n-1}+y_{n-1})^2}{4} \theta_1 +P(y'', y_{n-1},
x_{n-1}, \theta'').
\end{eqnarray*}
where $P(y'', y_{n-1}, x_{n-1}, \theta'')$ is
\begin{eqnarray}\label{old-43}
\tilde{P}(\cdot):= &-& \frac{1}{4 \theta_1
}(\partial_{z_n}N(\cdot))^2\\ &-&
(z_n+y_{n-1})^2\frac{(\partial_{z_{n-1}}Q)^2}{4
(\partial_{y_{n-1}}S_1(\cdot) + S_5(\cdot))} - (z_n+y_{n-1})
Q(\cdot)\nonumber
\end{eqnarray}
pushed forward under the previous change of variables.
\medskip
\par Letting ${\cal S}_1^{C_0}(z,y)$ denote the critical set in the $z,y$
variables, i.e., for
$C\subset T^*Z\times T^*Y$, note that
$\tilde{P}(\cdot)|_{{\cal S}_1^{C_0}(z,y)}=0$ and $\nabla
\tilde{P}(\cdot)|_{{\cal S}_1^{C_0}(z,y)}=0$, so that
\begin{equation}\label{fgp-1}
P|_{ \{
x_{n-1}=y_{n-1} \}}=0 \hbox{ and }\nabla P|_{ \{ x_{n-1}=y_{n-1} \}
}=0.
\end{equation}
\medskip
\par As in the model case, it follows that $K_{F^*F}(x,y)$ has an
oscillatory integral representation,
\[\int_{\mathbb R^n} e^{i\{(x''-y'')\cdot\theta'' +
(x_{n-1}-y_{n-1})\theta_{n-1} +\frac{\rho}{\theta_1}(\theta_{n-1}-
\frac{(x_{n-1}+y_{n-1})^2}{4} \theta_1 -P)\}} a(x,y, \theta, \rho)
d\theta d \rho,\]
where $a \in S^{2m,-1}(2n-2,n-1,1)$. On the region $\{|\rho|\le c
|\xi|\}$, the new
phase function, $\psi(x,y;\theta;\rho)$ is a multi-phase for a pair
$(\Delta', {\tilde C}')$ in the sense of Def.\ref{old-def3.6}:
$\psi(x,y;\theta;0)$
parametrizes the diagonal Lagrangian $\Delta'$ and $\psi(x,y;\theta;\rho)$
parametrizes a Lagrangian ${\tilde C}'$. Hence, the contribution to
$F^*F$ from this
region is in
$I^{p,l} (\Delta,\tilde C)$ for some
$p,l\in \mathbb R$, and the orders of $F^*F$ are computed using Remark 3.7
in the same way
as for $K^0$ in the model case in \S5, so that $p=2m-\frac12$ and
$l=\frac12$. On the
other hand, the contribution from $\{|\rho|\ge c |\xi|\}$ is handled
in the same way
as for
$K^1$ for the model case in \S5, giving an element of $I^{2m-\frac12}(\tilde
C)\subset I^{2m-\frac12,\frac12}(\Delta,\tilde C)$.
\par Next, we show that ${\tilde C}$ is a two-sided fold; the fact
that $\tilde C$ is
symmetric just follows from $\Delta\cup\tilde C= C^t\circ C$. We have:
\begin{eqnarray*}
\tilde C = \Big\{ \big(&x''&, x_{n-1}, \theta'', \theta_{n-1} -
\frac{x_{n-1}+y_{n-1}}{2} \rho -
\frac{\rho}{\theta_1} \partial_{x_{n-1}} P;\\
&y''&, y_{n-1},
\theta'', \theta_{n-1} + \frac{x_{n-1}+y_{n-1}}{2} \rho +
\frac{\rho}{\theta_1}
\partial_{y_{n-1}} P\big):\\
& &
\theta_{n-1}- \frac{(x_{n-1} + y_{n-1})^2}{4}\theta_{1} - P=0,
x_{n-1}-y_{n-1} + \frac{\rho}{\theta_1} =0,\\
& & x_i-y_i + \frac{\rho}{\theta_1} \partial_{\theta_i} P=0,\ \ 2
\leq i \leq n-2,\\
& & x_1-y_1 -\frac{\rho}{\theta_1^2}(\theta_{n-1}-
\frac{(x_{n-1}+y_{n-1})^2}{4} \theta_1 -P) \\
& &- \frac{\rho}{\theta_1}
\frac{(x_{n-1}+y_{n-1})^2}{4} -\frac{\rho}{\theta_1}
\partial_{\theta_1}P=0 \Big\}.
\end{eqnarray*}
\par The coordinates on $\tilde C$ are $(x'', x_{n-1}, y_{n-1},\theta_1,
\theta''')$. Note that
$\rho=-(x_{n-1}-\nolinebreak y_{n-1})\theta_1$ and $\theta_{n-1}=
\frac{(x_{n-1} + y_{n-1})^2}{4}\theta_{1} + P$; thus, $y_i$ and
$y_1$ become:
$$y_i= x_i-(x_{n-1}-y_{n-1}) \partial_{\theta_i}P, \ \ 2 \leq i
\leq n-2,$$
and
$$y_1=x_1 + \frac{(x_{n-1}-y_{n-1})(x_{n-1}+y_{n-1})^2}{4} +
(x_{n-1}-y_{n-1}) \partial_{\theta_1}P.$$
\par We now consider the projections $\pi_R$ and $\pi_L$.
We have
\begin{eqnarray*}
\pi_R \big(x_1, x''', y_{n-1}, &\theta_1&, \theta''', x_{n-1}\big)\\
&=&\Big(x_1+\frac{(x_{n-1}-y_{n-1})(x_{n-1}+y_{n-1})^2}{4} +
(x_{n-1}+y_{n-1})\partial_{\theta_1}P,\\
&\ \ &\ \ x''' -(x_{n-1}-y_{n-1})
\partial_{\theta'''} P, y_{n-1};\theta_1, \theta''', \\
&\ \ &
\frac{(x_{n-1} + y_{n-1})(3y_{n-1}-x_{n-1})}{4}\theta_{1} + P -
(x_{n-1}-y_{n-1})\partial_{y_{n-1}} P\Big)
\end{eqnarray*}
and
\begin{eqnarray*}
\pi_L\big( &x''& ,\ \ x_{n-1}\ \ ,\ \ \theta_1\ \ ,\ \ \theta'''\
\ ,
\ \ y_{n-1}\ \
\big)\\ &=& \left(x'', x_{n-1};
\theta_1,
\theta''',
\ \ \frac{(x_{n-1} +
y_{n-1})(3x_{n-1}-y_{n-1})}{4}\theta_{1} + P + (x_{n-1}-y_{n-1})
\partial_{x_{n-1}} P \right).
\end{eqnarray*}
\par One easily computes
\[{d \pi_L}= \left(\begin{array}{ccccc}
I_{n-1} & 0 & 0 & 0 & 0 \\
0 & 1 & 0 & 0 & 0\\
0 & 0 & 1 & 0 & 0\\
0 & 0 &0 & I_{n-3} & 0 \\
0 & \cdot & \cdot & \cdot & D
\end{array} \right)\]
where $D= \frac{x_{n-1}-y_{n-1}}{2} \theta_1 +
\partial_{y_{n-1}}P-\partial_{x_{n-1}}P +
(x_{n-1}-y_{n-1})\partial^2_{x_{n-1}y_{n-1}}P$ and
Ker\nolinebreak$\quad d \pi_L$
is spanned by $ \frac{\partial}{\partial y_{n-1}}$.
\par Similarly,
\[{d \pi_R}= \left(\begin{array}{cccccc}
1 & \cdot & \cdot & \cdot & \cdot & \cdot \\
A & B & \cdot & \cdot & \cdot & \cdot \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 0 \\
0 & 0 & 0 & 0 & I_{n-3} & 0 \\
\cdot & \cdot & \cdot & \cdot & \cdot & -D
\end{array} \right)\]
where $A=-(x_{n-1}-y_{n-1})\partial_{\theta''' x_1}^2P$ and
$B=I_{n-3}-(x_{n-1}-y_{n-1}) \partial_{\theta ''' x'''}^2P$.
Hence, det $d \pi_R= D$, as well, and $\textrm { Ker } d\pi_R$ is spanned
by $\frac{\partial\quad}{\partial x_{n-1}}$.
\par We have:
$$\partial_{y_{n-1}}D= -\frac{1}{2} \theta_1 +
\partial^2_{y^2_{n-1}}P - 2 \partial^2_{x_{n-1}y_{n-1}}P
+ (x_{n-1}-y_{n-1})\partial^3_{x_{n-1}y^2_{n-1}}P$$
and $$\partial_{x_{n-1}}D= \frac{1}{2} \theta_1 -
\partial^2_{x^2_{n-1}}P + 2 \partial^2_{x_{n-1}y_{n-1}}P
+ (x_{n-1}-y_{n-1})\partial^3_{x^2_{n-1}y_{n-1}}P$$
Thus, to show that $\pi_R$ and $\pi_L$ are both folds, with
${\cal S}_1^{\tilde{C}}=\{x_{n-1}=y_{n-1}\}=\Delta\cap\tilde{C}$, it
suffices to prove that
\begin{equation}\label{fgp-2}
|\partial^2_{y^2_{n-1}}P|+|
\partial^2_{x_{n-1}y_{n-1}}P|+|\partial^2_{x^2_{n-1}}P|\le\frac18
|\theta_1|
\end{equation}
on a neighborhood of $\tilde{c}_0$;
to do this, we need to show that
$\partial^2_{y^2_{n-1}}\tilde{P},
\partial^2_{x_{n-1}y_{n-1}}\tilde{P}$
and $\partial^2_{x^2_{n-1}}\tilde{P}$ satisfy the
same estimate near the corresponding point.
\par Let $T=\frac{(\partial_{z_{n-1}}Q)^2}{4
(\partial_{y_{n-1}}S_1(\cdot) + S_5(\cdot))} $.
From (\ref{old-43}), we see that
\begin{eqnarray*}
\partial_{y_{n-1}} \tilde{P}&=& -\frac{1}{2\theta_1} \partial_{z_n}N
\partial_{z_n y_{n-1}}N -2(z_n+y_{n-1})(\partial_{y_{n-1}}z_n
+1)T \\
& &- (z_n+y_{n-1})^2
\partial_{y_{n-1}} T-(z_n+y_{n-1})\partial_{y_{n-1}}Q-
(\partial_{y_{n-1}}z_n+1)Q
\end{eqnarray*}
and
\begin{eqnarray*}
\partial_{x_{n-1}} \tilde{P}&=& -\frac{1}{2\theta_1} \partial_{z_n}N
\partial_{z_n x_{n-1}}N -2(z_n+y_{n-1})(\partial_{x_{n-1}}z_n
)T\\
& &- (z_n+y_{n-1})^2
\partial_{x_{n-1}} T-(z_n+y_{n-1})\partial_{x_{n-1}}Q-
(\partial_{x_{n-1}}z_n)Q.
\end{eqnarray*}
Thus
\begin{eqnarray*}
\partial^2_{y^2_{n-1}}\tilde{P}& =&
-\frac{1}{2\theta_1}(\partial^2_{z_ny_{n-1}}N)^2-\frac{1}{2\theta_1}\partial_{z_n}N
\partial^3_{z_ny^2_{n-1}}N \\
& &-2 (\partial_{y_{n-1}}z_n+1)^2 T-2(z_n+y_{n-1})(\partial^2_{y^2_{n-1}}z_n
)T\\
& &- 4(z_n+y_{n-1})(\partial_{y_{n-1}}z_n
+1)\partial_{y_{n-1}}T -(z_n+y_{n-1})^2
\partial^2_{y^2_{n-1}} T\\
& & - 2(\partial_{y_{n-1}}z_n+1)\partial_{y_{n-1}}Q
-(z_n+y_{n-1})\partial^2_{y^2_{n-1}}Q -(\partial^2_{y^2_{n-1}}z_n)Q.
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
\partial^2_{x^2_{n-1}}\tilde{P} &=&
-\frac{1}{2\theta_1}(\partial^2_{z_nx_{n-1}}N)^2-\frac{1}{2\theta_1}\partial_{z_n}N
\partial^3_{z_nx^2_{n-1}}N \\
& &-2 (\partial_{x_{n-1}}z_n)^2 T-2(z_n+y_{n-1})(\partial^2_{x^2_{n-1}}z_n
)T\\
& &- 4(z_n+y_{n-1})(\partial_{x_{n-1}}z_n
)\partial_{x_{n-1}}T -(z_n+y_{n-1})^2
\partial^2_{x^2_{n-1}} T\\
& & - 2(\partial_{x_{n-1}}z_n)\partial_{x_{n-1}}Q
-(z_n+y_{n-1})\partial^2_{x^2_{n-1}}Q -(\partial^2_{x^2_{n-1}}z_n)Q
\end{eqnarray*}
and
\begin{eqnarray*}
\partial^2_{y_{n-1}x_{n-1}}\tilde{P} &=&
-\frac{1}{2\theta_1}(\partial^2_{z_ny_{n-1}}N)
\partial^2_{z_nx_{n-1}}N-\frac{1}{2\theta_1}\partial_{z_n}N
\partial^3_{z_ny_{n-1}x_{n-1}}N \\
& &-2 (\partial_{x_{n-1}}z_n)(\partial_{y_{n-1}}z_n+1) T-
2(z_n+y_{n-1})(\partial^2_{x_{n-1}y_{n-1}}z_n )T\\
& &-
2(z_n+y_{n-1})(\partial_{y_{n-1}}z_n
+1)\partial_{x_{n-1}}T - 2(z_n+y_{n-1})(\partial_{x_{n-1}}z_n
)\partial_{y_{n-1}}T\\
& &-(z_n+y_{n-1})^2
\partial^2_{y_{n-1}x_{n-1}} T - (\partial_{x_{n-1}}z_n)
\partial_{y_{n-1}}Q\\
& &-(z_n+y_{n-1})\partial^2_{y_{n-1}x_{n-1}}Q
-(\partial^2_{y_{n-1}x_{n-1}}z_n) Q -
(\partial_{y_{n-1}}z_n+1)\partial_{x_{n-1}}Q.
\end{eqnarray*}
\par We have
\begin{eqnarray*}
\partial_{z_n}N&=&z_{n-1}^2(\partial^2_{z_ny_{n-1}}S_1 + \partial_{z_n}S_5)\\
& &+ (z_n+y_{n-1})
[z_{n-1}\partial_{z_n}S_6 + S_7 + (z_n+y_{n-1})\partial_{z_n}S_7 +
\partial^2_{y_{n-1}z_n}M] + Q.
\end{eqnarray*}
from which it follows that
\begin{eqnarray}\label{old-49}
\partial^2_{z_n y_{n-1}}N&=& 2
z_{n-1}\partial_{y_{n-1}}z_{n-1}(\partial^2_{z_ny_{n-1}}S_1 +
\partial_{z_n}S_5)+ z^2_{n-1}\partial_{y_{n-1}}(\partial^2_{z_ny_{n-1}}S_1 +
\partial_{z_n}S_5)\nonumber\\
& &+ (\partial_{y_{n-1}}z_n+1)[z_{n-1}\partial_{z_n}S_6 + S_7 +
(z_n+y_{n-1})\partial_{z_n}S_7 +
\partial^2_{y_{n-1}z_n}M] \nonumber\\
& &+ (z_n+y_{n-1})\partial_{y_{n-1}}[z_{n-1}\partial_{z_n}S_6 + S_7 +
(z_n+y_{n-1})\partial_{z_n}S_7 +
\partial^2_{y_{n-1}z_n}M]\nonumber\\
& & +\partial_{y_{n-1}}Q
\end{eqnarray}
and
\begin{eqnarray}\label{old-50}
\partial^2_{z_n x_{n-1}}N&=& 2
z_{n-1}\partial_{x_{n-1}}z_{n-1}(\partial^2_{z_ny_{n-1}}S_1 +
\partial_{z_n}S_5)+ z^2_{n-1}\partial_{x_{n-1}}(\partial^2_{z_ny_{n-1}}S_1 +
\partial_{z_n}S_5)\nonumber\\
& &+ (\partial_{x_{n-1}}z_n)[z_{n-1}\partial_{z_n}S_6 + S_7 +
(z_n+y_{n-1})\partial_{z_n}S_7 +
\partial^2_{y_{n-1}z_n}M] \nonumber\\
& &+ (z_n+y_{n-1})\partial_{x_{n-1}}[z_{n-1}\partial_{z_n}S_6 + S_7\nonumber\\
& & + (z_n+y_{n-1})\partial_{z_n}S_7 +
\partial^2_{y_{n-1}z_n}M] +\partial_{x_{n-1}}Q.
\end{eqnarray}
Similarly,
\begin{eqnarray}\label{old-51}
\partial_{y_{n-1}}Q&=&\partial_{y_{n-1}}z_{n-1}S_6+
z_{n-1}\partial_{y_{n-1}}S_6+
(\partial_{y_{n-1}}z_n+1)S_7\nonumber\\
& &+(z_n+y_{n-1})\partial_{y_{n-1}}S_7+\partial^2_{y^2_{n-1}}M
\end{eqnarray}
and
\begin{eqnarray}\label{old-52}
\partial_{x_{n-1}}Q&=&\partial_{x_{n-1}}z_{n-1}S_6+
z_{n-1}\partial_{x_{n-1}}S_6\nonumber\\
& &+
(\partial_{x_{n-1}}z_n)S_7+(z_n+y_{n-1})\partial_{x_{n-1}}S_7+\partial^2_{y_{n-1}x_{n-1}}M.
\end{eqnarray}
\vskip.3in
Using (\ref{old-40}), we have
\begin{eqnarray}\label{old-53}
\partial^2_{y^2_{n-1}}M&=&2z_{n-1}(\partial_{y_{n-1}}z_{n-1})
\partial_{y_{n-1}}S_5+z^2_{n-1}\partial^2_{y^2_{n-1}}S_5+
\partial_{y_{n-1}}z_{n-1}S_6 \nonumber\\
& &+
z_{n-1}\partial_{y_{n-1}}S_6+\partial_{y_{n-1}}z_{n-1}(z_n+y_{n-1})
\partial_{y_{n-1}}S_6 \nonumber\\
& &+z_{n-1}(\partial_{y_{n-1}}z_n+1)
\partial_{y_{n-1}}S_6+z_{n-1}(z_n+y_{n-1})\partial^2_{y^2_{n-1}}S_6
\nonumber\\
& &+
2(\partial_{y_{n-1}}z_n+1)S_7+2(z_n+y_{n-1})\partial_{y_{n-1}}S_7\\
& &
+2(z_n+y_{n-1})(\partial_{y_{n-1}z_n}+1)\partial_{y_{n-1}}S_7
+(z_n+y_{n-1})^2\partial^2_{y^2_{n-1}}S_7 \nonumber
\end{eqnarray}
and
\begin{eqnarray}\label{old-54}
\partial^2_{y_{n-1}x_{n-1}}M&=&2z_{n-1}(\partial_{x_{n-1}}z_{n-1})
\partial_{y_{n-1}}S_5+z^2_{n-1}\partial^2_{y_{n-1}x_{n-1}}S_5+
\partial_{x_{n-1}}z_{n-1}S_6 \nonumber\\
& &+
z_{n-1}\partial_{x_{n-1}}S_6+\partial_{x_{n-1}}z_{n-1}(z_n+y_{n-1})
\partial_{y_{n-1}}S_6
\nonumber\\
& &+z_{n-1}(\partial_{x_{n-1}}z_n)
\partial_{y_{n-1}}S_6+z_{n-1}(z_n+y_{n-1})\partial^2_{y_{n-1}x_{n-1}}S_6
\nonumber\\
& &+2(\partial_{x_{n-1}}z_n)S_7+2(z_n+y_{n-1})\partial_{x_{n-1}}S_7
\\
& &+2(z_n+y_{n-1})(\partial_{x_{n-1}}z_n)\partial_{y_{n-1}}S_7
+(z_n+y_{n-1})^2\partial^2_{y_{n-1}x_{n-1}}S_7.\nonumber
\end{eqnarray}
Similarly,
\begin{eqnarray}\label{old-55}
\partial^2_{y_{n-1}z_n}M&=&z_{n-1}^2\partial_{y_{n-1}z_n}S_5+z_{n-1}\partial_{z_n}S_6
+
z_{n-1}\partial^2_{y_{n-1}}S_6
\nonumber\\
& &+z_{n-1}(z_n+y_{n-1})\partial^2_{y_{n-1}z_n}S_6+
2S_7+2(z_n+y_{n-1})\partial_{z
_n}S_7
\nonumber\\
&
&+2(z_n+y_{n-1})\partial_{y_{n-1}}S_7+(z_n+y_{n-1})^2\partial^2_{y_{n-1}z_n}S_7.
\end{eqnarray}
\par Using (\ref{old-49}) - (\ref{old-55}) and the fact that
$\partial_{z_n}N, Q, \partial_{y_{n-1}}M,
S_6$ and $S_7$ are all small near $c_0$, we obtain that
$\partial^2_{y_{n-1}x_{n-1}}M, \ \ \partial^2_{y^2_{n-1}}M,
\ \ \partial^2_{y_{n-1}z_n}M, \ \ \partial_{x_{n-1}}Q$,
$\partial_{y_{n-1}}Q, \ \ \partial^2_{z_ny_{n-1}}N$ and
$ \ \ \ \ \ \ \partial^2_{z_nx_{n-1}}N$ are as well. Thus,
$\partial^2_{y^2_{n-1}}\tilde{P},
\partial^2_{x^2_{n-1}}\tilde{P}$ and
$\partial^2_{x_{n-1}y_{n-1}}\tilde{P}$ are also small, from which
it follows that (\ref{fgp-2}) holds, and thus $\tilde{C}$ is a
symmetric, two-sided fold, with
$\Delta\cap\tilde{C}={\cal S}_1^{\tilde C}$.
\begin{remark}
{\rm An interesting question, particularly relevant for the marine
seismic imaging
problem, is whether (under an ellipticity assumption on $F$), } $F^*F$
{\rm can be inverted, at least modulo operators mapping} $H^s
\rightarrow H^{s-2m + \delta}$ {\rm for some} $\delta >0$. {\rm
Note that, by Remark \ref{old-rem3.4}}, {\rm the order of} $F^*F$ {\rm on}
$\tilde{C} \setminus \Delta$ {\rm is } $\frac{1}{2}$ {\rm
less than on} $\Delta \setminus \tilde{C} $. {\rm However, the
standard technique of parabolic decomposition (see \cite{gruh2})
only results in a decomposition} $F^*F= T_1 +T_2$ {\rm with} $T_1
\in I_{\frac{1}{2}, \frac{1}{2}}^{2m} (\Delta)$ {\rm and} $T_2 \in
I_{\frac{1}{2}, \frac{1}{2}}^{2m} (\tilde{C})$ {\rm .
Since} $\tilde{C}$ {\rm is a two-sided fold, there is further loss
of} $\frac{1}{6}$ {\rm derivatives in terms of Sobolev mapping
properties \cite{meta}. We will return to this question in \cite{fgp}.}
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,014 |
Chris Brown Welcomes Second Child With Ammika Harris
by Singersroom November 21, 2019
Looks like Grammy-winning R&B artist Chris Brown is a dad, again!
Brown and ex-girlfriend Ammika Harris welcomed a son on Wednesday (Nov. 20), which they seemly confirmed via social media.
The "Loyal" hitmaker shared a close-up black-and-white photo of himself gazing down at something adoringly with the caption, "11-20-2019," which lead followers to believe he was referring to the birth date of his new bundle of joy. In a second post, Brown is photographed from the side with a hoodie pulled up over his head to display the word "BORN" embroidered on the top.
Meanwhile, model Harris took to her Instagram Story timeline on Thursday with a gushing note: "I was in love, when I first saw you," she wrote, alongside a red heart emoji.
This is the first child for Harris, who will be a half-brother to Brown's 5-year-old daughter Royalty.
In June, Page Six revealed that the two were expecting their first child. Fans started speculating in May after Brown left comments such as "BM BAD" on Harris' Instagram photos with heart and kissy face emojis.
Brown shares his little girl with another ex, Nia Guzman, who fell pregnant while Brown was in an on/off relationship with model-turned-actress Karrueche Tran.
Taggedchris brown
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Omarion and Bow Wow to Launch 'Millennium Tour 2020' W/ Pretty Ricky, Lloyd, Sammie, More | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,168 |
{"url":"https:\/\/www.gradesaver.com\/textbooks\/science\/chemistry\/chemistry-a-molecular-approach-3rd-edition\/chapter-3-sections-3-1-3-12-exercises-problems-by-topic-page-131\/53a","text":"## Chemistry: A Molecular Approach (3rd Edition)\n\nHydrofluoric acid is $HF$\nBinary acids are named as : $hydro + 'base name' + ic + acid.$ Here, $fluor$ is the basename of fluorine ($F$). Hence, the formula of hydrofluoric acid is $HF$.","date":"2018-09-19 20:26:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9297279715538025, \"perplexity\": 5419.520544040882}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-39\/segments\/1537267156305.13\/warc\/CC-MAIN-20180919200547-20180919220547-00246.warc.gz\"}"} | null | null |
El es la lista de los sencillos digitales más vendidos proporcionada por la Recording Industry Association of Japan desde abril de 2009. La lista de la semana corre del miércoles al martes y es actualizada todos los viernes a las 11 a.m. (JST). La primera canción número uno en la lista fue "It's All Love!" por Kumi Kōda y Misono.
Canciones con más número uno
4 semanas
Greeeen - "Haruka"
3 semanas
Hilcrhyme - "Daijōbu"
Hilcrhyme - "Shunkashūtō"
Juju with Jay'ed - "Ashita ga Kuru Nara"
Infinity 16 welcomez Waka-danna from Shōnan no Kaze & Jay'ed - "Tsutaetai Koto ga Konna Aru noni"
Kana Nishino - "Best Friend"
Kana Nishino - "Kimi tte"
2 semanas
Exile - "Yasashii Hikari"
Masaharu Fukuyama - "Hatsukoi"
Ayumi Hamasaki - "Sunrise (Love Is All)"
Hilcrhyme - "Loose Leaf"
Kaela Kimura - "Butterfly"
Mika Nakashima - "Always"
Kana Nishino - "Aitakute Aitakute"
Kana Nishino - "Motto..."
Fuyumi Sakamoto - "Mata Kimi ni Koi Shiteru"
AKB48 - "Heavy Rotation"
Véase también
Anexo:Lista de artistas que alcanzaron el número uno en el RIAJ Digital Track Chart
Anexo:Sencillos digitales número uno de 2009 (Japón)
Anexo:Sencillos digitales número uno de 2010 (Japón)
Anexo:Sencillos digitales número uno de 2011 (Japón)
Referencias
Enlaces externos
RIAJ Digital Track chart (Japonés)
Listas musicales de Japón | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,768 |
{"url":"https:\/\/www.realmeye.com\/wiki\/centipede-poison","text":"# Centipede Poison\n\n Deadly poison of the swamp centipede.\n\nTier 0\nMP Cost 25 (4.8 damage\/MP)\nDamage Boost ${\\begin{cases}&space;5&space;&&space;\\text({wis}<50)\\\\&space;5+0.05(wis-50)&&space;({wis}\\geq50)\\\\&space;\\end{cases}&space;}$ \u00a0%\nThrow Time 0.8 second(s)\nImpact Damage 20\nDamage Over Time ${\\begin{cases}&space;100&space;&&space;\\text({wis}<50)\\\\&space;100+2(wis-50)&&space;({wis}\\geq50)\\\\&space;\\end{cases}&space;}$ \u00a0damage\nDuration 3 second(s)\nRadius ${\\begin{cases}&space;2&space;&&space;\\text({wis}<50)\\\\&space;2+0.025(wis-50)&&space;({wis}\\geq50)\\\\&space;\\end{cases}&space;}$ \u00a0 tiles\nFeed Power 5\n\nObtained Through Creating a new Assassin character\n\nNotes\nThis is a starter ability for the Assassin class. It does not drop from enemies.\n\nThis poison is very commonly used as a swapout used in place of other poisons because it has a very low MP cost, resulting in its user being able to be spam the item. It is perfect for players who are inaccurate and don\u2019t own a Plague Poison, because if spammed, there\u2019s a high likelihood of some poison projectile making contact with the enemy.\n\nPoisons\n T0. Centipede Poison\nLimited Edition Poisons\nTier 0 Abilities\n Centipede Poison","date":"2022-11-28 08:18:12","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 3, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2637450397014618, \"perplexity\": 11436.00403363792}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-49\/segments\/1669446710488.2\/warc\/CC-MAIN-20221128070816-20221128100816-00522.warc.gz\"}"} | null | null |
namespace native
{
namespace ui
{
class MenuAdapter : public IMenuAdapter
{
public:
MenuAdapter();
MenuAdapter(handle_t handle);
~MenuAdapter();
virtual void insert(size_t index, Action& action) override;
virtual void insertSeparator(size_t index) override;
virtual void insert(size_t index, Menu& menu) override;
virtual void remove(Menu& menu) override;
virtual void remove(Action& action) override;
virtual void update(Action& action) override;
virtual void update(Menu& menu) override;
virtual handle_t getHandle() const override { return _handle; }
private:
// Instance Variables
handle_t _handle;
};
}
}
#endif // _NATIVE_UI_MENU_ADAPTER_H_
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,754 |
Rosie was found as a stray duckling in Milwaukee, and fostered by one of Heartland's staff for a month before coming to the barn.
Loves people and has a sweet, gentle personality, but isn't afraid to keep the other ducks in line!
Sitting on a warm human lap, swimming and eating lettuce in her pool. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,039 |
Grace and George have again taken the spoils at Essex Junior Squash's grand prix event. Held on 25th November at Connaught Squash Club the pair travelled to take on the opposition.
George playing up in an older age group of under year 8 school children, children as old as 13 as he is too strong for his own age group now. George confidently took all his matches except for the last in which he took one game against the overall winner who is 13 years old, George at just 10 to take eventual silver medal position.
Grace played in the year 4/6 group but as the only year 4 she was playing against girls aged up to 11 years old. Grace even managed to take a game from Girl number one from Tiptree's team in the Lexden Cup. Looking like a promising prospect for Mersea in the future Grace today took another gold medal win for Girls under year 4 section. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,170 |
{"url":"https:\/\/www.ajpmonline.org\/article\/S0749-3797(21)00059-3\/fulltext","text":"Research Article| Volume 60, ISSUE 5, P614-620, May 01, 2021\n\nSubway Ridership, Crowding, or Population Density: Determinants of COVID-19 Infection Rates in New York City\n\nOpen AccessPublished:January 25, 2021\n\nIntroduction\n\nThis study aims to determine whether subway ridership and built environmental factors, such as population density and points of interests, are linked to the per capita COVID-19 infection rate in New York City ZIP codes, after controlling for racial and socioeconomic characteristics.\n\nMethods\n\nSpatial lag models were employed to model the cumulative COVID-19 per capita infection rate in New York City ZIP codes (N=177) as of April 1 and May 25, 2020, accounting for the spatial relationships among observations. Both direct and total effects (through spatial relationships) were reported.\n\nResults\n\nThis study distinguished between density and crowding. Crowding (and not density) was associated with the higher infection rate on April 1. Average household size was another significant crowding-related variable in both models. There was no evidence that subway ridership was related to the COVID-19 infection rate. Racial and socioeconomic compositions were among the most significant predictors of spatial variation in COVID-19 per capita infection rates in New York City, even more so than variables such as point-of-interest rates, density, and nursing home bed rates.\n\nConclusions\n\nPoint-of-interest destinations not only could facilitate the spread of virus to other parts of the city (through indirect effects) but also were significantly associated with the higher infection rate in their immediate neighborhoods during the early stages of the pandemic. Policymakers should pay particularly close attention to neighborhoods with a high proportion of crowded households and these destinations during the early stages of pandemics.\n\nINTRODUCTION\n\nNew York City (NYC) has been particularly hit hard by coronavirus disease 2019 (COVID-19). As of May 25, about a quarter of total COVID-19 deaths in the U.S. occurred in NYC. Research efforts to investigate the determinant factors of the COVID-19 outbreak in NYC mostly focused on socioeconomic factors and reported significant associations between socioeconomic and racial variations and the COVID-19 per capita infection rate, COVID-19 testing rates, and proportion of positive tests.\n\nBorjas GJ. Demographic determinants of testing incidence and COVID-19 infections in New York City neighborhoods. HKS Working Paper No. RWP20-008. SSRN. Online April 10, 2020. https:\/\/doi.org\/10.2139\/ssrn.3572329.\n\n\u2022 Credit K\nNeighborhood inequity: exploring the factors underlying racial and ethnic disparities in COVID-19 testing and infection rates using ZIP code data in Chicago and New York.\n\u2022 Lieberman-Cribbin W\n\u2022 Tuminello S\n\u2022 Flores RM\n\u2022 Taioli E\nDisparities in COVID-19 testing and positivity in New York City.\n\u2022 Almagro M\n\u2022 Orane-Hutchinson A\nJUE insight: the determinants of the differential exposure to COVID-19 in New York City and their evolution over time.\nHowever, there is very little empirical evidence on the effects of subway ridership on COVID-19. The only existing evidence is a non\u2013peer reviewed working paper released by the National Bureau of Economic Research entitled \u201cThe Subways Seeded the Massive Coronavirus Epidemic in New York City.\u201d Without any statistical analysis and largely based on observational data, the study argued that the New York subway system was a major disseminator and likely served as the transmission vehicle for the spread of the COVID-19 pandemic, particularly in early days during the first 2 weeks of March. The study concluded that ZIP codes that are located along the subway lines had a higher number of confirmed cases than ZIP codes that were not served by subway.\n\nHarris JE. The subways seeded the massive coronavirus epidemic in New York City. NBER working paper 27021. Cambridge, MA: National Bureau of Economic Research.https:\/\/doi.org\/10.3386\/w27021. Revised August 2020. Accessed March 31, 2021.\n\nIn the absence of data and statistical analysis, claims in this paper have fueled political debates on conservative media outlets and among policymakers. In NYC, 4 council members cited this paper in their letter to New York Governor Cuomo demanding the complete shutdown of the New York subway system. The petition was largely pushed back by the Metropolitan Transit Authority, emphasizing the critical role of public transit in providing mobility for the frontline essential workers during the pandemic.\n\nBliss L. The New York subway got caught in the coronavirus culture war. Bloomberg CityLab. April 21, 2020.https:\/\/www.bloomberg.com\/news\/articles\/2020-04-21\/the-tenuous-link-between-the-subway-and-covid-19. Accessed September 10, 2020.\n\n,\n\nSadik-Khan J, Solomonow S. Fear of public transit got ahead of the evidence. The Atlantic. June 14, 2020.https:\/\/www.theatlantic.com\/ideas\/archive\/2020\/06\/fear-transit-bad-cities\/612979\/. Accessed September 10, 2020.\n\nIn addition, there is very little evidence on the relationship between population density and crowding and spatial variations in COVID-19 infection rates at the ZIP code level in NYC. The effects of population density on COVID-19 have been at the center of attention; however, population density is distinct from crowding, which is defined as a large number of people gathered closely together. Crowding could happen in bars, restaurants, sport events, and any other destination that could attract visitors; in other words, points of interest (POIs).\n\u2022 Hamidi S\n\u2022 Sabouri S\n\u2022 Ewing R\nDoes density aggravate the COVID-19 pandemic? Early findings and lessons for planners.\n,\n\u2022 Hamidi S\n\u2022 Ewing R\n\u2022 Sabouri S\nLongitudinal analyses of the relationship between development density and the COVID-19 morbidity and mortality rates: early evidence from 1,165 metropolitan counties in the United States.\nThe Pearson correlation coefficient between population density and POIs per 1,000 population in NYC ZIP codes is <0.052, which also confirms the distinction between the 2 measures. Very little is known about the relationship between different types of crowding venues at the neighborhood level and the COVID-19 infection rate.\nAnother factor that has been largely missed by existing studies is the extent to which NYC neighborhoods have been emptying out to escape the pandemic. According to the New York Times, as of May 1, in many neighborhoods in Manhattan, between 30% and 50% of residents were gone.\n\nQuealy K. The richest neighborhoods emptied out most as coronavirus hit New York City. The New York Times. May 15, 2020. https:\/\/www.nytimes.com\/interactive\/2020\/05\/15\/upshot\/who-left-new-york-coronavirus.html. Accessed September 10, 2020.\n\nIt is impossible to contract the virus in NYC if a person is not physically living there. Similarly, nursing homes facilities have been major COVID-19 hotspots in NYC and other parts of the country. In the State of New York, nursing home facilities accounted for >20% of all COVID-19 death cases.\n\nThe New York Times. More than one-third of U.S. coronavirus deaths are linked to nursing homes. The New York Times. Updated November 20, 2020. https:\/\/www.nytimes.com\/interactive\/2020\/us\/coronavirus-nursing-homes.html. Accessed September 10, 2020.\n\nThis study is the first to conceptualize and integrate 3 dimensions of crowding, including households, businesses, and subways, in a comprehensive framework. The major aim of this study is to investigate the relationship among these 3 crowding variables, population density, and other confounding factors and the COVID-19 (per capita) infection rate during the early stages (as of April 1) and after the epidemic curve was flattened (as of May 25) at the ZIP code level in NYC. Spatial autoregressive modeling techniques were employed to control for the spatial dependency of observations (ZIP codes) in the sample. The authors hypothesize that, during the early stages, crowding-related factors such as POIs and crowded housing explain the spatial distributions of infection rates, whereas on May 25, racial and socioeconomic characteristics had the strongest relationship with the per capita infection rate.\n\nMETHODS\n\nStudy Sample\n\nThe sample in this study consisted of 177 ZIP Code Tabulation Areas (ZCTAs) in 5 boroughs of NYC. Data on the cumulative number of COVID-19 tests and the cumulative number of confirmed cases were downloaded from the NYC Department of Health from March 2, 2020 through April 1 and May 25, 2020.\n\nNew York City Department of Health. Confirmed and probable COVID-19 deaths. New York, NY: New York City Department of Health.https:\/\/www1.nyc.gov\/site\/doh\/covid\/covid-19-data-archive.page. Accessed January 8, 2021.\n\n,\n\nBuchanan L, Patel JK, Rosenthal BM, Singhvi A. A month of coronavirus in New York City: see the hardest-hit areas. The New York Times. April 1, 2020. https:\/\/www.nytimes.com\/interactive\/2020\/04\/01\/nyregion\/nyc-coronavirus-cases-map.html. Accessed on September 10, 2020.\n\nThe outcome variables were the\u00a0cumulative COVID-19\u00a0per capita infection rates at 2 points in time to account for the different nature of the pandemic spread at early stages (April 1) and after the epidemic curve was flattened (from March 2 to May 25). The 2 outcome variables were mapped using quantile categorization in ArcMap, version 10.7.1. In addition, hotspot analyses were performed using the Getis\u2013Ord method to identify clusters of ZCTAs with a high concentration of infection rates (hotspots) and a low concentration of infection rates (coldspots) in NYC (Figure 1).\n\nMeasures\n\nThe independent variable of greatest interest is subway ridership. Raw data on transit ridership were obtained from the Metropolitan Transit Authority.\n\nMetropolitan Transit Authority. Turnstile data. http:\/\/web.mta.info\/developers\/turnstile.html. Accessed July 6, 2020.\n\nThe Metropolitan Transit Authority releases\u00a0daily subway ridership data based on entrees and exits for each turnstile by station, which were downloaded and cleaned to compute 3 ridership variables. The first ridership variable represented the prepandemic baseline ridership and was computed as the average weekday ridership in the last week of February, before the first COVID-19 case was confirmed in NYC on February 29. The second and third ridership variables represented the percentage changes in subway ridership relative to the baseline during 2-week time periods before the confirmed positive cases in each model (April 1 and May 25). These 2 variables were estimated backward from observed confirmed cases to estimate transmission that occurred several weeks previously, allowing for the time lag between infection and positive COVID-19 test.\n\u2022 Flaxman S\n\u2022 Mishra S\n\u2022 Gandy A\n\u2022 et al.\nEstimating the effects of non-pharmaceutical interventions on COVID-19 in Europe.\nThis analysis also accounted for the number of POIs within each ZCTA in NYC, utilizing data from SafeGraph.\n\nSafeGraph. Places schema.https:\/\/docs.safegraph.com\/docs. Accessed July 6, 2020.\n\nThe SafeGraph database measures foot traffic patterns to POIs based on GPS data from >45 million smartphones in the U.S. POIs include restaurants, cafes, retail shops, movie theaters, parks, and other public places that could attract visitors. Initially, 2 sets of POI variables representing the level of crowding at the baseline and in March were computed for each ZCTA. However, checking the face validity of these variables\n\u2022 Duke C\n\u2022 Hamidi S\n\u2022 Ewing R\nValidity and reliability.\nvia ArcMap and Google Maps showed that the most reliable and accurate variable was the number of POIs in each ZCTA per 1,000 population, which was computed and used as a proxy for business crowding in this study.\nIn addition, analyses controlled for the percentage of residents in each ZCTA who left NYC to escape the pandemic in March and April. The data were\u00a0borrowed from the New York Times based on aggregated smartphone location data from Descartes Lab and measured the proportion of population who lived in NYC during the last 2 weeks of February but were not living there on May 1.\n\nQuealy K. The richest neighborhoods emptied out most as coronavirus hit New York City. The New York Times. May 15, 2020. https:\/\/www.nytimes.com\/interactive\/2020\/05\/15\/upshot\/who-left-new-york-coronavirus.html. Accessed September 10, 2020.\n\nThe population-weighted average of Census tracts was calculated to obtain the ZCTA-level variable.\nEmploying the same methodology as Yost et al.,\n\u2022 Yost K\n\u2022 Perkins C\n\u2022 Cohen R\n\u2022 Morris C\n\u2022 Wright W\nSocioeconomic status and breast cancer incidence in California for different race\/ethnic groups.\nan SES index was developed for each ZCTA based on the following variables from the 2018 American Community Survey (5-year estimates)\nAmerican Community Survey\nAmerican Community Survey 5-Year Estimates. U.S. Census Bureau.\n: median household income in the past 12 months; median gross rent; median home value; percentage unemployed (aged \u226516 years); percentage working class (aged \u226516 years); percentage living <150% of poverty line; and education index, which is a weighted combination of the percentage below high school education, high school graduates, and more than high school degrees (adults aged \u226525 years). Higher value of education index represents higher educational attainments. Using principal component analysis, these variables were combined into 1 score for each ZCTA with an eigenvalue of 5.2, which explains 74.8% of the variance following this equation:\n$(medianhomevalue\u00d70.141)+(mediangrossrent\u00d70.17)+(medianhouseholdincome\u00d70.181)+(percentagebelowpoverty\u00d7\u22120.161)+(percentageunemployed\u00d7\u22120.147)+(percentageworkingclass\u00d7\u22120.174)+(educationscore\u00d70.177).$\n\nThe score was standardized to have a mean of 100 and an SD of 25.\nMeasures of racial composition characteristics, including percentage Black, percentage Hispanic, and average household size, were computed based on data from the 2018 American Community Survey (5-year estimates).\nAmerican Community Survey\nAmerican Community Survey 5-Year Estimates. U.S. Census Bureau.\nIn addition, population density was computed by dividing the ZCTA's total population by the land area in square miles. Finally, using ArcMap, the number of beds in nursing homes and assisted living facilities for each ZCTA was calculated based on data from the Homeland Infrastructure Foundation-level Data\n\nU.S. Department of Homeland Security. Homeland Infrastructure Foundation-Level Data (HIFLD). https:\/\/hifld-geoplatform.opendata.arcgis.com\/. Accessed January 8, 2021.\n\nand was converted to a per capita rate variable by dividing the number of beds in each ZCTA by ZCTA population. Pearson correlation coefficients between explanatory variables are presented in Appendix Table 1, available online.\n\nStatistical Analysis\n\nThe nature of virus spread is a spatialized phenomenon, meaning that the per capita rate of infection rate in a ZCTA is not independent of the infection rate in surrounding ZCTAs. People move beyond the boundary of ZIP codes and so does the virus. The spatial relationship between ZCTAs violates the assumption of ordinary least squares, which requires the unexplained error term to be randomly distributed across observations.\n\u2022 Anselin L\nSpatial Econometrics: Methods and Models.\nThis was also confirmed with Moran's I analysis of ordinary least squares regression residuals with a coefficient value of 0.38, which was statistically significant at <0.001 level.\nTwo forms of spatial autoregressive modeling methods, spatial lag and spatial error, are used to account for spatial dependency among observations.\n\u2022 Anselin L\nSpatial Econometrics: Methods and Models.\nBased on the results of Lagrange multiplier tests, the spatial lag model was selected and performed using R, version 4.0.2 software. The spatial lag model estimates both direct and indirect effects of explanatory variables on COVID-19 infection rates. The indirect effects are through the spatial relationship between observations (ZCTAs). The total effect is the sum of direct and indirect effects, which is also presented in the Results tables. Except for subway ridership variables and nursing home bed rate, all other variables were log-transformed to achieve a better fit with the data, reduce the influence of outliers, and adjust for nonlinearity of the data. Therefore, the coefficients in the Results tables are interpreted as elasticities. The collinearity diagnostic test was also performed, and the tolerance values of explanatory variables, in both models, were higher than the 0.2 threshold,\n\u2022 O'brien RM\nA caution regarding rules of thumb for variance inflation factors.\nwhich suggested no issue of multicollinearity.\n\nRESULTS\n\nThe results of spatial lag models for the COVID-19 infection rate per 1,000 population as of April 1 and May 25 are shown in Tables 2 and 3, respectively. The comparison between the 2 tables shows noticeable differences between factors that significantly explained the infection rate at these 2 times during the COVID-19 pandemic.\nTable 1Variable Descriptions, Data Sources, and Descriptive Statistics\nVariable\/descriptionData sourcesMean (SD)\nDependent variables\nln of confirmed cases per 1,000 (as of April 1)NYC Department of Health 20204.59 (1.7)\nln of confirmed cases per 1,000 (as of May 25)NYC Department of Health 202021.9 (8.5)\nIndependent variables\nln of percent Black populationACS 201821.7 (24.9)\nln of percent Hispanic populationACS 201826.1 (19.5)\nln of average household sizeACS 20182.6 (0.51)\nln of standardized SES indexDeveloped by authors based on data from ACS 2018100 (25)\nln of the number of POIs per 1,000SafeGraph 202011.28 (13.1)\nln of population densityACS 2018 (5-year estimates)39,886 (25,067)\nln of percent emptying outThe New York Times 202011.0 (10.4)\nNumber of nursing home beds per 1,000HIFLD 20195.7 (10.4)\nSubway ridership in 1,000s (baseline)MTA 2020172.1 (254.6)\n% change in subway ridership (March 1\u2013March 14, relative to the baseline)MTA 2020\u22121.82 (5.94)\n% change in subway ridership (April 27\u2013May 10, relative to the baseline)MTA 2020\u221260.1 (40.41)\nln of tests per 1,000 (as of April 1)NYC Department of Health 20209.1 (2.6)\nln of tests per 1,000 (as of May 25)NYC Department of Health 202072.2 (21.9)\nNote: Descriptive statistics were calculated before log-transformation.\nACS, American Community Survey; HIFLD, Homeland Infrastructure Foundation-Level Data; ln, natural logarithm; MTA, Metropolitan Transit Authority; NYC, New York City; POI, point of interest.\nTable 2Results of the Spatial Lag Model as of April 1\nVariablesbSEt-ratiop-valueTotal effects\nIntercept\u22120.18470.6710\u22120.27520.783\nln of percent Black0.02390.01221.95740.0470.0245\nln of percent Hispanic0.00670.02670.24990.8030.0068\nln of average household size0.71580.11386.2890<0.0010.7350\nln of SES index\u22120.33980.1083\u22123.13780.002\u22120.3488\nln of POI per 1,000 population0.07220.03721.96320.0490.0742\nln of population density0.00540.02260.23870.8110.0055\nSubway ridership per 1,000 population (baseline)0.0000860.000071.12020.2630.000088\n% change in subway ridership (March 1\u2013March 14, relative to the baseline)\u22120.00390.0030\u22121.30590.192\u22120.0040\nNumber of nursing home beds per 1,000 population0.00180.00141.26470.2060.0018\nln of tests per 1,000 population (April 1)1.07780.053820.0433<0.0011.1066\nNote: Boldface indicates statistical significance (p<0.05). Outcome variable is the natural log of the number of confirmed cases per 1,000 population as of April 1.\nln, natural logarithm; POI, point of interest.\nTable 3Results of the Spatial Lag Model as of May 25\nVariablesbSEt-ratiop-valueTotal effects\nIntercept10.7850.47022.94<0.001\nln of percent Black0.02720.00803.39<0.0010.0269\nln of percent Hispanic0.04320.01732.490.0130.0428\nln of average household size0.36210.07644.74<0.0010.3592\nln of SES index\u22120.24360.0713\u22123.42<0.001\u22120.2417\nln of POI per 1,000 population\u22120.04180.0244\u22121.710.087\u22120.0414\nln of population density\u22120.02590.0168\u22121.540.123\u22120.0257\nSubway ridership (baseline)0.0000820.000071.670.0940.000081\n% change in subway ridership (April 27\u2013May 10, relative to the baseline)0.00020.00030.750.4530.00024\nNumber of nursing home beds per 1,000 population0.00270.00102.810.0050.0027\nln of % emptying out\u22120.11140.0187\u22125.97<0.001\u22120.110\nln of tests per 1,000 population (as of May 25)0.95810.043322.15<0.0010.951\nNote: Boldface indicates statistical significance (p<0.05). Outcome variable is natural log of the number of confirmed cases per 1,000 population as of May 25.\nln, natural logarithm; POI, point of interest.\nThe comparison between the 2 models revealed that, at early stages of the pandemic and before NYC reached the apex, ZCTAs with the higher number of POIs (per capita) as potential venues for crowding reported significantly higher per capita infection rates. The concentration of POIs in a ZIP code facilitates social interactions and closer contacts and could lead to the transmission of disease in the immediate neighborhood. This was no longer the case on May 25, possibly because of business closures and the implementation of stay-at-home orders.\nHowever, the average household size, representing the level of crowding in households, was the only crowding variable that was significant in both the April 1 and May 25 models. On April 1, doubling household size was associated with a 36% increase in COVID-19 infection rate per 1,000 population. The spread of COVID-19 may begin in schools, workplaces, or POIs, but eventually neighborhoods with relatively larger households are the most vulnerable to the possibilities of transmission. These findings suggest that neighborhoods with relatively larger households, such as immigrant communities, are more vulnerable to the spread of virus during the pandemic.\nAfter controlling for the variables that represented the level of crowding, population density had no significant relationship with the COVID-19 infection rate on April 1 and May 25. These findings indicate that variables representing different dimensions of crowding might be better predictors of the per capita infection rate than population density. Recent national polls show that residents in dense places are more likely to voluntarily engage in social distancing, being more cognizant of the threat.\n\nSaad L. Americans rapidly answering the call to isolate, prepare. Gallup. March 20, 2020.https:\/\/news.gallup.com\/poll\/297035\/americans-rapidly-answering-call-isolate-prepare.aspx. Accessed September 10, 2020.\n\nAfter controlling for the level of crowding and population density, the baseline subway ridership per 1,000 population had no significant relationship with the cumulative ZCTA per capita infection rates on April 1 and May 25. Similarly, the changes in subway ridership relative to baseline were not significantly related to the COVID-19 infection rates on April 1 and May 25. These findings were confirmed with follow-up t-tests, which showed no significant differences between ZCTAs with no subway station and ZCTAs that were served by subway in terms of the per capita infection rate on April 1 and May 25 (p-values of 0.685 and 0.735, respectively).\nIn contrast, from the list of control variables, racial and socioeconomic compositions were among the most significant predictors of the spatial variation in COVID-19 per capita infection rates in NYC, even more so than variables such as POI rates, density, and nursing home bed rates. These findings align with recent findings about the increased prevalence of COVID-19 in low-income, Hispanic-, and Black-majority neighborhoods in NYC, possibly because of their greater risk of occupational exposure and other key social determinants of health.\n\nBorjas GJ. Demographic determinants of testing incidence and COVID-19 infections in New York City neighborhoods. HKS Working Paper No. RWP20-008. SSRN. Online April 10, 2020. https:\/\/doi.org\/10.2139\/ssrn.3572329.\n\n,\n\u2022 Credit K\nNeighborhood inequity: exploring the factors underlying racial and ethnic disparities in COVID-19 testing and infection rates using ZIP code data in Chicago and New York.\n,\n\u2022 Yancy CW\nCOVID-19 and African Americans.\n\u2022 Quinn SC\n\u2022 Kumar S\nHealth inequalities and infectious disease epidemics: a challenge for global health security.\n\u2022 Kumar S\n\u2022 Quinn SC\n\u2022 Kim KH\n\u2022 Daniel LH\n\u2022 Freimuth VS\nThe impact of workplace policies and other social factors on self-reported influenza-like illness incidence during the 2009 H1N1 pandemic.\n\nDISCUSSION\n\nThis study found no evidence that subway ridership was related to the COVID-19 infection rate in NYC. The recent experience of a few developed countries in tracing infection clusters confirms this finding. In Japan, since the state of emergency was lifted in late May, the majority of infection clusters were traced to gyms, bars, music clubs, and karaoke rooms, whereas not even a single infection cluster, defined as \u22653 COVID-19 infections linked by contact, were associated with its highly popular and often crowded commuter trains.\n\u2022 Normile D\nJapan ends its COVID-19 state of emergency.\nSimilarly, according to the National Public Health Institute in France, between May 9 and June 15, from 150 clusters of new COVID-19 infections, none were traced to the nation's public transit system, consisting of 6 subway systems, trams, light rail, and bus networks. In fact, most of these clusters had emerged in hospitals, workplaces, and homeless shelters.\n\nO'Sullivan F. In Japan and France, riding transit looks surprisingly safe. Bloomberg CityLab. June 9, 2020.https:\/\/www.bloomberg.com\/news\/articles\/2020-06-09\/japan-and-france-find-public-transit-seems-safe. Accessed September 10, 2020.\n\nIn addition, findings about the insignificant link between population density and the per capita COVID-19 infection rate run counterintuitive to recent dialogues in news media outlets and among policymakers that highlight the role of density on the COVID-19 spread, particularly in NYC.\n\nRosenthal BM. Density is New York City's big \u201cenemy\u201d in the coronavirus fight. The New York Times. March 23, 2020.https:\/\/www.nytimes.com\/2020\/03\/23\/nyregion\/coronavirus-nyc-crowds-density.html. Accessed September 10, 2020.\n\nCNN, for instance, quoted Governor Cuomo of New York in an article on May 2, 2020 and wrote \u201cIt's very simple. It's about density. It's about the number of people in a small geographic location allowing that virus to spread.... Dense environments are its feeding grounds.\u201d\n\nShoichet CE, Jones A. Coronavirus is making some people rethink where they want to live. CNN. May 2, 2020.https:\/\/www.cnn.com\/2020\/05\/02\/us\/cities-population-coronavirus\/index.html. Accessed September 10, 2020.\n\nBefore the COVID-19 pandemic, extensive research has confirmed the environmental and public health benefits of dense, compact, and transit-accessible developments.\n\u2022 Hamidi S\nUrban sprawl and the emergence of food deserts in the USA.\n\u2022 Hamidi S\n\u2022 Ewing R\n\u2022 Tatalovich Z\n\u2022 Grace JB\n\u2022 Berrigan D\nAssociations between urban sprawl and life expectancy in the United States.\n\u2022 Ewing R\n\u2022 Hamidi S\n\u2022 Grace JB\nUrban sprawl as a risk factor in motor vehicle crashes.\n\u2022 Ewing R\n\u2022 Hamidi S\nCosts of Sprawl.\nThis study found no evidence that population density was associated with a higher per capita COVID-19 infection rate. Indeed, crowding (and not density) was associated with the higher infection rate on April 1.\n\nLimitations\n\nOne limitation of this study is that the analyses were based on ZCTA-level aggregated data and did not control for the individual-level variations and interactions among variables. Therefore, findings could not draw individual-level conclusions, particularly related to socioeconomic factors. In addition, the aggregated nature of this study limits the ability to control for individual-level factors, such as underlying health conditions that might be associated with the severity of disease and the likelihood of testing. Also, the transit ridership variables only represent the subway ridership, and findings are not generalizable to other modes of public transit, such as bus or ride-hailing services. It is possible that other modes of public transportation, such as bus transit, which are more widely accessible across all ZCTAs in the study area have a significant relationship with the COVID-19 per capita infection rate. In addition, the POI variables were computed based on GPS data from smartphones and may underrepresent those who do not have a smartphone or opt to turn off the location feature of their smartphone. Furthermore, NYC is the densest U.S. city, has the highest transit ridership, and may not represent a typical American city. Finally, the socioeconomic and demographic variables in this study are mainly based on Census data and may underrepresent noncitizens and undocumented immigrants. Therefore, the SES and racial composition of ZCTAs may not have been fully captured with measures in this study.\n\nCONCLUSIONS\n\nThis study offers empirical evidence that distinguishes between population density and different forms of crowding and shows that crowded households, measured in terms of household size, are associated with the significantly higher per capita infection rate across NYC ZIP codes. In addition, destinations (POIs) that could attract visitors not only could facilitate the spread of virus to other parts of the city (through indirect effects) but also are significantly associated with the higher per capita infection rate in their immediate neighborhoods, particularly during the early stages of the pandemic. Policymakers should pay particularly close attention to neighborhoods with a high proportion of crowded households and these destinations (or POIs) during the early stages of pandemics.\nAnother major takeaway of this study is that investigators found no evidence that a higher per capita subway ridership and percentage changes in subway ridership are related to the COVID-19 infection rate across the NYC ZIP codes. These findings challenge Harris,\n\nHarris JE. The subways seeded the massive coronavirus epidemic in New York City. NBER working paper 27021. Cambridge, MA: National Bureau of Economic Research.https:\/\/doi.org\/10.3386\/w27021. Revised August 2020. Accessed March 31, 2021.\n\nwho argued that the ZCTAs along the subway lines had significantly higher infection rates than ZIP codes that were not served by subway. Still, it may be too early to draw a definitive conclusion, and more studies are needed to further investigate the role of the transit system (including other transit modes) on COVID-19 pandemic spread through contact tracing.\n\nACKNOWLEDGMENTS\n\nThis research was supported by the Bloomberg American Health Initiative at the Johns Hopkins Bloomberg School of Public Health.\nSH contributed to conceptualization, formal analysis, methodology, validation, supervision, visualization, writing\u2013original draft, and writing\u2013review and editing. IH contributed to data curation.\n\nREFERENCES\n\n1. Borjas GJ. Demographic determinants of testing incidence and COVID-19 infections in New York City neighborhoods. HKS Working Paper No. RWP20-008. SSRN. Online April 10, 2020. https:\/\/doi.org\/10.2139\/ssrn.3572329.\n\n\u2022 Credit K\nNeighborhood inequity: exploring the factors underlying racial and ethnic disparities in COVID-19 testing and infection rates using ZIP code data in Chicago and New York.\nReg Sci Policy Pract. 2020; 12: 1249-1271https:\/\/doi.org\/10.1111\/rsp3.12321\n\u2022 Lieberman-Cribbin W\n\u2022 Tuminello S\n\u2022 Flores RM\n\u2022 Taioli E\nDisparities in COVID-19 testing and positivity in New York City.\nAm J Prev Med. 2020; 59: 326-332https:\/\/doi.org\/10.1016\/j.amepre.2020.06.005\n\u2022 Almagro M\n\u2022 Orane-Hutchinson A\nJUE insight: the determinants of the differential exposure to COVID-19 in New York City and their evolution over time.\nJ Urban Econ. 2020; (In press. Online October 28, 2020)https:\/\/doi.org\/10.1016\/j.jue.2020.103293\n2. Harris JE. The subways seeded the massive coronavirus epidemic in New York City. NBER working paper 27021. Cambridge, MA: National Bureau of Economic Research.https:\/\/doi.org\/10.3386\/w27021. Revised August 2020. Accessed March 31, 2021.\n\n3. Bliss L. The New York subway got caught in the coronavirus culture war. Bloomberg CityLab. April 21, 2020.https:\/\/www.bloomberg.com\/news\/articles\/2020-04-21\/the-tenuous-link-between-the-subway-and-covid-19. Accessed September 10, 2020.\n\n4. Sadik-Khan J, Solomonow S. Fear of public transit got ahead of the evidence. The Atlantic. June 14, 2020.https:\/\/www.theatlantic.com\/ideas\/archive\/2020\/06\/fear-transit-bad-cities\/612979\/. Accessed September 10, 2020.\n\n\u2022 Hamidi S\n\u2022 Sabouri S\n\u2022 Ewing R\nDoes density aggravate the COVID-19 pandemic? Early findings and lessons for planners.\nJ Am Plann Assoc. 2020; 86: 495-509https:\/\/doi.org\/10.1080\/01944363.2020.1777891\n\u2022 Hamidi S\n\u2022 Ewing R\n\u2022 Sabouri S\nLongitudinal analyses of the relationship between development density and the COVID-19 morbidity and mortality rates: early evidence from 1,165 metropolitan counties in the United States.\nHealth Place. 2020; 64102378https:\/\/doi.org\/10.1016\/j.healthplace.2020.102378\n5. Quealy K. The richest neighborhoods emptied out most as coronavirus hit New York City. The New York Times. May 15, 2020. https:\/\/www.nytimes.com\/interactive\/2020\/05\/15\/upshot\/who-left-new-york-coronavirus.html. Accessed September 10, 2020.\n\n6. The New York Times. More than one-third of U.S. coronavirus deaths are linked to nursing homes. The New York Times. Updated November 20, 2020. https:\/\/www.nytimes.com\/interactive\/2020\/us\/coronavirus-nursing-homes.html. Accessed September 10, 2020.\n\n7. New York City Department of Health. Confirmed and probable COVID-19 deaths. New York, NY: New York City Department of Health.https:\/\/www1.nyc.gov\/site\/doh\/covid\/covid-19-data-archive.page. Accessed January 8, 2021.\n\n8. Buchanan L, Patel JK, Rosenthal BM, Singhvi A. A month of coronavirus in New York City: see the hardest-hit areas. The New York Times. April 1, 2020. https:\/\/www.nytimes.com\/interactive\/2020\/04\/01\/nyregion\/nyc-coronavirus-cases-map.html. Accessed on September 10, 2020.\n\n9. Metropolitan Transit Authority. Turnstile data. http:\/\/web.mta.info\/developers\/turnstile.html. Accessed July 6, 2020.\n\n\u2022 Flaxman S\n\u2022 Mishra S\n\u2022 Gandy A\n\u2022 et al.\nEstimating the effects of non-pharmaceutical interventions on COVID-19 in Europe.\nNature. 2020; 584: 257-261https:\/\/doi.org\/10.1038\/s41586-020-2405-7\n10. SafeGraph. Places schema.https:\/\/docs.safegraph.com\/docs. Accessed July 6, 2020.\n\n\u2022 Duke C\n\u2022 Hamidi S\n\u2022 Ewing R\nValidity and reliability.\nin: Ewing R Park K Basic Quantitative Research Methods for Urban Planners. Routledge, New York, NY2020: 88-106https:\/\/doi.org\/10.4324\/9780429325021-6\n\u2022 Yost K\n\u2022 Perkins C\n\u2022 Cohen R\n\u2022 Morris C\n\u2022 Wright W\nSocioeconomic status and breast cancer incidence in California for different race\/ethnic groups.\nCancer Causes Control. 2001; 12: 703-711https:\/\/doi.org\/10.1023\/a:1011240019516\n\u2022 American Community Survey\nAmerican Community Survey 5-Year Estimates. U.S. Census Bureau.\n2020 (https:\/\/www.census.gov\/programs-surveys\/acs\/data.html. Updated March 30, 2020. Accessed January 8, 2021)\n11. U.S. Department of Homeland Security. Homeland Infrastructure Foundation-Level Data (HIFLD). https:\/\/hifld-geoplatform.opendata.arcgis.com\/. Accessed January 8, 2021.\n\n\u2022 Anselin L\nSpatial Econometrics: Methods and Models.\nSpringer Science\u202f+\u202fBusiness Media, Dordrecht, Netherlands1988\n\u2022 O'brien RM\nA caution regarding rules of thumb for variance inflation factors.\nQual Quant. 2007; 41: 673-690https:\/\/doi.org\/10.1007\/s11135-006-9018-6\n12. Saad L. Americans rapidly answering the call to isolate, prepare. Gallup. March 20, 2020.https:\/\/news.gallup.com\/poll\/297035\/americans-rapidly-answering-call-isolate-prepare.aspx. Accessed September 10, 2020.\n\n\u2022 Yancy CW\nCOVID-19 and African Americans.\nJAMA. 2020; 323: 1891-1892https:\/\/doi.org\/10.1001\/jama.2020.6548\n\u2022 Quinn SC\n\u2022 Kumar S\nHealth inequalities and infectious disease epidemics: a challenge for global health security.\nBiosecur Bioterror. 2014; 12: 263-273https:\/\/doi.org\/10.1089\/bsp.2014.0032\n\u2022 Kumar S\n\u2022 Quinn SC\n\u2022 Kim KH\n\u2022 Daniel LH\n\u2022 Freimuth VS\nThe impact of workplace policies and other social factors on self-reported influenza-like illness incidence during the 2009 H1N1 pandemic.\nAm J Public Health. 2012; 102: 134-140https:\/\/doi.org\/10.2105\/AJPH.2011.300307\n\u2022 Normile D\nJapan ends its COVID-19 state of emergency.\nScience. May 26, 2020; (https:\/\/doi.org\/10.1126\/science.abd0092. Accessed March 31, 2021)\n13. O'Sullivan F. In Japan and France, riding transit looks surprisingly safe. Bloomberg CityLab. June 9, 2020.https:\/\/www.bloomberg.com\/news\/articles\/2020-06-09\/japan-and-france-find-public-transit-seems-safe. Accessed September 10, 2020.\n\n14. Rosenthal BM. Density is New York City's big \u201cenemy\u201d in the coronavirus fight. The New York Times. March 23, 2020.https:\/\/www.nytimes.com\/2020\/03\/23\/nyregion\/coronavirus-nyc-crowds-density.html. Accessed September 10, 2020.\n\n15. Shoichet CE, Jones A. Coronavirus is making some people rethink where they want to live. CNN. May 2, 2020.https:\/\/www.cnn.com\/2020\/05\/02\/us\/cities-population-coronavirus\/index.html. Accessed September 10, 2020.\n\n\u2022 Hamidi S\nUrban sprawl and the emergence of food deserts in the USA.\nUrban Stud. 2020; 57: 1660-1675https:\/\/doi.org\/10.1177\/0042098019841540\n\u2022 Hamidi S\n\u2022 Ewing R\n\u2022 Tatalovich Z\n\u2022 Grace JB\n\u2022 Berrigan D\nAssociations between urban sprawl and life expectancy in the United States.\nInt J Environ Res Public Health. 2018; 15: 861https:\/\/doi.org\/10.3390\/ijerph15050861\n\u2022 Ewing R\n\u2022 Hamidi S\n\u2022 Grace JB\nUrban sprawl as a risk factor in motor vehicle crashes.\nUrban Stud. 2016; 53: 247-266https:\/\/doi.org\/10.1177\/0042098014562331\n\u2022 Ewing R\n\u2022 Hamidi S\nCosts of Sprawl.\nRoutledge, New York, NY2017","date":"2022-08-08 01:05:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 1, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2941898703575134, \"perplexity\": 7039.915909264616}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882570741.21\/warc\/CC-MAIN-20220808001418-20220808031418-00277.warc.gz\"}"} | null | null |
From electric and hybrid cars through to safety features that protect you and your loved ones on the road, every year brings improvements. Buying a car manufactured prior to 2015 could be cheaper, but it may be lacking in the innovative safety features the automotive industry has seen over the past five years. From features that help you avoid accidents in the first place, to those that keep you and your family safe if the unfortunate does happen, cars today are safer than they have ever been.
In today's blog, we'll be examining the range of safety features that come standard in new cars, so that you know what to look for when you're buying your next vehicle for those anticipated family road trips.
Understanding ANCAP safety ratings is simple - the more stars awarded to a car, the better the car performed in ANCAP's safety tests. These ratings indicate the level of safety provided to both passengers of the car and the driver, as well as pedestrian safety. Their ratings also cover the ability of a car to avoid a crash. As you'll see, a number of modern vehicle safety features are equally focused on crash prevention as well as crash safety.
The easiest way to determine the ANCAP rating of a car is to head to their website. Simply enter in the make and model of the car you are looking at to see their ratings at a glance, as well as a more in-depth report that lists particular shortcomings. You can also compare classes of cars, for example, medium-sized SUVs against other medium SUVs, or big cars against small cars.
Essentially, adaptive cruise control uses radar (or other technology like laser and optical systems) to track traffic. Its aim is to keep a consistent following distance between your car and the one in front, slowing your car down or speeding it up to suit. Typically, a driver sets a maximum speed, as with cruise control, then sets the desired following distance. The system does the rest.
Autonomous braking is part of a broader set of technology aimed at collision avoidance. Through using sensor and radar technology, modern cars can detect objects in the vehicle's path and automatically brake if the driver fails to respond. Autonomous braking support then kicks in in the final stages of a potential collision in order to prevent or reduce the severity of accidents.
Lane support systems use radar or imaging technology to recognise lane markings in order to provide support for the driver. Lane support systems can range from autonomous driving that can keep your car tracking through the corner according to markings, through to warnings that alert the driver with visual warnings in the dash or noises.
Through visual or auditory warnings, modern cars will let you know when other vehicles enter your blind zones, helping you avoid collision. This is crucial for driver safety, as well as pedestrian safety. Likewise, reversing cameras help us for those tricky reverse parallel parks, so we can avoid collisions with stationary objects or worse, children.
To avoid fatigue-caused accidents, vehicles are equipped with in-car fatigue detection and assistance by using sensors that detect fatigue symptoms. When these symptoms are detected, vehicle software can then prompt drivers, either through auditory alerts, vibrations of the seat or steering wheel, or messages in the dash. The driver is then able to assess and take appropriate action.
Make the safety of your family a sure thing. | {
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\section{Introduction}
The weather is a climatic state of the atmosphere that includes wind speed, temperature, and humidity.
One of the factors that influences the weather the most is air temperature.
The air temperature is an important meteorological parameter that has a direct relationship with other meteorological parameters \textit{viz.} solar radiation, air humidity, and atmospheric pressure~\cite{manandhar2018systematic}.
Due to the increased green house emissions in the past few decades, the air temperature has continued to increase, which is a telltale sign of abnormal climatic change~\cite{wu2021ontology}. Climate change is a phenomenon that can affect many departments including health, development, and planning~\cite{brandao2019quantifying}. Therefore, it is of paramount importance to accurately predict the air temperature values. This in turn would assist in understanding its impact on other
meteorological parameters. Also, an accurate prediction of temperature will lead to better heating and cooling management of buildings.
In this paper, we propose the use of triple exponential smoothing for an accurate prediction of
air temperature.
The main contributions of this paper include:
\begin{itemize}
\item We present a robust framework to forecast ground-based air temperature values using historical data;
\item In the spirit of reproducible research, we share the source-code of our approach to the community for further benchmarking\footnote{The code related to this paper is available here: \url{https://github.com/Soumyabrata/temperature-forecasting}.}.
\end{itemize}
\section{Prediction of Air Temperature}
\subsection{Related works}
In \cite{chen2018time}, the authors used SARIMA (Seasonal Autoregressive Integrated Moving Average) techniques to predict the monthly mean air temperature in Nanjing city of china from 1951–2017. The
forecasting accuracy of the SARIMA model was acceptable for most practical purposes. In \cite{murat2018forecasting}, the daily temperature between 1980–2010 for four different European cities was forecasted. Murat \textit{et al.} used Box-Jenkins and Holt Winters seasonal auto regressive integrated moving-average to forecast the future temperature values.
In addition to ground-based weather station data, satellite data were also explored to estimate temperature\cite{astsatryan2021air} and analyze other atmospheric events~\cite{manandhar2017correlating}. Astsatryan \textit{et al.} in \cite{astsatryan2021air} implemented several neural network architectures to predict the hourly air temperature for up to $24$ hours in the Ararat valley of Armenia. Therefore, statistical time series techniques and neural networks offer stable frameworks in forecasting ground-based air temperature.
\subsection{Our proposed approach}
We represent the air temperature values upto time $t$ as $a_1$, $a_2$, \ldots , $a_t$. We use triple exponential smoothing technique~\cite{manandhar2019predicting} to model the seasonality of the temperature values. We model the future air temperature values $a_{t+m}$ as:
\begin{dmath}
a_{t+m} =s_t+mb_t + c_{t-L+1+(m-1) \mod L},
\end{dmath}
Here, $L$ is the season length, $s_t$ is the smooth version, $b_t$ is the linear trend estimate, and $c_t$ is seasonal corrections. In this paper, we benchmark our proposed method with persistence model and average model. The persistence model assumes that the forecasted temperature value is same as latest value, indicated by $a_{t+m} = a_t$. The average model forecasts the future temperature value as the average of the historical values, indicated by $a_{t+m} = \frac{1}{t} \sum_{t}^{} a_t$.
\section{Results \& Discussions}
In this section, we provide a detailed analysis of the prediction of temperature using triple exponential smoothing.
\begin{figure*}
\xdef\xfigwd{\textwidth}
\centering
\includegraphics[height=0.3\textwidth]{prediction-index20.PDF}
\label{1a
\includegraphics[height=0.3
\textwidth]{prediction-index146.PDF}
\label{1d}
\caption{We demonstrate sample illustrations of prediction of ground-based temperature. We observe that our proposed technique can clearly capture the fluctuations of the air temperature.
}
\label{fig1}
\vspace{-0.4cm}
\end{figure*}
\subsection{Dataset}
We obtain the air temperature data from National Oceanic and Atmospheric Administration (NOAA) Climate Data Online service (CDO\footnote{https://www.ncdc.noaa.gov/cdo-web/}). We choose the weather station that is situated at Alpena Regional Airport, based in Michigan, United States. This data is the daily averaged value of the air temperature measured from the ground-based weather station. We use $6$ years worth of data for the period $2015$-$2020$.
\subsection{Qualitative evaluation}
Our proposed method can effectively capture the temperature values and provide a basis for short-term and long-term prediction. Figure~\ref{fig1} provides a subjective evaluation of our proposed method.
We observe that our proposed method can capture the peaks and troughs of the variation of the ground-based air temperature. We use historical data of $5$ years to forecast the
future temperature values. We observe that our proposed technique can accurately capture both the rising and falling trends of temperature values in the two subplots of Fig.~\ref{fig1}.
\subsection{Quantitative evaluation}
We compute the root mean square error (RMSE) value between the measured data and the forecasted temperature value in order to
provide an objective evaluation of our proposed method. The performance of the temperature prediction is determined by two primary factors -- the amount of historical data for training and the length of the lead time.
We use historical data of $5$ years in our benchmarking experiments to forecast the future temperature values. We perform $50$ distinct experiments in order to remove any sampling bias.
We benchmark our proposed method with two popular baseline models -- persistence model and average model.
Table~\ref{tab1} shows the RMSE value (measured in K) averaged across $50$ experiments for the benchmarking methods.
We observed that the average model performs the worst. Our proposed method shows a consistent improvement over the persistence model for the varying lead times. The reason for this is due to the fact that temperature values remain fairly constant for shorter lead times. We also observe that the RMSE value gradually increases for larger lead times for all the methods, owing to the gradual propagation of the prediction error.
\begin{table}[htb]
\centering
\small
\caption{We compute the RMSE values (measured in K) of the air temperature for varying lead times.}
\begin{tabular}{c|ccc}
\hline
\textbf{Lead Time} & \textbf{Proposed} & \textbf{Persistence} & \textbf{Average}\\
\hline
1 day & 3.141 & 5.320 & 15.979 \\
2 days & 4.098 & 5.669 & 14.721 \\
3 days & 4.618 & 6.819 & 15.123 \\
4 days & 5.322 & 7.835 & 16.055 \\
\hline
\end{tabular}
\label{tab1}
\vspace{-0.3cm}
\end{table}
\section{Conclusion \& Future Work}
In this paper, we use triple exponential smoothing method for predicting future temperature values using past temperature data. Our proposed method shows better performance as compared to the other models. In the future, we intend to evaluate the impact of the length of historical data on the forecasts estimates. We also plan to further improve the forecasting performance by
incorporating other sensor data in addition to temperature data.
\bibliographystyle{IEEEtran}
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Exceptions
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{"url":"http:\/\/data8.org\/datascience\/_autosummary\/datascience.tables.Table.with_columns.html","text":"# datascience.tables.Table.with_columns\u00b6\n\nTable.with_columns(*labels_and_values)[source]\n\nReturn a table with additional or replaced columns.\n\nArgs:\nlabels_and_values: An alternating list of labels and values or\na list of label-value pairs. If one of the labels is in existing table, then every value in the corresponding column is set to that value. If label has only a single value (int), every row of corresponding column takes on that value.\nRaises:\nValueError: If\n\u2022 any label in labels_and_values is not a valid column\nname, i.e if label is not of type (str).\n\u2022 if any value in labels_and_values is a list\/array and\ndoes not have the same length as the number of rows in the table.\nAssertionError:\n\u2022 \u2018incorrect columns format\u2019, if passed more than one sequence\n(iterables) for labels_and_values.\n\u2022 \u2018even length sequence required\u2019 if missing a pair in\nlabel-value pairs.\nReturns:\nCopy of original table with new or replaced columns. Columns added in order of labels. Equivalent to with_column(label, value) when passed only one label-value pair.\n>>> players = Table().with_columns('player_id',\n... make_array(110234, 110235), 'wOBA', make_array(.354, .236))\n>>> players\nplayer_id | wOBA\n110,234 | 0.354\n110,235 | 0.236\n>>> players = players.with_columns('salaries', 'N\/A', 'season', 2016)\n>>> players\nplayer_id | wOBA | salaries | season\n110,234 | 0.354 | N\/A | 2,016\n110,235 | 0.236 | N\/A | 2,016\n>>> salaries = Table().with_column('salary',\n... make_array('$500,000', '$15,500,000'))\n>>> players.with_columns('salaries', salaries.column('salary'),\n... 'years', make_array(6, 1))\nplayer_id | wOBA | salaries | season | years\n110,234 | 0.354 | $500,000 | 2,016 | 6 110,235 | 0.236 |$15,500,000 | 2,016 | 1\n>>> players.with_columns(2, make_array('$600,000', '$20,000,000'))\nTraceback (most recent call last):\n...\nValueError: The column label must be a string, but a int was given\n>>> players.with_columns('salaries', make_array('\\$600,000'))\nTraceback (most recent call last):\n...\nValueError: Column length mismatch. New column does not have the same number of rows as table.","date":"2019-03-18 22:24:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3066179156303406, \"perplexity\": 11701.133990909224}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-13\/segments\/1552912201707.53\/warc\/CC-MAIN-20190318211849-20190318233849-00294.warc.gz\"}"} | null | null |
/* %TITLE "Subroutine to Scroll Window One Column"
*/
#pragma module dspl_scroll "V3.6 Calico"
/*
*
* COPYRIGHT (C) 1987 BY
* Computer Management Center, Inc.
* Idaho Falls, Idaho.
*
* This software is furnished under a license and may be used and
* copied only in accordance with terms of such license and with
* the inclusion of the above copyright notice. This software or
* any other copies thereof may not be provided or otherwise made
* available to any other person. No title to and ownership of
* the software is hereby transferred.
*
* The information in this software is subject to change without
* notice and should not be construed as a commitment by
* Computer Management Center, Inc.
*
* CMC assumes no responsibility for the use or reliability of
* its software on equipment which is not supported by CMC.
*
*++
*
* Abstract:HELP
* .b
* .lm +5
* This function is used to scroll text within a
* virtual window. It will scroll up, down, go to
* next screen, previous screen, top of text,
* bottom of text, and find a line in the text.
* .b
* SMG_SCROLL::SMG_FLAG MEANING( If Set )
* .table 3,25
* .te
* 1% Encode the lines
* .te
* 2% Draw lines
* .te
* 8% No Begin/End updates on scroll
* .te
* 128% Advance one line at a time if no prompt
* .end table
* .lm -5
*
* Index:
*
* Parameters:
*
* SMG_SCROLL
* A structure defining the scrolling window.
*
* SMG_ARRAY()
* The returned array that holds the Fun Scroll values.
*
* SMG_SCOPE.EXIT
* A function key that is assigned a command.
*
* SMG_SCROLL_OPTION
* The returned user's options.
*
* Example:
*
* Compile:
*
* $ CC/G_FLOAT FUNC_SOURCE:DSPL_SCROLL
* $ LIB FUNC_LIB:CMC_3VECTOR/REP DSPL_SCROLL
* $ DELETE DSPL_SCROLL.OBJ;*
*
* AUTHOR:
*
* 04/23/87 - B. Craig Larsen
*
* MODIFICATION HISTORY:
*
* 11/17/89 - Kevin Handy
* Turned off error trapping because all it did
* was an on error goto 0 in effect.
*
* 11/17/89 - Kevin Handy
* Put in sharable library (Possible now because
* of disabling ENTR_MESSAGE call)
*
* 08/06/91 - Kevin Handy
* Removed goofball code that would print a '+'
* on the screen. I don't know who added this
* junk code (Frank?) but they were too embarrased
* to say they did it in the edit history.
*
* 03/14/92 - Kevin Handy
* Clean up vars (checkvar)
*
* 06/01/93 - Kevin Handy
* Converted to C.
*
* 04/17/95 - Kevin Handy
* (V3.6)
* Update to V3.6 coding standards.
*
* 05/27/99 - Kevin Handy
* Modified to compile with DEC-C without errors
* (module, sspace, prompt, smg$routines.h, casts)
*--
*/
/*
* Include files
*/
#include <descrip.h>
#include <string.h>
#include <ctype.h>
#include <smg$routines.h>
#include <smgdef.h>
#include "func_include:cmcfun.h"
/* #include "func_include:scroll.h" */
/*
* Defined functions
*/
#define max(x,y) ((x)>(y)?(x):(y))
#define min(x,y) ((x)<(y)?(x):(y))
/*
* Local functions
*/
#if 0
static int calcscroll(struct smg_scroll_struct *smg_scroll, int moveflag, int lines);
#else
#define calcscroll(smg_scroll, moveflag, lines) \
dofind(smg_scroll, moveflag, smg_scroll->cur_line + lines);
#endif
static int dofind(struct smg_scroll_struct *smg_scroll, int moveflag, int line);
static void doscroll(struct smg_scroll_struct *smg_scroll, int newtop);
static void dopaint(struct smg_scroll_struct *smg_scroll,
struct dsc$descriptor_a *smg_array, long leader);
static struct dsc$descriptor_s *getelement(struct dsc$descriptor_a *array,
long item);
static void drawline(struct smg_scroll_struct *smg_scroll,
int height, int lineloop);
/*
* Local constants
*/
static $DESCRIPTOR(nullstring, "");
static char sspace[] = " ";
/*
* Main function
*/
long dspl_scroll(
struct smg_scroll_struct *smg_scroll,
struct dsc$descriptor_a *smg_array,
long *smg_scope_exit,
struct dsc$descriptor_s *smg_scroll_option)
{
/*
* Local variables
*/
char prompt[8]; /* Prompting character */
int leader; /* Length of prompt */
int win_siz; /* Lines in scrolling region */
struct dsc$descriptor_s spaces;
long line;
int scrollamount;
int newtop;
int moveflag; /* Movement flag (curent fixed at top?) */
/*
* Play games with the prompt to see if we have one (length > 0)
* after stripping off spaces
*/
strncpy(prompt, smg_scroll->prompt, sizeof(smg_scroll->prompt));
leader = sizeof(smg_scroll->prompt);
while ((leader > 0) && (prompt[leader - 1] == ' '))
{
leader--;
}
prompt[leader] = '\0';
/*
* Determine moveflag
*/
if ((leader != 0) || ((smg_scroll->smg_flag & 128) != 0))
{
moveflag = 1;
}
else
{
moveflag = 0;
}
/*
* Determine size of scrolling region
*/
win_siz = smg_scroll->scroll_bot - smg_scroll->scroll_top + 1;
/*
* Remove the Prompt
*/
if (leader != 0)
{
spaces.dsc$a_pointer = sspace;
spaces.dsc$w_length = leader;
spaces.dsc$b_class = DSC$K_CLASS_S;
spaces.dsc$b_dtype = DSC$K_DTYPE_T;
line = smg_scroll->scroll_top +
(smg_scroll->cur_line - smg_scroll->top_line);
smg$put_chars(&smg_scroll->window, &spaces,
&line, &1l);
}
/*
* Handle options
*/
switch (*smg_scope_exit)
{
case SMG$K_TRM_DOWN:
scrollamount = 1;
newtop = calcscroll(smg_scroll, moveflag, scrollamount);
doscroll(smg_scroll, newtop);
dopaint(smg_scroll, smg_array, leader);
break;
case SMG$K_TRM_UP:
scrollamount = -1;
newtop = calcscroll(smg_scroll, moveflag, scrollamount);
doscroll(smg_scroll, newtop);
dopaint(smg_scroll, smg_array, leader);
break;
case SMG$K_TRM_PREV_SCREEN:
scrollamount = -win_siz + 1;
newtop = calcscroll(smg_scroll, moveflag, scrollamount);
doscroll(smg_scroll, newtop);
dopaint(smg_scroll, smg_array, leader);
break;
case SMG$K_TRM_NEXT_SCREEN:
scrollamount = win_siz - 1;
newtop = calcscroll(smg_scroll, moveflag, scrollamount);
doscroll(smg_scroll, newtop);
dopaint(smg_scroll, smg_array, leader);
break;
case SMG$K_TRM_F18:
newtop = dofind(smg_scroll, moveflag, smg_scroll->top_array);
doscroll(smg_scroll, newtop);
dopaint(smg_scroll, smg_array, leader);
break;
case SMG$K_TRM_F19:
newtop = dofind(smg_scroll, moveflag, smg_scroll->bot_array);
doscroll(smg_scroll, newtop);
dopaint(smg_scroll, smg_array, leader);
break;
case SMG$K_TRM_SELECT:
if ((smg_scroll->v_select_line <= 0) && (leader != 0))
{
smg_scroll->v_select_line = smg_scroll->cur_line;
}
dopaint(smg_scroll, smg_array, leader);
break;
case SMG$K_TRM_REMOVE:
if (smg_scroll->v_select_line > 0)
{
smg_scroll->v_select_line = 0;
}
newtop = smg_scroll->top_line;
dopaint(smg_scroll, smg_array, leader);
break;
default:
if (smg_scroll_option->dsc$w_length != 0)
{
switch (smg_scroll_option->dsc$a_pointer[0])
{
case 'P': /* 'PAINT' */
dopaint(smg_scroll, smg_array, leader);
break;
case 'F': /* 'FIND' */
newtop = dofind(smg_scroll, moveflag, smg_scroll->find_line);
doscroll(smg_scroll, newtop);
dopaint(smg_scroll, smg_array, leader);
break;
}
}
}
/*
* Calculate current line position
*/
smg_scroll->cur_w_row = smg_scroll->scroll_top +
smg_scroll->find_line - smg_scroll->top_line;
/*
* Paint the Prompt
*/
if (leader != 0)
{
spaces.dsc$a_pointer = prompt;
spaces.dsc$w_length = leader;
spaces.dsc$b_class = DSC$K_CLASS_S;
spaces.dsc$b_dtype = DSC$K_DTYPE_T;
line = smg_scroll->scroll_top +
smg_scroll->cur_line - smg_scroll->top_line;
smg$put_chars(&smg_scroll->window, &spaces,
&line, &1l, 0l, &SMG$M_BOLD);
}
return(0);
}
#if 0
/*
* Calculations for scrolling the screen.
* sets current line in scroll structure.
* returns new top line.
*/
static int calcscroll(struct smg_scroll_struct *smg_scroll, int moveflag, int lines)
{
/*
* Adjust current line pointer
*/
return(dofind(smg_scroll, moveflag, smg_scroll->cur_line + lines));
}
#endif
/*
* Calculations for scrolling the screen.
* sets current line in scroll structure.
* returns new top line.
*/
static int dofind(struct smg_scroll_struct *smg_scroll, int moveflag, int line)
{
/*
* local variables
*/
int newtop;
int newline;
int screenlines;
/*
* Handle differently if doing with pointer, or without
*/
if (moveflag != 0)
{
/*
* A pointer is used, so calculate position of pointer
* on the screen.
*/
/*
* Adjust current line pointer
*/
newline = line;
newline = max(newline, smg_scroll->top_array);
newline = min(newline, smg_scroll->bot_array);
smg_scroll->cur_line = newline;
smg_scroll->find_line = newline;
/*
* Calculate a new top line
*/
if (newline < smg_scroll->top_line)
{
newtop = newline;
}
else
{
screenlines = (smg_scroll->scroll_bot - smg_scroll->scroll_top);
if (newline > smg_scroll->top_line + screenlines)
{
newtop = newline - screenlines;
}
else
{
newtop = smg_scroll->top_line;
}
}
}
else
{
/*
* No pointer is used, so scroll as much as possible
*/
screenlines = (smg_scroll->scroll_bot - smg_scroll->scroll_top);
newline = line;
newline = max(newline, smg_scroll->top_array);
newline = min(newline,
max(smg_scroll->top_array, smg_scroll->bot_array - screenlines));
smg_scroll->cur_line = newline;
smg_scroll->find_line = newline;
newtop = newline;
}
return(newtop);
}
/*
* Repaint the screen,
* changing the top line.
*/
static void doscroll(struct smg_scroll_struct *smg_scroll, int newtop)
{
/*
* Local variables
*/
int change; /* Number of lines to scroll */
long screenlines;
/*
* Decide how to scroll
*/
change = newtop - smg_scroll->top_line;
screenlines = (smg_scroll->scroll_bot - smg_scroll->scroll_top) + 1;
/*
* Push lines upward
*/
if ((change > 0) && (change <= screenlines))
{
smg$scroll_display_area(&smg_scroll->window,
&smg_scroll->scroll_top,
&1l,
&screenlines,
0l,
&SMG$M_UP,
&change);
}
/*
* Push lines downward
* Must keep at least two lines, because backwards scroll
* can look really funny (scrolls forewards sometimes)
*/
if ((change < 0) && (-change <= screenlines))
{
change = -change;
smg$scroll_display_area(&smg_scroll->window,
&smg_scroll->scroll_top,
&1l,
&screenlines,
0l,
&SMG$M_DOWN,
&change);
}
/*
* Set the top line
*/
smg_scroll->top_line = newtop;
}
static void dopaint(struct smg_scroll_struct *smg_scroll,
struct dsc$descriptor_a *smg_array,
long leader)
{
/*
* Local variables
*/
int lineloop; /* Loop for lines */
int bottomline; /* Bottom line of screen */
long video_set; /* Special display mode */
struct dsc$descriptor_s *text; /* Pointer to array element */
long leaderone = leader + 1; /* Leader + 1 */
long row; /* row to display on */
long height;
long width;
long length;
long status;
/*
* Turn on display update
*/
smg$begin_display_update(&smg_scroll->window);
/*
* Paint all lines
*/
bottomline = (smg_scroll->scroll_bot - smg_scroll->scroll_top);
for (lineloop = 0; lineloop <= bottomline; lineloop++)
{
/*
* Get text to display
*/
if (lineloop + smg_scroll->top_line <= smg_scroll->bot_array)
{
text = getelement(smg_array, lineloop + smg_scroll->top_line);
}
else
{
text = &nullstring;
}
/*
* Calculate position
*/
row = lineloop + smg_scroll->scroll_top;
/*
* Handle video set
*/
video_set = 0;
if (smg_scroll->v_select_line != 0)
{
/*
* If line is in select range
*/
if ((lineloop + smg_scroll->top_line >=
smg_scroll->v_select_line) &&
(lineloop +smg_scroll->top_line <=
smg_scroll->cur_line))
{
video_set = smg_scroll->video_set;
}
if ((lineloop + smg_scroll->top_line <=
smg_scroll->v_select_line) &&
(lineloop +smg_scroll->top_line >=
smg_scroll->cur_line))
{
video_set = smg_scroll->video_set;
}
}
/*
* Output the line
*/
status = smg$erase_line(&smg_scroll->window, &row, &1l);
if (((smg_scroll->smg_flag & 1) != 0) && (text->dsc$w_length != 0))
{
length = text->dsc$w_length;
smg_put_virtual_display_encoded(
&smg_scroll->window,
&length,
text,
&row,
&leaderone,
&0l,
&0l,
&video_set);
}
else
{
status = smg$put_chars(&smg_scroll->window,
text,
&row,
&leaderone,
&0l,
&video_set,
&smg_scroll->video_comp,
&smg_scroll->charset);
}
}
/*
* Paint vertical lines if so desired
*/
if (smg_scroll->draw_cols[0] != ' ')
{
smg$get_display_attr(&smg_scroll->window, &height, &width);
for (lineloop = 0;
(lineloop < sizeof(smg_scroll->draw_cols)) &&
(smg_scroll->draw_cols[lineloop] != ' ');
lineloop += 4)
{
drawline(smg_scroll, height, lineloop);
}
}
/*
* Turn off display update
*/
smg$end_display_update(&smg_scroll->window);
}
/*
* Get a descriptor element
*/
static struct dsc$descriptor_s *getelement(
struct dsc$descriptor_a *array,
long item)
{
/*
* Static local variables
*/
static struct dsc$descriptor_s temp_desc;
/*
* Local Variables
*/
struct dsc$descriptor_s *str;
long length;
long elem;
/*
* Can we fit it in?
*/
elem = item * (long)array->dsc$w_length;
if (elem < array->dsc$l_arsize)
{
if (array->dsc$b_dtype == DSC$K_DTYPE_DSC)
{
/*
* Calculate position of descriptor
*/
str = (struct dsc$descriptor_s*)(array->dsc$a_pointer + elem);
}
else
{
temp_desc.dsc$a_pointer = array->dsc$a_pointer + elem;
temp_desc.dsc$w_length = array->dsc$w_length;
temp_desc.dsc$b_class = DSC$K_CLASS_S;
temp_desc.dsc$b_dtype = array->dsc$b_dtype;
str = &temp_desc;
}
}
else
{
str = &nullstring;
}
return(str);
}
static void drawline(struct smg_scroll_struct *smg_scroll,
int height,
int lineloop)
{
/*
* Local variables
*/
int column;
/*
* Calculate column to display on
*/
column = (smg_scroll->draw_cols[lineloop] - '0') * 100 +
(smg_scroll->draw_cols[lineloop+1] - '0') * 10 +
(smg_scroll->draw_cols[lineloop+2] - '0');
/*
* Display line
*/
smg$draw_line(&smg_scroll->window, &1l, &column, &height, &column);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,944 |
\section{Introduction} \label{section-introduction}
Machine Learning (ML) is expanding rapidly in numerous applications. In parallel with this rapid growth, the expansion of ML towards dependability-critical applications raises societal concern regarding the reliability and safety assurance of ML. For instance, ML in medicine by \cite{begoli2019need,wiens2019no,qayyum2020secure}, in autonomous systems e.g. self-driving cars by \cite{burton2020mind,du2020ai}, in military \cite{sharkey2019autonomous}, and in economic applications by \cite{Davenport2019}. In addition, different organizations and governmental institutes are trying to establish new rules, regulations and standards for ML, such as in \cite{ISO_AI,UK_GOV,alexander2020safety}.
While ML is a powerful tool for enabling data-driven applications, its unfettered use can pose risks to financial stability, privacy, the environment and in some domains even life. Poor application of ML is typically characterized by poor design, misspecification of the objective functions, implementation errors, choosing the wrong learning process, or using poor or non-comprehensive datasets for training. Thus, safety for ML can be defined as a set of actions to prevent any harm to humanity by ML failures or misuse. However, there are many perspectives and directions to be defined for ML Safety. In fact, \cite{amodei2016concrete} have addressed different research problems of certifying ML systems operating in the field. They have categorized safety issues into five categories: a) safe exploration, b) robustness to distributional shift, c) avoiding negative side effects, d) avoiding "reward hacking" and "wire heading", e) scalable oversight. This categorization is helpful for an adequate assessment of the applicability a concept for a given (safety) problem. In the work presented here, we will be focusing on addressing distributional shift, however using a non-standard interpretation. Distributional shift is usually interpreted as the gradual deviation of the initial state of learning of an ML component and its ongoing state as it performs online learning. As will be shown later, distributional shift will instead be used by our approach to evaluate the distance between the training and observed data of an ML component.
Statistical distance measures can be considered as a common method to measure distributional shift. Furthermore, in modern ML algorithms like Generative Adversarial Nets (GANs), statistical distance or divergence measures are applied as a loss function, such as the Jensen-Shannon divergence \cite{goodfellow2014generative}, the Wasserstein distance \cite{gulrajani2017improved}, and the Cramer distance \cite{bellemare2017cramer}. For dimension reduction, the t-SNE (t-distributed stochastic neighbour embedding) algorithm uses the Kullback-Leibler divergence as a loss function \cite{van2014accelerating}.
\subsection{Contributions}
This paper studies the applicability of safety-security monitoring based on statistical distance measures on the robustness of ML systems in the field.The basis of this work is a modified version of the statistical distance concept to allow the comparison of the data set during the ML training procedure and the observed data set during the use of the ML classifier in the field. The calculation of the distance is carried out in a novel controller-in-the-loop procedure to estimate the accuracy of the classifier in different scenarios. By exploiting the accuracy estimation, applications can actively identify situations where the ML component may be operating far beyond its trained cases, thereby risking low accuracy, and adjust accordingly. The main advantage of this approach is its flexibility in potentially handling a large range of ML techniques, as it is not dependent on the ML approach. Instead, the approach focuses on the quality of the training data and its deviation from the field data. In a comprehensive case study we have analyzed the possibilities and limitations of the proposed approach.
\subsection{Overview of the Paper}
The rest of the paper is organised as follows: In Section \ref{related-work}, previous work related to this publication is discussed. In Section \ref{section-problem-definition}, the problem definition is provided. The proposed method is addressed in Section \ref{section-method}. Numerical results are demonstrated in Section \ref{section-case-studies} with a brief discussion. Explainable AI is introduced and discussed as a highly relevant topic in Section \ref{section-xai}. The capabilities and limitations of the proposed method are summarised in Section \ref{section-discussion} and the paper terminates with a conclusion in Section \ref{section-conclusion}.
\subsection{Related Work} \label{related-work}
Our analysis of the research literature did not reveal any reference to existing publications dealing with the safety, security and accuracy of ML-based classification using statistical measures of difference. Nevertheless, there are publications that provide a basis for comparison with the current study.
A Resampling Uncertainty Estimation (RUE)-based algorithm has been proposed by \cite{schulam2019can} to ensure the point-wise reliability of the regression when the test or field data set is different from the training dataset. The algorithm has created prediction ensembles through the modified gradient and Hessian functions for ML-based regression problems. An uncertainty wrapper for black-box models based on statistical measures has been proposed by \cite{klas2019uncertainty}. Hobbhahn. M. et al. \cite{hobbhahn2020fast} have proposed a method to evaluate the uncertainty of Bayesian Deep Networks classifiers using Dirichlet distributions. The results were promising but to a limited class of classifiers (Bayesian Network-based classifiers). A new Key Performance Index (KPI), the Expected Odds Ratio (EOR) has been introduced in \cite{finlay2019empirical}. This KPI was designed to estimate empirical uncertainty in deep neural networks based on entropy theories. However, this KPI has not yet been applied to other types of machine learning algorithms. A comprehensive study on dataset shift has been provided by \cite{quionero2009dataset} and the dataset issues such as projection and projectability, simple and prior probability shift are discussed there. However, the mentioned study does not address the use of statistical distance and error bound to evaluate the dataset shift, in contrast to the work presented here.
\section{Problem Definition} \label{section-problem-definition}
Classification ML algorithms are typically employed to categorize input samples into predetermined categories. For instance, abnormality detection can be performed by detecting whether a given sample falls within known ranges i.e. categories. A simple example of a classifier for 1-dimensional input can be a line or threshold. Consider a hypothetical measurement t (e.g. time, temperature etc.) and a classifier D based on it, as shown in Figure \ref{fig1}-(a) and defined as (\ref{eq1}). Note that Figure \ref{fig1} shows the true classes of the input.
\begin{equation}
D(t) =
\begin{cases}
Class1, & \mbox{if } 0 < n \leq 100 \\
Class2, & \mbox{if } 100 < n \leq 200
\end{cases}
\label{eq1}
\end{equation}
The classifier $ D\left(t\right)\ $ can predict two classes which represent, in this example, the normal and abnormal state of a system. From measurement input 0 to 100, the sample is considered to fall under class 1 and from above 100 to 200 under class 2. The Probability Density Functions (PDFs) of the (true) classes can be estimated as shown in Figure \ref{fig1}-(b). In this figure, the threshold of the classifier has been represented with a red vertical dash-line and value of four. The area with an overlap in this figure can cause false detection to occur, as the classifier misclassifies the input belonging to the opposite class. These type of misclassifications are also known as false positive/type I errors (e.g. when misclassifying input as being class 1) and false-negative/type II errors (e.g. when misclassifying input as not being class 1).
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/OneDv2.png}
\caption{(a) A hypothetical measurement (i.e. from 0 to 100 is Class 1 and from 101 to 200 is Class 2) (b) The estimated probability density function for both Class 1 and Class 2 with a classifier threshold equal to four} \label{fig1}
\end{figure}
Considering Figure \ref{fig1}-(b) of probability density functions, we notice that in the area where the two probability density functions merge, the misclassifications and thus the errors can occur. The probability of the error or misclassification can be calculated with (\ref{eq2}) \cite{Theodoridis2009}. Note that the error probability is also related to the threshold value (x considered as the threshold value), (for more details see \cite{aslansefat2020performance}).
\begin{equation}
P\left(error\right)\ =\ \int_{-\infty}^{+\infty}P\left(error|x\right)P\left(x\right)dx
\label{eq2}
\end{equation}
In listing (2), the $P\left(error|x\right)$ can be calculated as the minimum of both PDFs as (\ref{eq3}). The minimization is subject to variation of threshold value from $-\infty$ to $+\infty$.
\begin{equation}
P\left(error|x\right)\ =\ min\left[P\left(Class\ 1|x\right),\ P\left(Class\ 2|x\right)\right]
\label{eq3}
\end{equation}
By dividing the space into two regions as $R_1$ and $R_2$, the probability of error can be written in two parts.
\begin{equation}
\begin{split}
P\left(error\right)\ =\ P\left(x\in\ R_1,Class\ 1\right)+P\left(x\in R_2,Class\ 2\right) \\ =\int_{R_1} P\left(x|Class\ 1\right)P\left(Class\ 1\right)dx+\int_{R_2} P\left(x|Class\ 2\right)P\left(Class\ 2\right)dx
\label{eq4}
\end{split}
\end{equation}
To ease the minimization problem, consider the following inequality rule \cite{fukunaga2013introduction}.
\begin{equation}
min\left[a,b\right]\le a^\lambda b^{1-\lambda}\ where\ a,b\ \geq0\ and\ 0\le\alpha\le1
\label{eq5}
\end{equation}
Equation (\ref{eq3}) can be rewritten as (\ref{eq6}). Note that in (\ref{eq5}) the $"\le"$ can considered as $"="$ when we consider the worst-case scenario or upper bound error.
\begin{equation}
\begin{split}
P\left(error|x\right)\ =\ min\left[P\left(Class\ 1|x\right),\ P\left(Class\ 2|x\right)\right]= \\
min\left[\frac{P\left(x|Class\ 1\right)P\left(Class\ 1\right)}{P\left(x\right)},\frac{P\left(x|Class\ 2\right)P\left(Class\ 2\right)}{P\left(x\right)}\right]
\end{split}
\label{eq6}
\end{equation}
Using the inequality rule and equation (\ref{eq6}), the conditional probability of error can be derived as (\ref{eq7}).
\begin{equation}
\small
P\left(error|x\right)\ \le\left(\frac{P\left(x|Class\ 1\right)P\left(Class\ 1\right)}{P\left(x\right)}\right)^\lambda\left(\frac{P\left(x|Class\ 2\right)P\left(Class\ 2\right)}{P\left(x\right)}\right)^{1-\lambda}
\label{eq7}
\end{equation}
The equation (\ref{eq8}) can be obtained using equations (\ref{eq2}) and (\ref{eq7}).
\begin{equation}
\begin{split}
P\left(error\right)\ \le\left(P\left(Class\ 1\right)\right)^\lambda\left(P\left(Class\ 2\right)\right)^{1-\lambda}\
\\
\int_{-\infty}^{+\infty}{\left(P\left(x|Class\ 1\right)\right)^\lambda\left(P\left(x|Class\ 2\right)\right)^{1-\lambda}dx}
\end{split}
\label{eq8}
\end{equation}
In safety assurance, it is important to consider the worst-case scenario which can lead us to (\ref{eq9}), known as the Chernoff upper bound of error \cite{fukunaga2013introduction}.
\begin{equation}
\begin{split}
P\left(error\right)\ = P\left(Class\ 1\right)^\lambda P\left(Class\ 2\right)^{1-\lambda}\
\\
\int_{-\infty}^{+\infty}{P\left(x|Class\ 1\right)^\lambda P\left(x|Class\ 2\right)^{1-\lambda}dx}
\end{split}
\label{eq9}
\end{equation}
If the probability distributions of the features obey normal or exponential distribution families, the integral part of (\ref{eq9}) can be solved through (\ref{eq10}) \cite{fukunaga2013introduction}.
\begin{equation}
\int_{-\infty}^{+\infty}{P\left(x|Class\ 1\right)^\lambda P\left(x|Class\ 2\right)^{1-\lambda}dx}=e^{-\theta\left(\lambda\right)}
\label{eq10}
\end{equation}
The $\theta\left(\lambda\right)$ can be calculated using (\ref{eq11}) where $\mu$ and $\Sigma$ are mean vector and variance matrix of each class respectively.
\begin{equation}
\begin{split}
\theta\left(\lambda\right)=\frac{\lambda\left(1-\lambda\right)}{2}\left[\mu_2-\mu_1\right]^T\left[\lambda\Sigma_1+\left(1-\lambda\right)\Sigma_2\right]^{-1}\left[\mu_2-\mu_1\right]
\\
+0.5\ log\frac{\left|\lambda\Sigma_1+\left(1-\lambda\right)\Sigma_2\right|}{\left|\Sigma_1\right|^\lambda\left|\Sigma_2\right|^{\left(1-\lambda\right)}}
\label{eq11}
\end{split}
\end{equation}
Considering $\alpha=0.5$ the equation (\ref{eq11}) effectively becomes the Bhattacharyya distance. It can be proven that this value is the optimal value when $\Sigma_1=\Sigma_2$ \cite{fukunaga2013introduction,nielsen2018chord}. In this study, for simplicity, the Bhattacharyya distance will be used to demonstrate the approach. It should be noted that there may be cases where the calculated error bound is higher than the real value. However, this is acceptable as an overestimation of the classifier error would not introduce safety concerns (although it may impact performance). As the $P\left(error\right)$ and $P\left(correct\right)\ $ are complementary, the probability of having a correct classification can be calculated using (\ref{eq12}).
\begin{equation}
P\left(correct\right)\ =1\ -\ \sqrt{P\left(Class\ 1\right)P\left(Class\ 2\right)}{\ e}^{-\theta\left(\lambda\right)}
\label{eq12}
\end{equation}
The Chernoff upper bound of error is usually used as a measure of separability of two classes of data, but in the above context, equation (\ref{eq12}) measures the similarity between two classes. In other words, in an ideal situation, by comparing the $P\left(error\right)\ $ of a class, with itself, the response should be equal to one while $P\left(correct\right)$ should be zero. The intuitive explanation is to determine whether the distribution of the data during training is the same as the distribution observed in the field (or not). \\
Assuming $P\left(Class\ 1\right) = P\left(Class\ 2\right)$, the integral part of $P\left(error\right)$ can be converted to the cumulative distribution function as (\ref{eq13}).
\begin{equation}
\begin{split}
P\left(error\right)\ =\ \left(\int_{-\infty}^{T}{P_{Class\ 1}\left(x\right)}dx+\int_{T}^{\infty}{P_{Class\ 2}\left(x\right)}dx\right)
\\
=\left(\int_{-\infty}^{T}{P_{Class\ 1}\left(x\right)}dx+\int_{T}^{+\infty}{P_{Class\ 2}\left(x\right)}dx\right)
\\
=\ \left(F_{Class\ 1}\left(T\right)+\left({1-F}_{Class\ 2}\left(T\right)\right)\right)
\\
=\ 1\ -\left(F_{Class\ 2}\left(T\right)-F_{Class\ 1}\left(T\right)\right)
\end{split}
\label{eq13}
\end{equation}
Equation (\ref{eq13}) shows that there is relation between probability of error (and also accuracy) and statistical difference between two Cumulative Distribution Functions (CDF) of two classes. Using this fact and considering that the Empirical CDFs of each class is available, ECDF-based statistical measures such as the Kolmogorov-Smirnov distance (equation \ref{eq14}) and similar distance measures can be used \cite{deza2014distances,raschke2011empirical}.
\begin{equation}
P(error) \approx \sup_{x} { \left(F_{Class\ 2}\left(x\right)-F_{Class\ 1}\left(x\right)\right) }
\label{eq14}
\end{equation}
It should be mentioned that such ECDF-based distances are not bounded between zero and one and, in some cases, need a coefficient to be adjusted as a measure for accuracy estimation. In section \ref{subsection-SHT}, the correlation between ECDF-based distance and accuracy will be discussed.
\section{SafeML Method} \label{section-method}
To begin with, we should note that while this study focuses on ML classifiers, the proposed approach does not prohibit application on ML components for regression tasks either. Figure \ref{fig2} illustrates how we envision the approach to be applied practically. In this flowchart, there are two main sections; the training phase and the application phase.
A) The 'training' phase is an offline procedure in which a trusted dataset is used to train the ML algorithm. Once training is complete, the classifier's performance is measured with user-defined KPIs. Meanwhile, the PDF and statistical parameters of each class are also computed and stored for future comparison in the second phase.
B) The second or 'application' phase is an online procedure in which real-time and unlabelled data is provided to the system. For example, consider an autonomous vehicle's machine vision system. Such a system has been trained to detect obstacles (among other tasks), so that the vehicle can avoid collisions with them. A critical issue to note in the application phase is that the incoming data is unlabeled. So, it cannot be assured that the classifier will perform as accurately as it had in during the training phase. As input samples are collected, the PDF and statistical parameters of each class can be estimated. The system requires enough samples to reliably determine the statistical difference, so a buffer of samples may have to be accumulated before proceeding. Using the modified Chernoff error bound in \ref{eq12}, the statistical difference of each class in the training phase and application phase is compared. If the statistical difference is very low, the classifier results and accuracy can be trusted. In the example mentioned above, the autonomous vehicle would continue its operation in this case. Instead, if the statistical difference is greater, the classifier results and accuracy are no longer considered valid (as the difference between the training and observed data is too large). In this case, the system should use an alternative approach or notify a human operator. In the above example, the system could ask the driver to takeover control of the vehicle.
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/FlowChart.png}
\caption{Flowchart of the proposed approach} \label{fig2}
\end{figure}
\section{Case Studies} \label{section-case-studies}
In this section, the proposed method described in Section \ref{section-method} is applied on typical synthetic benchmarks for ML classification.
The proposed method has been implemented in three different programming languages including R, Python and MATLAB. Regarding R programming, three well-known benchmarks have been selected: a) the XOR dataset, b) the Spiral dataset and c) the Circle dataset. Each dataset has two features (i.e. input variables) and two classes. Figure \ref{fig_Bench} illustrates the scatter plots of the selected benchmarks. More examples and benchmarks are available at \href{https://github.com/ISorokos/SafeML}{SafeML Github Repository}.
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/XOR_Spiral_Circle.png}
\caption{Scatter plot of XOR, Spiral and Circle Benchmarks} \label{fig_Bench}
\end{figure}
\subsection{Methodology for Evaluation against Benchmark Datasets}
To start the ML-based classification, 80 percent of each dataset was used for training and testing and 20 percent of the dataset has been used for validation, with 10-fold cross-validation. Both linear and nonlinear classifiers have been selected for classification. The Linear discriminant analysis (LDA) and the Classification And Regression Tree (CART) are used as linear methods. Moreover, The Random Forest (RF), K-Nearest Neighbours (KNN) and Support Vector Machine (SVM) are applied as nonlinear methods. As KPIs, the accuracy and Kappa measure are used to measure the performance of each classifier. Finally, as Empirical Cumulative Distribution Function (ECDF)-based statistical distance measures, the Kolmogorov-Smirnov Distance (KSD), Kuiper Distance, Anderson-Darling Distance (ADD), Wasserstein Distance (WD), and a combination of ADD and Wasserstein-Anderson-Darling Distance (WAD) have been selected for evaluation.
\textbf{XOR Dataset:}
The XOR dataset has two features and two classes in which features have the same mean and variance characteristics. Table \ref{Tb1l} compares the estimated accuracy based on the ECDF measures with the Minimum True Accuracy (MTA) and the Average True Accuracy (ATA) over 10 folds. For instance, the second column of this table provides the estimated accuracy based on the KSD measure. As a matter of safety, MTA is more important because it represents the worst-case scenarios, where the lowest accuracy may be experienced and impact safety. We observe that the KSD measure reports low accuracy for the LDA classifier (~.77). Instead, the ADD and WAD measures significantly overestimate the accuracy of the LDA.
\begin{comment}
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/res1.png}
\caption{Comparison of estimated accuracies vs minimum true accuracy for XOR dataset} \label{fig4}
\end{figure}
\end{comment}
\begin{table}[htbp]
\centering
\caption{Comparison of estimated accuracies vs minimum true accuracy for XOR dataset}
\scalebox{0.85}{
\begin{tabular}{lcccccccccccccccc}
\hline \hline
Method & & KSD & & Kuiper & & ADD & & WD & & WAD & & BD & & MTA & & ATA \\
\hline
LDA & & 0.772217 & & 0.770600 & & 0.902818 & & 0.755064 & & 0.985666 & & 0.154506 & & 0.50833 & & 0.59121 \\
CART & & 0.928179 & & 0.921982 & & 0.987722 & & 0.92545 & & 0.995211 & & 0.497243 & & 0.98744 & & 0.99415 \\
KNN & & 0.93057 & & 0.913063 & & 0.993151 & & 0.958768 & & 0.997076 & & 0.497102 & & 0.97489 & & 0.98666 \\
SVM & & 0.931045 & & 0.917586 & & 0.993489 & & 0.95819 & & 0.997064 & & 0.496731 & & 0.97916 & & 0.98791 \\
RF & & 0.92962 & & 0.910749 & & 0.992742 & & 0.957821 & & 0.997018 & & 0.496856 & & 0.99583 & & 0.99833 \\
\hline \hline
\end{tabular}}
\label{Tb1l}%
\end{table}%
Based on Table \ref{Tb1l}, Table \ref{TB_XOR} represents the (absolute) difference between accuracy estimations of each measure and the MTA of each classifier. The ADD, WD and WAD measures have the best accuracy estimations overall. In particular, when a LDA classifier is used, the WD measure provides an estimated accuracy with comparatively less error.
\begin{table}[htbp]
\centering
\caption{Difference between Distance Measures and MTA for XOR dataset}
\begin{tabular}{lrcrcrcrcrcrr}
\hline \hline
Method & & KSD & & Kuiper & & ADD & & WD & & WAD & & \multicolumn{1}{c}{BD} \\
\hline
LDA & & 0.263883 & & 0.262267 & & 0.394484 & & 0.246731 & & 0.477333 & & 0.353828 \\
CART & & 0.059269 & & 0.065466 & & 0.000274 & & 0.06199 & & 0.007763 & & 0.490205 \\
KNN & & 0.044320 & & 0.061833 & & 0.018256 & & 0.016127 & & 0.02218 & & 0.477793 \\
SVM & & 0.048122 & & 0.061580 & & 0.014322 & & 0.020976 & & 0.017897 & & 0.482310 \\
RF & & 0.066207 & & 0.085084 & & 0.003092 & & 0.038012 & & 0.001184 & & 0.499102 \\
\hline \hline
\end{tabular}%
\label{TB_XOR}%
\end{table}%
\textbf{Spiral Dataset:} Similar to the XOR dataset, the proposed method can be applied for the spiral dataset. Table \ref{TB_Spiral} presents difference between ECDF-based distance measures and minimum true accuracy for this dataset. For brevity, for this dataset and the next one, only the difference table is provided. Based on this table, the KSD and Kuiper distance have better estimation for accuracy of the classifiers for the spiral dataset.
\begin{table}[htbp]
\centering
\caption{Difference between Distance Measures and MTA for Spiral dataset}
\begin{tabular}{lrcrcrcrcrcrc}
\hline \hline
Method & & KSD & & Kuiper & & ADD & & WD & & WAD & & BD \\
\hline
LDA & & 0.099447 & & 0.088252 & & 0.269975 & & 0.248396 & & 0.528852 & & 0.043445 \\
CART & & 0.056131 & & 0.031092 & & 0.149191 & & 0.09477 & & 0.158529 & & 0.355675 \\
KNN & & 0.047526 & & 0.075598 & & 0.001468 & & 0.014756 & & 0.002734 & & 0.496559 \\
SVM & & 0.047526 & & 0.075598 & & 0.001468 & & 0.014756 & & 0.002734 & & 0.496608 \\
RF & & 0.024471 & & 0.050261 & & 0.018778 & & 0.003885 & & 0.019643 & & 0.479893 \\
\hline \hline
\end{tabular}%
\label{TB_Spiral}%
\end{table}%
\textbf{Circle dataset:} The circle dataset has similar statistical characteristics with the spiral dataset. Table \ref{TB_Circle} provides the difference between ECDF-based distance measures and MTA for this dataset. As can be seen, the worst accuracy estimation is related to the accuracy estimation of the LDA classifier. For the LDA, the Kuiper distance estimates with less error, with the KSD and WD being in second and third place respectively.
\begin{table}[htbp]
\centering
\caption{Difference between Distance Measures and MTA for Circle dataset}
\begin{tabular}{lrcrcrccccccc}
\hline \hline
Method & & KSD & & Kuiper & & ADD & & WD & & WAD & & BD \\
\hline
LDA & & 0.329391 & & 0.250345 & & 0.412382 & & 0.347450 & & 0.49826 & & 0.236670 \\
CART & & 0.114312 & & 0.019111 & & 0.168596 & & 0.099549 & & 0.24322 & & 0.455675 \\
KNN & & 0.004833 & & 0.037554 & & 0.027649 & & 0.010871 & & 0.02775 & & 0.498459 \\
SVM & & 0.016133 & & 0.043604 & & 0.019147 & & 0.001695 & & 0.01935 & & 0.498808 \\
RF & & 0.004663 & & 0.034529 & & 0.027776 & & 0.012814 & & 0.02782 & & 0.468893 \\
\hline \hline
\end{tabular}%
\label{TB_Circle}%
\end{table}%
\subsection{Security dataset} \label{case-study-security}
This case-study applies the proposed method towards the CICIDS2017 dataset, which was originally produced by \cite{securitydataset} at the Canadian Institute for Cyber Security (CICS) as an aide to the development and research of anomaly-based intrusion detection techniques for use in Intrusion Detection Systems (IDSs) and Intrusion Prevention Systems (IPSs) \cite{datasetanalysis}.
The labelled dataset includes both benign (Monday) and malicious (Tuesday, Wednesday, Thursday, Friday) activity. The benign network traffic is simulated by abstraction of typical user activity using a number of common protocols such as HTTP, HTTPS, FTP and SHH. Benign and malicious network activity is included as packet payloads in packet capture format (PCAPS).
\\
\textbf{Wednesday Attack:} This attack occurred on Wednesday, July 5, 2017, and different types of attacks on the availability of the victim's system have been recorded, such as DoS / DDoS, DoS slowloris (9:47 – 10:10 a.m.), DoS Slowhttptest (10:14 – 10:35 a.m.), DoS Hulk (10:43 – 11 a.m.), and DoS GoldenEye (11:10 – 11:23 a.m.). Regarding the cross-validation, a hold-out approach has been used, in which 70 percent of data has been randomly extracted for testing and training and the rest has been used for accuracy estimation. Additionally, traditional classifiers including 'Naive Bayes','Discriminant Analysis','Classification Tree', and 'Nearest Neighbor' have been used. Figure \ref{fig_W01} shows the confusion matrix when Naive Bayes classifier is used.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Pictures/Security_Dataset_W01.png}
\caption{Confusion matrix for Wednesday Security Intrusion Detection in CICIDS2017 Dataset} \label{fig_W01}
\end{figure}
\begin{comment}
Figure \ref{fig_W02} shows a sample result of six statistical measures (one PDF-based and five ECDF-based) vs. the classifier's accuracy. It is expected to have low distance (from the ECDF measures) when the accuracy is high and vice versa. As can be seen in \ref{fig_W02}, the Kuiper distance measure performs better. However, these results were derived from only one iteration of the random hold-out process. Therefore, the process should be iterated over several times and the performance of each distance measure on average should be reviewed.
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/Security_Dataset_W02v2.png}
\caption{Sample statistical distance measures vs. accuracy} \label{fig_W02}
\end{figure}
\end{comment}
Figure \ref{fig_W03} has been generated over 100 iterations. For each iteration, 70 percent of the data has been randomly extracted for testing and training and the rest has been used for accuracy estimation. Figure \ref{fig_W03} shows the box plot of the statistical distance measurements vs. the evaluated accuracy over 100 iterations. By observing the average values (red lines) of each box plot, the relationship between each measure and the average change in accuracy can be understood. In addition, this plot shows which method has less variation. For instance, the Kuiper distance and WD have the best performance while Chernoff has the least performance.
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/Security_Dataset_W03v3.png}
\caption{Box plot of statistical distance measures vs. accuracy over 100 iterations} \label{fig_W03}
\end{figure}
\begin{comment}
\todo[inline]{Figures \ref{fig_W03} and the following are not readable.}
\end{comment}
\textbf{Thursday Attack:} This attack occurred on Thursday, July 6, 2017, and various attacks, such as the Web Attack – Brute Force (9:20 – 10 a.m.), Web Attack – XSS (10:15 – 10:35 a.m.), and Web Attack – Sql Injection (10:40 – 10:42 a.m.) have been recorded. Figure \ref{fig_Th01} shows the confusion matrix for Thursday morning's security intrusion in the CICIDS2017 dataset when the Naive Bayes classifier is applied. Similar to Wednesday, 70 percent hold-out cross validation is used for this dataset. As can be seen, this dataset has four classes and the classifier has problem to detect the last class or last type of intrusion.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{Pictures/Security_Dataset_Th01.png}
\caption{Confusion matrix for Thursday Security Intrusion Detection in CICIDS2017 dataset} \label{fig_Th01}
\end{figure}
Figure \ref{fig_Th02} shows a sample result of six statistical measures (Chern-off and five ECDF-based measures) vs. accuracy of the classifier. In this sample, the Kolmogorov-Smirnov and Kuiper measures have better performance.
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/Security_Dataset_Th02v3.png}
\caption{Sample statistical distance measures vs. accuracy for Thursday Security Intrusion Detection in CICIDS2017 dataset} \label{fig_Th02}
\end{figure}
Similar to the previous example, Figure \ref{fig_Th02} has been generated over 100 times and the box plot of Figure \ref{fig_Th03} can be seen. In this figure, the Kolmogorov-Smirnov, Kuiper and Wassertein distance measures have a better performance, however, their decision variance is a bit high.
\begin{figure}
\includegraphics[width=\textwidth]{Pictures/Security_Dataset_Th03v3.png}
\caption{Box plot of statistical distance measures vs. accuracy over 100 iterations for Thursday Security Intrusion Detection in CICIDS2017 dataset} \label{fig_Th03}
\end{figure}
The rest of results for Security Intrusion Detection in CICIDS2017 dataset are available in the \href{https://github.com/ISorokos/SafeML}{SafeML Github Repository}.
\subsection{Correlation Analysis} \label{subsection-SHT}
Figure \ref{fig_corr} shows Pearson's correlation between the classes of Wednesday's data and the statistical ECDF-based distances. As can be seen, the WD and WAD distances have more correlation with the classes. This figure also shows the correlation between the measures themselves. The KSD and KD appear to be correlated. The WD and WAS also seem to be correlated. These correlations can be explained due to the similarity in their formulation. P-values for the above correlations were evaluated to be zero, thereby validating the correlation hypotheses above.
\begin{figure}
\centering
\includegraphics[width=0.7\textwidth]{Pictures/Corr_Plot.png}
\caption{Correlation between class label numbers and statistical ECDF-based Distance Measures} \label{fig_corr}
\end{figure}
\section{Towards Explainable AI} \label{section-xai}
In this section, we discuss a relevant topic to our proposed approach, to explain how the proposed approach could be applied for this purpose as well. Explainable AI (XAI) can be defined as a tool or framework that increases interpretability of ML algorithms and their outputs \cite{cutillo2020machine}. Our proposed approach can also be used to improve the interpretability of ML classifiers using the statistical ECDF-based distance measures seen previously. We shall discuss a small example here and intend to delve further on this topic in our future works. For the example, the Wednesday data from the security dataset mentioned previously is chosen and its class labels vs. the sample time has been plotted in Figure \ref{fig_XAI}. This dataset has six different classes with variable number of occurrence. In this figure a sliding window with the size of $d=1500$ is used. In the beginning, $1500$ samples of class one are considered as reference and then compared with the rest of the samples for each window using the statistical ECDF-based distance measures. It should be mentioned that the smoothness of the output is related to the sliding window's size. As can be see in the figure, the change in the average distance vs. the class shows the existing high correlation. In addition, it seems that class number five is slightly robust to statistical change and class number six has a low number of samples, that cannot produce meaningful statistical difference. The problem of detecting class six can be solved by decreasing the size of the sliding window.This figure can be generated for different classifiers and show how their decisions are correlated to the ECDF-based distance measures. As an future work, we aim to investigate ECDF-based distance inside different algorithms to better understand their actions. This section is just a hint for future works.
\begin{comment}
\todo[inline]{I cannot understand this section completely. As an XAI technique, the approach needs to improve the understandability of the ML system. I am not certain, if this can be derived from the Fig. 9. It rather seems to me that high correlations stand for a high quality of the classifier, although we are not able to interpret the ML classifier, i.e., explain the reason for the improved quality.}
\end{comment}
\begin{figure}
\includegraphics[width=0.95\textwidth]{Pictures/XAI.png}
\caption{Plot of class label and statistical ECDF-based Distances vs. time (Security dataset: Wednesday)} \label{fig_XAI}
\end{figure}
\section{Discussion} \label{section-discussion}
Overall, our preliminary investigation indicates that statistical distance measures offer the potential for providing a suitable indicator for ML performance, specifically for accuracy estimation. In particular, we further denote the following capabilities and limitations for the proposed approach.
\subsection{Capabilities of SafeML}
\begin{itemize}
\item By modifying the existing statistical distance and error bound measures, the proposed method enables estimation of the accuracy bound of the trained ML algorithm in the field with no label on the incoming data.
\item A novel human-in-loop monitoring procedure is proposed to certify the ML algorithm during operation. The procedure has three levels of operation: I) nominal operation allowed with assured ML-accuracy based on the distance estimation, II) buffering data samples to generate estimation, and III) low estimated accuracy estimated, leading to external intervention by automated/human controller being needed.
\item The proposed approach is easy to implement, and can support a variety of distributions (i.e. exponential and normal distribution families).
\item The outcome of the proposed approach can be used as an input for runtime safety analysis in adaptive systems \cite{kabir2019runtime,papadopoulos2019model}
\end{itemize}
\subsection{Limitations of the proposed method}
\begin{itemize}
\item The proposed algorithm is currently only tackling the safety evaluation problem of the machine-learning-based classification. However, we believe it can be easily expanded for clustering, dimension reduction or any problem that can be evaluated through statistical difference.
\item Some of the machine learning algorithms can be robust to a certain distributional shift or variation in the dataset distribution. This may limit the effectiveness of the discussed distance measures. That being said, the proposed measures can then be used as additional confirmation of the robustness, contributing to certification arguments.
\end{itemize}
\section{Conclusion} \label{section-conclusion}
The expansion of ML applications to safety-critical domains is a major research question. We investigate the problem of context applicability of an ML classifier, specifically the distributional shift between its training and observed data. We have identified and evaluated sets of statistical distance measures that can provide estimated upper error bounds in classification tasks based on the training and observed data distance. Further, we have proposed how this approach can be used as part of safety and security-critical systems to provide active monitoring and thus improve their robustness. The overall most effective distance measure was identified to be the Kolmogorov-Smirnov. The proposed human-in-the-loop procedure uses this statistical distance measure to monitor the estimated accuracy of the ML component and notify its AI or human controller when the deviation exceeds specific boundaries. The study is still in its early stages, but we believe the results to offer a promising starting point. The strengths and weaknesses of the proposed approach are discussed in the previous section.
\nocite{*}
\section*{Acknowledgements} \label{section-acknowledgements}
This work was supported by the DEIS H2020 Project under Grant 732242. We would like to thank EDF Energy R$\&$D UK Centre, AURA Innovation Centre and the University of Hull for their support.
\bibliographystyle{unsrt}
| {
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} | 3,898 |
'Why would he take a gun to school?': Parents concerned after student brings gun to Lumberton elementary school
By Kristin Nelson
LUMBERTON, N.C. (WMBF) – Parents and grandparents are concerned after the school district said an 8-year-old student brought a gun to a Lumberton elementary school.
Dr. Glen Burnette with the Public Schools of Robeson County said the 8-year-old was on the playground at Knuckles Elementary School and showed another student the gun.
The second student told a teacher and the teacher recovered the firearm. The teacher then informed the principal and the principal alerted the school resource officer about the gun, Burnette explained.
Burnette said that the school district is concerned about the number of guns that have been found on school campuses this semester but they are doing everything they can to keep students and staff members safe.
Barbara McLaurin's grandson attends Knuckles Elementary School. She said that all she can do at this point is to assure her grandson that he is safe in the classroom.
"He said, 'Grandma I'm scared. You know I'm scared to go back to school.' I said, 'Well, Trey, I'm pretty sure it's safer to go back to school you know. You got good teachers and that take care of you and love you,' McLaurin said.
McLaurin added that her grandson said, "Grandma, why would he take a gun to school, he's just a little boy."
RELATED COVERAGE:
Lumberton High School student charged after bringing loaded gun to school, officials say
Student found with gun at Lumberton High School during medical seizure, officials say
Relative mistakenly gives backpack with gun inside to Lumberton kindergarten student, district explains
Burnette said that the student faces a 365-day suspension.
Some parents said they're happy with the decision to suspend the student for a year, while others, like Queenetta London whose son goes to the elementary school, believes the punishment is too harsh for an 8-year-old child.
"He is not mature enough to know what to do with a gun and every child needs his education. No child left behind as they say," London said.
There is an investigation underway into the gun being brought on the campus. No charges have been filed in the case.
This is the 6th gun found on a school campus in the Public Schools of Robeson County district, and the second one on an elementary school campus.
Winter storm response continues Monday, with focus on power, roads
DHEC updates teacher quarantine guidelines for 'crisis staffing'
VIDEO: 5 Pee Dee residents sentenced to federal prison for drug conspiracy | {
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} | 7,443 |
Oakland var en amerikansk biltillverkare som byggde bilar i Pontiac, Michigan mellan 1907 och 1931.
Oakland köptes upp av General Motors 1909.
Historia
Den första Oakland-bilen byggdes av Alanson P Brush, som även varit med och byggt de första Cadillac-bilarna. Den hade tvåcylindrig motor och planetväxellåda. 1909 fick bilarna fyrcylindrig motor och normal växellåda med skjutdrev. Samma år köptes företaget upp av William C. Durants nybildade General Motors. 1913 tillkom en sexcylindrig modell och två år senare även en V8-motor. Mellan 1919 och 1923 såldes bara en sexcylindrig modell med toppventiler. Den ersattes av en sexa med sidventiler. 1926 introducerades det billigare systermärket Pontiac, som snart sålde betydligt bättre än Oakland. 1930 gick Oakland tillbaka till V8-motor, men försäljningen sjönk obevekligt till följd av den stora depressionen och året därpå lade GM ned märket till förmån för det mer framgångsrika Pontiac.
Källor
Stora billexikonet, red. G N Georgano, svensk övers. Björn-Eric Lindh 1982.
Externa länkar
Ej längre existerande amerikanska biltillverkare
Ej längre existerande amerikanska fordonstillverkare
Företag bildade 1907
Företag upplösta 1931
Fordonsmärken tillhörande General Motors | {
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} | 4,030 |
package org.cuacfm.members.model.configurationservice;
import org.cuacfm.members.model.configuration.Configuration;
/** The Interface ConfigurationService. */
public interface ConfigurationService {
/**
* Save.
*
* @param configuration the configuration
* @return the configuration
*/
public Configuration save(Configuration configuration);
/**
* Update.
*
* @param configuration the configuration
* @return the configuration
*/
public Configuration update(Configuration configuration);
/**
* Get Configuration.
*
* @return the configuration
*/
public Configuration getConfiguration();
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,900 |
Black Katana Sword on our Colinzi website today.
Find More The Hottese Black Katana Sword on our Colinzi website today.
How do you want your notebook on latest 2010. Smaller? Smarter? high tech? It's a fact. The Notebook Computer is shrinking. Most of today's computer customers probably don't remember back when a portable computer was the size of a large briefcase, complete with handle. Many of them probably don't even remember the days when a 12″ LCD screen was considered large, and seven or eight pounds was thought of as light weight.
These days, Notebook Computers come as small as seven or eight inches across, and many average three pounds or less. Computers are getting smaller, lighter, and simpler to go, for a society on the go. Here's a look at the phenomenon of Small Notebook Computers, and what you should look for when you buy.
Why Buy Small Notebook Computers?
So, why should you look at a small notebook, anyhow? If you're one of the thousands of people on the market for a new computer, there are three main factors. The first one is price. These smaller computers are lower performing than many of their competitors. They aren't going to run all the latest games or do any processing intensive work.
But, for most people who just want to do some word processing, chat with friends, and play a few low intensity casual games on the go, the price is right. You can get a Small Notebook Computer (netbook) for five or six hundred dollars, or even less. A larger notebook will start at double that.
Portability is another huge plus. Five or six pounds of laptop computer might not seem like much, until you've walked a mile with it hanging off one shoulder. Then you realize that lighter Small Notebook Computers have a real advantage. Many of them don't even require their own computer case. They come with small foam cases to protect them inside your messenger bag, backpack or purse. Just place the computer inside and go. Of course, for longer trips or more secure carrying, a proper laptop bag or backpack is probably a smart thought.
Last, but not least, Small Notebook Computers are convenient. Unlike a larger notebook or a desktop, they can be there when you need them. That means it's simpler than ever to take your work with you, whether it's to a friend's house, on the plane, or down the street to the local coffee shop. You don't have to wonder if that email has arrived, wait to get home to check on your package, or sit bored when you have work to do, just because your ancient computer is too inconvenient. Small laptop computers are the answer to all your problems.
Students, freelance writers, and business people all like these small computers. But they need to know that their machines are going to do what they want them to, and so do you. That means that before you buy, you need to take a look at the technical specifications for the computer you're thinking about. Take the time to check out a few laptop reviews, too.
They'll tell you what problems people have had with a given model and what works well for them. There are listings for keyboard size, graphics and sound card type, processor, memory and many other tech specs. All you have to do is a small comparison to find the one you want. Look into construction quality, too – some have complained about lower end machines feeling "flimsy" or too light. A few dollars more could get you a machine that performs comparably but lasts a lot longer.
A Small Notebook Computer isn't everyone's perfect choice. After all, some things have to be sacrificed for small size and light weight. These include a small bit of performance, screen size, and sometimes keyboard size. The tiniest computers might have cramped keyboards, which typists can find annoying. They also feature relatively small screens that make them sub-optimal for group movie watching, and don't have the graphics cards or the processor speed to play the latest games.
They're never going to replace your top of the line desktop, with its giant screen and fantastic processing capability. If you need to do lots of typing, require excellent processor performance and lots of memory, or want to spend a lot of your time watching movies with friends, this isn't going to be a primary computer for you.
Of course, that doesn't mean it's not the perfect second computer, when you can't bring your huge one along. It's also fantastic as a first computer for people who prioritize light weight and easily mobility, without a lot of need to engage in heavy processing. Sure, a small notebook isn't perfect for everyone, but it still has an dreadful lot to offer. Try one out today and see what a difference they can make.
A tiny computer could be your best friend if you need to work on the go. Light, convenient, attractive and durable, these small machines have it all. From Asus's tiny 7″ Eee (which debuted on the market a few years ago) to larger 12″ models from companies like Samsung, there are all kinds of Small Notebook Computers out there.
They're not a replacement for your desktop, and they don't have the fastest processors on the market. That's a fact. But, a Small Notebook Computer has a whole lot to offer to people who need to work on the go, without a lot of inconvenience. Check them out and see how a small notebook could change how you work.
The second rumor I've heard about Windows Phone 7 base smartphone, its the Dell Lightning. Microsoft seems pretty optimistic about this latest mobile OS, I hope it's not going to be just like Vistagate:).
November 2010, Blackpad Blackberry Tablet by RIM, iPad like?
The rumors started when there was a domain name registration by the Ontario RIM headquarter with the name of blackpad(dot)com.
Steven Tyler as a new panel judge in American Idol?
iPhone 4 White, July or late of this year?
I am still waiting for the iPhone 4 white for my girl, but, with the problems with the regular iPhone, I presume they will delay the launch and, maybe, just maybe, this iPhone white will be shipped with the fix, hopefully, at least with the antenna problem and the screen manufacture defective fixed.
Samsung is heading for more, engaging other rivals for more premium phone, this one is still a rumour, but it came with a spec.
The Samsung Galaxy Q. SMaller, impoved, QWERTY style of Samsung Galaxy S.
HTC Windows Phone 7, upcoming or hoax?
Linked from the article in engadget.com, i made my own research abut this rumored newest HTC model. A candybar model from HTC with SLCD and 1GHz Snapdragon processor, it surely sound like the new HTC Desire rumor, but this one use Windows Phone 7, certainly a developer build, that still hidden from public. There are only three touch sensitive button under its giant screen, nearly the same as Nexus One, with a 8 megapixels camera. Is it the rumored Desire HD? Or HTC scorpion? We'll see more about this soon. | {
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} | 6,373 |
Шарль Куапо д'Ассуси (16 октября 1605, Париж — 29 октября 1677 (по другим данным, 1679), Париж) — французский поэт, актёр, певец и композитор. Музыка его преимущественно не сохранилась, до наших дней дошли лишь несколько его куртуазных четырёхголосных песен.
Биография
Родился в Париже 16 октября 1605 г. в семье адвоката и музыкантши. Начальное музыкальное образование получил от матери. В возрасте восьми или девяти лет сбежал из дома, после чего отец определил его в иезуитский Клермонский коллеж, откуда мальчик убегал смотреть фарсовые представления на Новом мосту. В семнадцать лет покинул Париж и отправился в странствования по французским провинциям, где пел для местной аристократии и преподавал игру на лютне; к сер. 1620-х гг. оказался в Италии, освоив редкий для Франции инструмент, теорбу. В 1630 г. выступал в Англии при дворе Карла I и в Нидерландах при дворе Маргариты Лотарингской, будущей супруги Гастона Орлеанского. В 1636 г. возвращается в Париж и поступает на службу к королю Людовику XIII, получив придворную должность «рядового музыканта Короля» ().
В 1642 г. попадает в кружок Гассенди. Знакомство с Сент-Аманом, Скарроном и Сирано де Бержераком толкают его на создание первых опытов в жанре бурлеска, — поэм «Суд Париса» (1646-47) и «Овидий в прекрасном настроении» (1649). При этом д'Ассуси продолжает играть в придворном оркестре (в частности, он вместе с итальянскими музыкантами сопровождает игрой на теорбе оперу Луиджи Росси «Орфей»), а также сочиняет арии к балету «Брак Орфея и Эвридики» для Театра на болотах (Марэ). В 1650 г. он сочиняет музыку к трагедии Корнеля «Андромеда» и «Любовь Аполлона и Дафны», которую считают первым французским опытом в оперном жанре (партитура не сохранилась).
В том же году д'Ассуси отправляется в Турин, где намеревается получить пост учителя теорбы при дворе герцогства Савойя, но к декабрю 1651 г. опять возвращается во Францию и в Лангедоке присоединяется к труппе Мольера. Сотрудничество с Мольером не было успешным. Д'Ассуси покидает труппу и вновь поступает на королевскую службу. Он играет для короля, сочиняет стихи и песни и даёт уроки игры на лютне и теорбе. В 1655 г. он спешно покидает Париж и начинает сочинение двухтомных «Путешествий сира д'Ассуси». В Лионе вновь встречает Мольера и сопровождает его в Лангедок. В Монпелье д'Ассуси заключается в тюрьму по обвинению в гомосексуализме (также в 1652, 1667 и 1672 гг.)
В 1657 г. в сопровождении тринадцатилетнего пажа Пьера Валентена (, или ) по прозвищу Пьеротен приезжает в Мантую, где местный герцог, очарованный голосом мальчика, похищает его и отдаёт на кастрацию. В течение года д'Ассуси, по следам Пьеротена, посещает Венецию, Модену, Флоренцию и Рим, где он оседает на шесть лет. В Риме он сочиняет посвящения и получает за это хорошее вознаграждение. За эти годы он встречается с такими людьми, как королева Кристина и Марк Антуан Шарпантье, которому даёт «хлеб и кров».
В 1670 г. д'Ассуси возвращается в Париж и возобновляет дружбу с Мольером, который предлагает д'Ассуси сочинить музыку к комедии «Мнимый больной». Позже, однако, Мольер отказывается от этих планов и отдаёт предпочтение Шарпантье.
В 1673 г. д'Ассуси вновь заключён в тюрьму, откуда освобождён благодаря вмешательству Людовика XIV, назначившего ему придворную должность и пенсию.
Шарль Куапо д'Ассуси умер 29 октября 1677 в своём доме на острове Сите.
Примечания
Источники
Charles E. Scruggs, Charles Dassoucy: Adventures in the Age of Louis XIV (Lanham, MD: University Press of America, 1984)
Henri Prunières, «Les singulières aventures de M. Dassoucy, musicien et poëte burlesque», La Revue musicale, 1820 (1937-39)
Henri Prunières, «Le Page de Dassoucy, Contribution à l'histoire des moeurs musicales au XVIIe siècle», Feschrift für Guido Adler, Studien zur Musikgeschichte (Vienna, 1930, pp. 153-60
Patricia M. Ranum, Portraits around Marc-Antoine Charpentier (Baltimore, 2004), «Dassoucy the Poet-Composer», pp. 126-31; and «Molière», 141-49
Claude Alberge, Voyage de Molière en Languedoc (1647—1657) (Presses du Languedoc, 1988) | {
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{"url":"https:\/\/www.sarthaks.com\/2817757\/the-electric-field-electro-magnetic-wave-vacuum-described-emax-sin-where-emax-100-and-speed","text":"# The electric field in an electro-magnetic wave (in vacuum) is described by E = Emax sin(Kx - \u03c9t) Where Emax = 100 N\/C and K = 1 \u00d7 107 m-1, speed of li\n\n44 views\nin General\nclosed\n\nThe electric field in an electro-magnetic wave (in vacuum) is described by E = Emax sin(Kx - \u03c9t)\n\nWhere\n\nEmax = 100 N\/C and K = 1 \u00d7 107 m-1, speed of light is 3 \u00d7 108 m\/s.\n\nWhat is the amplitude of the corresponding magnetic wave?\n\n1. 4.57 \u00d7 10-7 T\n2. 2.99 \u00d7 10-7 T\n3. 3.33 \u00d7 10-7 T\n4. 2.99 \u00d7 107 T\n\nby (60.0k points)\nselected\n\nCorrect Answer - Option 3 : 3.33 \u00d7 10-7 T\n\nConcept:\n\nThe intrinsic impedance of the wave is defined as the ratio of the electric field and the magnetic field phasor (complex amplitude), i.e.\n\n$\u03b7 =\\frac{E}{H}$\n\n\u03b7\u00a0 =\u00a0\u03b70\u00a0= 120\u03c0\n\nAnalysis:\n\nE = Emax\u00a0sin(Kx - \u03c9t)\n\nWith Emax =\u00a0100 N\/C, the magnetic field will be:\n\nWith\u00a0$\u03b7 =\\frac{E}{H}$:\n\n$H =\\frac{E}{\\eta}$\n\n$H =\\frac{100}{120\\pi}A\/m$\n\nB = \u03bc0 H\n\n$B=4\\pi \u00d7 10^{-7}\u00d7 \\frac{100}{120\\pi }$\n\nB = 3.33\u00a0\u00d7 10-7 T","date":"2023-03-31 07:35:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7529411315917969, \"perplexity\": 3394.2698112494177}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-14\/segments\/1679296949573.84\/warc\/CC-MAIN-20230331051439-20230331081439-00785.warc.gz\"}"} | null | null |
#pragma once
#include <aws/migrationhubstrategy/MigrationHubStrategyRecommendations_EXPORTS.h>
#include <aws/core/utils/memory/stl/AWSString.h>
namespace Aws
{
namespace MigrationHubStrategyRecommendations
{
namespace Model
{
enum class AntipatternReportStatus
{
NOT_SET,
FAILED,
IN_PROGRESS,
SUCCESS
};
namespace AntipatternReportStatusMapper
{
AWS_MIGRATIONHUBSTRATEGYRECOMMENDATIONS_API AntipatternReportStatus GetAntipatternReportStatusForName(const Aws::String& name);
AWS_MIGRATIONHUBSTRATEGYRECOMMENDATIONS_API Aws::String GetNameForAntipatternReportStatus(AntipatternReportStatus value);
} // namespace AntipatternReportStatusMapper
} // namespace Model
} // namespace MigrationHubStrategyRecommendations
} // namespace Aws
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,949 |
Copyright © 2014 by Michelle Babb
All rights reserved. No portion of this book may be reproduced or utilized in any form, or by any electronic, mechanical, or other means, without the prior written permission of the publisher.
Published by Sasquatch Books
Editor: Gary Luke
Project editor: Em Gale
Design: Anna Goldstein
Photography: Hilary McMullen
Food styling: Julie Hopper
Copy editor: Rachelle Longé McGhee
Library of Congress Cataloging-in-Publication Data is available.
eBook ISBN: 978-1-57061-934-2
Trade Paperback ISBN: 978-1-57061-933-5
Sasquatch Books
1904 Third Avenue, Suite 710
Seattle, WA 98101
(206) 467-4300
www.sasquatchbooks.com
custserv@sasquatchbooks.com
v3.1
To my husband, who is my number one recipe tester, and my mom, who is my biggest cheerleader
# Table of Contents
Cover
Title Page
Copyright
Dedication
Recipe List
Foreword by Dr. Katherine Oldfield
Introduction
Recognizing and Reducing Inflammation
Tools for Success
21-Day Nutritional Cleanse to Combat Inflammation
Breakfasts
Healthy Snacks
Soups and Stews
Salads and Sides
Vegetarian Main Dishes
Pescatarian Main Dishes
Hint-of-Meat Main Dishes
Desserts
Acknowledgments
Index
Conversions
About the Author
# Recipe List
BREAKFASTS
Breakfast Burrito with Chickpeas and Avocado
Smoked Salmon and Avocado Tartine
Breakfast Rice with Crumbled Nori
Sweet Potato Hash with Lamb Sausage
Sweet or Savory Quinoa Crepes
Mango Muesli with Brazil Nut Topping
Power-Packed Granola with Currants and Chia Seeds
Fresh Berry Parfait with Coconut Cashew Cream
Berry Green Power Smoothie
HEALTHY SNACKS
Black Bean and Artichoke Hummus
Crispy Curried Chickpeas
White Bean and Kalamata Olive Hummus
Shiitake Mushroom and Walnut Pâté
Creamy Avocado Spinach Dip
Artichoke and Basil Tapenade
Anti-Inflammatory Trail Mix
Nutty Coconut Energy Truffles
Tropical Quinoa Power Bars
SOUPS AND STEWS
Mediterranean White Bean Soup
Roasted Cauliflower Soup with Gremolata
Creamy Asparagus and Sunchoke Soup
Butternut Squash and White Bean Soup
Slow-Cooked Black Bean and Broccoli Stew
Three-Bean Stew with Red Quinoa
Caramelized Carrot and Ginger Soup
Black-Eyed Pea and Escarole Soup
Lentil and Spinach Stew with Roasted Garlic
Vegetable and Chicken Pho
SALADS AND SIDES
Wilted Kale Salad with Shredded Beets and Carrots
Spring Pea and Jicama Salad
Rainbow Quinoa with Roasted Asparagus and Adzuki Beans
Wild Rice and Roasted Vegetables
Brussels Sprout Slaw
Bhutanese Rice and Flageolet Bean Salad
Warm Brussels Sprout Salad with Pecans and Currants
Kale and Kohlrabi Salad with Creamy Avocado Vinaigrette
Oven-Roasted Beets with Sautéed Greens
Braised Greens and Roasted Fennel with Silky Walnut Sauce
Super Greens Salad with Pomegranate and Toasted Hazelnuts
Shredded Cabbage and Snow Pea Sauté
VEGETARIAN MAIN DISHES
Spaghetti Squash Primavera with Basil Walnut Pesto
Black-Eyed Peas and Forbidden Rice with Crispy Kale
Zucchini Noodles with Pistachio Pesto and Black Lentils
Portobello Mushrooms with Samosa Filling
Puttanesca-Style Beans and Greens
Toasted Pecan Quinoa Burgers
Hearty Mushroom and Lentil Stew
Southwestern-Style Buckwheat Polenta Stacks
Quinoa-Stuffed Collard Rolls
Golden Beet and Mushroom Faux Gratin
Veggie Pizza with Cauliflower-Yam Crust
PESCATARIAN MAIN DISHES
Hazelnut-Encrusted Halibut with Dipping Sauce
Poached White Fish with Mango Lime Chutney
Pan-Fried Sardines with Sautéed Kale and Chard
Salmon en Papillote with Silky Celery Root Puree
Mediterranean Salmon Skewers
Pumpkin Coconut Curry with White Fish
Sizzling Salmon and Quinoa Skillet
Nori-Wrapped Mackerel with Wasabi "Mayo"
Fish Taco Salad with Strawberry Avocado Salsa
Oven-Roasted Black Cod with Smashed Sweet Peas
HINT-OF-MEAT MAIN DISHES
Moroccan Lamb Tagine with Chickpeas and Apricots
Bison Lettuce Cups with Garnet Yam Home Fries
Spring Lamb Stew
Steak Salad with Massaged Kale
Veggie Beef Burger with Rocket Salad
Sweet Potato Shepherd's Pie
DESSERTS
Pumpkin Coconut Pie with Almond Crust
Mixed Berry Walnut Crumble
Rustic Pear and Fig Crostatas
So-Easy Coconut Mango Sorbet
Baked Pears or Apples with Cashew Cream
Strawberry Rhubarb Crumble
Vanilla Wafer Pudding
No-Bake Peach Pie
# Foreword
If you are lucky enough to be a friend of Michelle's, she has fed you delicious meals. If you are lucky enough to be her client, she has taught you how to feed yourself well. I am lucky enough to be both a friend and colleague. We've eaten many meals together, have co-taught anti-inflammatory classes, and have shared hundreds of patients. Why do I send my patients to Michelle? Because she gets them better, and her tools are simple: real food. It's a prescription that I write often: "Michelle Babb, RD, Anti-Inflammatory Diet." We have observed over the years that our patients who adopt a Mediterranean-style anti-inflammatory diet feel better. We note tremendous improvements in joint pain, inflammation, cardiovascular disease, and weight. We encourage our shared patients to eat this way because it is sustainable, delicious, celebrates seasonal food, and is supported by research showing that this is the way we should all be eating.
We know it can be intimidating to change what you eat. If you are concerned that switching to an anti-inflammatory diet means a sacrifice of sorts, you will be thrilled to try out the recipes in this book! Embrace them fully and serve them to dinner guests and your family. Take Michelle's Creamy Avocado Spinach Dip to your next gathering, and I guarantee you will be asked for the recipe. Serve Mediterranean Salmon Skewers and Warm Brussels Sprout Salad with Pecans and Currants to your family and see that we all love eating this way.
Michelle and I are part of a revolution of getting people back to eating real, whole foods. This book will not only give you the tools to heal your body but will also expand your palate and introduce you to new ingredients, flavors, and cooking techniques that aren't difficult. I'm thrilled to see Michelle weave together in one book her knowledge of nutrition and mindful eating, her years of feeding her friends and family, and her insights into what people are looking for when they ask, "What can I eat?"
—Dr. Katherine Oldfield, naturopathic physician and owner of the West Seattle Natural Medicine Clinic
# Introduction
Inflammation has become the latest buzzword in health and wellness circles, and it's making a name for itself as the underlying cause of a plethora of diseases. Having been in the nutrition field for over seventeen years, I've seen a lot of fad diets come and go, so I'm never too quick to jump on the latest diet bandwagon. What I find compelling about anti-inflammatory eating is that it's grounded in science and it's proven itself effective over, and over, and over again. That's because it's based on a Mediterranean-style diet. We can look back through decades of research and hundreds of studies that confirm that a Mediterranean diet, rich in plant-based foods and healthy fats and oils, helps reduce the risk of diabetes, cardiovascular disease, and cancer.
Every day in my private practice I see patients who are suffering from chronic conditions that are rooted in inflammation. Sometimes it manifests in obvious ways, like joint pain and general achiness in the body; other times it shows up as digestive distress, weight gain, or skin issues. Whatever the case may be, food can be a potent remedy to cool down the inflammation and restore balance in the body.
As a holistic-minded nutritionist, I'm always interested in helping patients uncover the underlying cause of chronic dysfunction. I consider it my role to empower people to use diet, exercise, sleep, stress management, and attitude to heal themselves. Making lifestyle changes isn't always easy for people, but when they start to feel the undeniable transformations, it becomes its own motivation. Even making relatively small changes to the diet can start to produce significant results.
Weight loss is frequently one of those results that make it easier for people to want to continue following an anti-inflammatory eating plan. My clients who are feeling stuck at an undesirable weight are often surprised to learn that when the body is in a state of chronic inflammation, it's often difficult to lose weight. Not to mention that carrying extra fat actually produces more inflammation. So instead of using calorie-restrictive diets, I simply guide my clients toward the types of foods that help them deal with inflammation and, low and behold, they begin to shed the pounds.
All of the recipes in this book contain wholesome ingredients that play a key role in reversing the body's inflammatory processes. I use evidence-based nutrition to make decisions about which foods to incorporate or omit, but I don't need to bore you with the details (although I couldn't resist weaving in a few nuggets for your intellectual enjoyment!). Just know that while the combination of ingredients may be therapeutic, the goal was to create delicious and simple recipes that can be enjoyed by anyone.
I've included information about the 21-day nutritional cleanse that I use with my clients to help them reduce inflammation and restore balance in their bodies. If you're suffering from chronic inflammation or you're drawn to the idea of a "detox" or a "cleanse," this is a very safe and effective protocol that includes plenty of nutrient-rich foods and a good balance of protein, fat, and carbs. All of the recipes in this book are suitable for someone on the cleanse as well as anyone who is on an elimination diet or has multiple food allergies. Doing the cleanse is absolutely optional. Simply eating more of the foods featured in this book would be enough to put you on the path toward a healthier, more anti-inflammatory lifestyle.
You'll find a number of useful tools in this book including sample menu plans, shopping lists, guidelines for essential pantry items, and seventy-five recipes to support your health and wellness goals. If you allow yourself to embrace the concepts in this book, you'll quickly discover that this is not just another diet, but rather a way of life. May you learn to nourish yourself well, savor every delicious morsel, and feel the joy of restoring balance in your body.
# Recognizing and Reducing Inflammation
WHAT IS INFLAMMATION?
Inflammation is your body's way of responding to injury and insults. It's a natural and healthy response when your body is in a state of alarm, and it helps aid in the healing process. For example, if you stumbled and twisted your ankle, you would probably experience all of the hallmarks of inflammation (swelling, redness, heat, and some loss of function). That's your body's clever way of sounding the alarm, recruiting the repair and restoration team, and preventing you from doing more harm. The important part about this kind of inflammatory response is that it has a beginning (time of injury) and an end (injury is healed).
It's the lack of an endpoint that turns this natural, healthy process into one that is dysfunctional. When the body loses its regulatory control and continues to sound the alarm bell indefinitely, the repair and restoration team becomes overzealous. It would be like using a fire hose to water the delicate plants in your garden. Sure, the plants get water, but they're uprooted and destroyed in the process.
This type of prolonged, maladaptive inflammation is a common thread in many disease states, including:
» Allergies
» Arthritis
» Asthma
» Autoimmune conditions
» Cancer
» Cardiovascular disease
» Depression
» Digestive disorders
» Eczema
» Metabolic syndrome (prediabetes)
» Neurodegenerative diseases
» Obesity
» Psoriasis
Anti-inflammatory medications are often prescribed, but they can cause unpleasant side effects and don't address the underlying cause of the symptoms that are related to inflammation, which is why an anti-inflammatory diet can be a great place to start. What follows are some other common questions about inflammation.
How Does Inflammation Become Chronic?
Genetics can certainly play a role in chronic inflammation. People who have a family history of inflammatory conditions, such as rheumatoid arthritis or cardiovascular disease, may be predisposed to chronic inflammation. However, while genetics may be what loads the gun, it's really diet and lifestyle habits that pull the trigger. Someone with a family history of cardiovascular disease who follows a Mediterranean-style diet, exercises regularly, and does not smoke may never stoke that inflammatory flame.
Even without a genetic predisposition, it's possible to create an environment of chronic inflammation in the body. This can happen for a number of reasons, including:
» Pro-inflammatory dietary choices
» Reliance on processed foods and refined carbohydrates
» Excess sugar
» Too much meat, pork, poultry, eggs, and dairy
» Not enough anti-inflammatory foods, like fish, vegetables, avocados, olives, nuts, seeds, and legumes
» Elevated stress
» Smoking
» Physical activity extremes
» Lack of some regular form of exercise
» Over exercising to the point of injury or not giving the body adequate recovery time between sessions
» An undiagnosed condition that's causing chronic inflammation (such as food allergies, irritable bowel syndrome, or an autoimmune disorder)
What Are the Symptoms of Chronic Inflammation?
The most obvious symptom associated with inflammation is joint pain, but there are many other ways in which inflammation can manifest in the body. For example, when clients complain of ongoing digestive problems accompanied by abdominal pain, bloating, and bowel irregularity, I know that there is inflammation in the gut. Skin issues like eczema and psoriasis are other ways in which inflammation shows up more overtly.
Some forms of inflammation are much less obvious and may go unnoticed altogether. For example, being overweight or obese puts the body in an inflammatory state because fat cells actually send out chemical messages that increase inflammation. Having high cholesterol and/or high blood sugar also initiates inflammatory processes in the body. It's really the combination of high cholesterol with inflammation that significantly increases the risk for heart disease.
Is There a Way to Be Tested for Inflammation?
The most common lab test for inflammation is a blood test for C-reactive protein, which is produced in the liver in response to inflammation. Practitioners may also measure sedimentation rate, which is the rate at which your red blood cells settle to the bottom of a test tube over a given time period. If there are inflammatory proteins present, the blood becomes sticky and takes more time to settle.
What Is an Anti-Inflammatory Diet?
An anti-inflammatory diet is based on a Mediterranean-style eating plan that includes plenty of vegetables, fruit, legumes, nuts, seeds, healthy oils, and fish. This may also be known as a pescatarian diet, because the main source of animal protein is fish and the majority of the diet is made up of plant-based foods. There are a lot of reasons why this way of eating keeps inflammation at bay, but the most notable (and well researched) superstars in a pescatarian diet are:
OMEGA-3 FATTY ACIDS: Found mostly in seafood but also in smaller amounts in walnuts, flaxseeds, and some leafy greens like kale, omega-3s are a group of unsaturated fats that protect the body in various ways from inflammation.
FIBER: Found in plant-based foods like fruit, vegetables, whole grains, and legumes, fiber is the indigestible portion of these foods, and it's important for healthy digestion, blood sugar balance, and cholesterol management. Studies confirm a relationship between high-fiber diets and lower levels of inflammatory markers in the blood.
CAROTENOIDS: Found in carrots, garnet yams, squash, and cantaloupe, beta-carotene and other carotenoids appear to significantly reduce the risk of inflammatory arthritis.
VITAMIN K: Found in leafy greens like kale and chard, brussels sprouts, broccoli, cabbage, and seafood, vitamin K is a key nutrient that helps regulate the body's inflammatory processes. It's also a powerful antioxidant, so it reduces oxidative stress that can lead to inflammation.
MAGNESIUM: Found in green leafy vegetables, nuts, seeds, legumes, and whole grains, magnesium ranks among the highest on the list of minerals that appear to protect against inflammation.
Another important aspect of an anti-inflammatory diet is that it is a low glycemic way of eating, which means it doesn't cause blood sugar to spike and crash throughout the day. The typical American diet tends to be high in refined grains and sugar, and low in fiber. That combination leads to an excess of sugar (in the form of glucose) in the blood. When that occurs over a long period of time, our bodies can become insensitive to the insulin signal that helps us take the glucose up into our cells for fuel. As a result, there's excess glucose floating around in places it doesn't belong, sounding those alarm bells that trigger the inflammatory response.
What other lifestyle factors help reduce inflammation? Research suggests that moderate physical activity, meditation, and good sleep hygiene can reduce inflammation. Conversely, smoking and alcohol abuse increase inflammation.
LESSONS FROM THE MEDITERRANEAN
There's a lot to be learned from the traditional dietary habits of the people living in regions bordering the Mediterranean Sea. Most notably, meals consisting primarily of fresh fruits, vegetables, whole grains, legumes, nuts, seeds, and plenty of olive oil appear to have a tremendous impact on longevity and quality of life.
In the 1950s, an American researcher named Ancel Keys decided to embark on a cross-cultural study involving seven countries in four regions (United States, Northern Europe, Southern Europe, and Japan). He and his team of researchers looked at dietary and lifestyle habits and incidences of cardiovascular disease and mortality. The research supported his hypothesis that there is a correlation between phytonutrient-rich, low-cholesterol diets and a decreased risk of cardiovascular disease.
Continued research on various aspects of the Mediterranean diet provides us with more good reasons to adopt this style of eating in order to combat inflammation. In 2013, a long-term study on the Mediterranean diet was actually terminated after five years because the benefits were so clear that it was considered unethical to continue. The study compared a low-fat diet to a Mediterranean-style diet supplemented with either olive oil or nuts. Both of the Mediterranean-diet study groups had 30 percent reduced risk of heart attacks, strokes, and deaths from cardiovascular disease. And participants were even allowed to drink one glass of wine per day!
Dr. Walter Willett, chair of nutrition at Harvard School of Public Health, conducted a meta-analysis on the Mediterranean diet and concluded that "over 80 percent of coronary heart disease, 70 percent of stroke, and 90 percent of type 2 diabetes can be avoided by healthy food choices that are consistent with the traditional Mediterranean diet."
There is also research supporting the additional benefits of adopting an anti-inflammatory diet in relation to some other prevalent conditions, including:
METABOLIC SYNDROME: This is a precursor to diabetes and cardiovascular disease and a person is diagnosed when three out of five of the following symptoms are present: high blood glucose, high blood pressure, high triglycerides, low HDL (good cholesterol) levels, and excess abdominal fat. Some recent studies have demonstrated a 25 percent reduction in the prevalence of metabolic syndrome with adherence to a Mediterranean-style diet, which is similar to the effects of prescription medications (without any of the downsides).
ALZHEIMER'S DISEASE: In a study of over 1,400 seniors, those who adhered most closely to a Mediterranean-style diet had 48 percent lower risk of mental degeneration progressing to Alzheimer's.
CANCER: A study in southern Europe revealed that adherence to a traditional Mediterranean diet was associated with a 6 percent reduction in cancer (along with a 9 percent decrease in cardiovascular mortality and a 13 percent reduction in Parkinson's disease).
It's becoming clear that some of the most widely feared diseases share a common inflammatory component, which can be largely prevented through an anti-inflammatory diet. As we continue to learn more about the compounds in foods that either initiate or prevent an inflammatory response, we'll be able to make dietary decisions that help offset our genetic risk. In the meantime, we have lots of evidence that the foundational foods of a Mediterranean diet (olive oil, fish, nuts, seeds, fruits, vegetables, and whole grains) can stave off inflammatory diseases and keep us healthy and well.
# Tools for Success
The key to a successful food plan is having the best foods available and knowing how to enjoy them in the right portions. The Anti-Inflammatory Plate will help you keep portions in check, and the pantry list provides the template you need to create a food environment that sets you up for success. These essential tools make it easier for you to eat in a way that nourishes and satisfies without feeling like you're on a diet.
THE ANTI-INFLAMMATORY PLATE
Forget about the food pyramids from your youth—the Anti-Inflammatory Plate visual that follows will give you guidance on how to construct the layout of your plate and meals in a way that supports your anti-inflammatory goals.
As you can see, vegetables occupy the most real estate on the plate. These comprise your most powerful delivery system for vitamins, minerals, antioxidants, and other phytonutrients that have been shown to decrease inflammation. There's also a place for colorful, antioxidant-rich fruits, and you can set aside concerns of too much sugar from these sources because most fruit is loaded with fiber, which helps slow down the blood sugar response. Just try to eat twice as many vegetables as fruits throughout the day.
Your portion of complex carbohydrates can be some of the starchier vegetables, like sweet potatoes and yams, carrots, and beets, or it can be from whole grains like brown rice or quinoa. Complex carbs are useful because they are what give us energy and help fuel our brain. It's also important when eating a more plant-based diet to include some whole grains in combination with beans to get your protein needs met. Speaking of protein, that section of the plate is smaller than what's customary for most Americans. The best bet for anti-inflammatory protein is fish, but grass-fed beef, bison, lamb, and free-range organic poultry are acceptable in small amounts. Vegetarian protein sources include legumes, nuts, seeds, and soy. Whole soy foods, such as edamame, tofu, tempeh, and miso, can be great additions to a plant-based diet. However, soy foods are not featured in this cookbook because all of the recipes provided are hypoallergenic, and soy is one of the top food allergens (but certainly not a problem for everyone!).
Finally, note that there's a place on the plate for fats and oils. It's time to let go of those fat phobias from the '80s and enjoy the delicious, satisfying healthy fats that complete an anti-inflammatory meal. These are primarily unsaturated fats and oils that often contain omega-3s and other phytosterols that are part of the anti-inflammatory pathways in the body. Including fat in your diet can also help you feel satiated and can help significantly decrease overeating.
The Anti-Inflammatory Plate
THE ANTI-INFLAMMATORY PANTRY
Keeping your pantry and fridge well stocked with foods that support your health and wellness goals is a big step in the right direction. By having the following items on hand, you'll be primed to quickly prepare anti-inflammatory recipes, and you'll avoid having to make a special trip to the store at the last minute. Many of the things listed can be stored for several weeks or more, so you shouldn't have any problem using them up well before the expiration dates. Please refer to the tables in the following pages.
For most people, just stocking the pantry with the ingredients listed here and using the Anti-Inflammatory Plate as a model will be enough to start restoring balance to their bodies and transforming their health. Others need a more intensive overhaul to get them started on a new dietary path. The next chapter covers how to embark on a 21-day anti-inflammatory cleanse to do just that. While the majority of people report feeling better while on the cleanse, it's particularly useful for those who suffer from conditions like irritable bowel syndrome or digestive distress, osteo- or rheumatoid arthritis, congestion or sinusitis, and/or skin conditions such as eczema or psoriasis. The cleanse is definitely an optional course of action—read through the instructions and determine if it's the right fit for you. Otherwise feel free to skip straight ahead to the recipes and start cooking!
Fats and Oils
---
MUST-HAVES | BENEFITS | USES | STORAGE
Avocado oil | Great source of monounsaturated fat; more heat stable than olive oil | Frying, sautéing, roasting, grilling; dressings, marinades | Pantry
Coconut oil | Body uses this type of fat as energy; cholesterol-free | Sautéing; butter or shortening replacement | Pantry
Flaxseed oil | Plant-based source of omega-3s | Finishing only (do not heat); dressings, smoothies | Refrigerator
Grapeseed oil | Naturally stable oil that does not oxidize at higher temperatures | Frying, sautéing, roasting, grilling | Pantry
Extra-virgin olive oil | Great source of monounsaturated fat with unique antioxidant polyphenols that have anti-inflammatory properties | Light sautéing, finishing; dressings, marinades | Refrigerator or cool, dark pantry
Nuts and Seeds
---
MUST-HAVES | BENEFITS | USES | STORAGE
Almonds | Contain healthy fats that decrease inflammation, help lower cholesterol; rich source of vitamin E | Snacking, topping (baked goods, salads, etc.); almond butter, almond milk | Airtight container in pantry
Flaxseeds | Excellent plant-based source of omega-3s, unique forms of fiber that improve digestion | Topping (baked goods, salads, etc.); smoothies; egg replacer | Sealed bag in refrigerator or freezer
Pumpkin seeds | Provide a very diverse blend of antioxidants; good source of magnesium, zinc, iron | Snacking, topping (baked goods, salads, etc.); pumpkin seed butter | Airtight container in pantry
Walnuts | Contains higher amounts of omega-3s than other nuts; rich in anti-inflammatory phytonutrients | Snacking, topping (baked goods, salads, etc.); dips, spreads | Airtight container in refrigerator or cool, dark pantry
Grains and Legumes
---
MUST-HAVES | BENEFITS | USES | STORAGE
Buckwheat (flour or groats) | Not actually a grain but a very nourishing fruit seed; gluten-free, so a good flour alternative for those avoiding wheat or gluten | Bean or grain dishes, soups, stews, salads; flour replacement | Flour: Sealed bag in freezer; Groats: Pantry
Legumes (adzuki, black, chickpea, lentils, navy, pinto) | Excellent source of fiber, particularly soluble fiber; valuable plant-based source of protein, essential nutrients; gluten-free | Bean or grain dishes, soups, stews, salads, dips | Pantry
Quinoa (flour or whole) | The only grain considered a complete protein; contains small amounts of omega-3s; gluten-free | Bean or grain dishes, soups, stews, salads | Flour: Sealed bag in freezer; Whole: Pantry
Rice (flour or whole: black, brown, purple, red) | Excellent source of fiber, antioxidants that protect against type 2 diabetes and heart disease; gluten-free | Bean or grain dishes, soups, stews, salads | Flour: Sealed bag in freezer; Whole: Pantry
Herbs and Spices
---
MUST-HAVES | BENEFITS | USES | STORAGE
Cinnamon | Inhibits release of pro-inflammatory messengers in the body; helps regulate blood sugar | Fruit, oatmeal, smoothies, chili, stews | Pantry
Cumin | Stimulates digestive enzymes; contains cancer-preventing compounds | Beans, vegetables, chili, dips, marinades | Pantry
Ginger (dried or fresh) | Contains the anti-inflammatory compounds gingerols; immune-boosting properties | Smoothies, dressings, vegetables, desserts, teas | Dried: Pantry; Fresh: Refrigerator
Oregano (dried or fresh) | Excellent source of vitamin K, which helps regulate the body's inflammatory processes | Beans, vegetables, chili, dips, marinades | Dried: Pantry; Fresh: Refrigerator
Rosemary (dried or fresh) | Contains compounds that stimulate immune system, improve circulation, decrease inflammation | Beans, vegetables, chili, dips, marinades | Dried: Pantry; Fresh: Refrigerator
Turmeric (dried or fresh) | Contains curcumin and volatile oils that have powerful anti-inflammatory effects | Curries, soups, stews, rice, vegetables, lentils | Dried: Pantry; Fresh: Refrigerator
# 21-Day Nutritional Cleanse to Combat Inflammation
This is where the rubber meets the road. If you've been a victim of chronic inflammation for as long as you can remember, or you just want to get a handle on it before it becomes a problem, you can benefit from an anti-inflammatory cleanse. The key to a good cleanse is that it's supported with nutritionally robust foods that provide the body with key nutrients to heal. I'm not a proponent of fasting or juice cleanses as a method for reducing inflammation in the body. Results from those types of cleanses are fleeting at best.
In contrast, by following this 21-day plan, you'll discover how to nourish yourself with delicious whole foods that support optimal health. You'll be omitting foods that are most commonly associated with inflammation and those that are among the eight most allergenic foods. If you're having an immune response to something you're eating regularly, such as bread, cheese, or eggs, then you're accidentally keeping your body in an inflammatory state.
The 21-day program that I recommend is based on an elimination diet, which is the gold standard for diagnosing food allergies. In addition to the usual omissions, this plan also has you taking a break from vegetables in the nightshade category, including potatoes, tomatoes, peppers, and eggplant. While these foods are not pro-inflammatory for everyone, they do contain a compound called alkaloids, which some people have difficulty breaking down. I've had a number of patients with arthritis who claim that their symptoms get significantly worse during tomato season when they're harvesting their beautiful crops and eating tomatoes to their hearts' content.
The obvious reaction to an elimination diet of any kind is to focus on the foods that you can't eat. Those who have the best results and enjoy the lasting benefits of this nutritional cleanse are those who focus more on all of the fabulous, nourishing, anti-inflammatory foods that they can eat. Within Phase II, you'll find the list of foods to include and exclude during the cleanse, and I've highlighted some "best bet" foods that have specific properties known to squelch the inflammatory processes in the body. If you did nothing more than just add more of these best-bet foods to your daily diet, it's likely that you'll start to see a positive shift occurring.
PHASE I: PREPARATION
First, look at your calendar and find a one-month block without travel or too many social distractions—this is the ideal time to fit a cleanse into your schedule for optimal results. Then it's time to start preparing for the cleanse. I usually recommend giving yourself a full week to get your body and your pantry ready before you begin. For example, one of the most challenging parts of this cleanse for most people is giving up caffeine and sugar, so use this preparation week to help reduce dependence, which will in turn minimize unpleasant withdrawal symptoms.
What follows are some useful tips to help you feel stronger and more confident once you officially start your 21-day cleanse.
Reduce Caffeine
If you're consuming caffeine in any form, start weaning yourself now. If you drink coffee, go from caffeinated to half-decaf, to full decaf (preferably Swiss Water decaf), to green or black tea, and eventually to herbal tea. If you drink soda, switch to caffeine-free versions and then replace with water. For any other caffeine sources, gradually decrease the amount each day until you've removed it from your diet completely.
Start Drinking (Water, That Is!)
Start increasing your water intake so that you're drinking at least sixty to seventy ounces of water and/or herbal tea every day. This flushes out toxins and keeps you well hydrated, which can help abate some withdrawal symptoms.
WEANING TIPS
» Decrease caffeine consumption by 25 percent in the first two days and drink two extra glasses of water per day. Decrease caffeine by another 25 percent on the third and fourth days, continuing to drink plenty of water. On the fifth day, omit all caffeine, transitioning entirely to herbal tea and water.
» Refrain from adding sugar to your food or beverages and start avoiding processed foods where sugar shows up in the first five ingredients.
» Schedule a massage or acupuncture appointment to help with headaches and other withdrawal symptoms.
» Allow yourself time during the week to adjust to these changes and understand that you might experience mild withdrawal symptoms and brain fog. It may be helpful to take a day or two off work if you're having a particularly hard time focusing.
Tame the Sugar Beast
Cut back on refined sugar and start reading the nutrition labels and ingredient lists on all packaged food products you buy to get in the habit of knowing exactly what's in your food. You'll be avoiding items that contain sugar, sucrose, evaporated cane juice, high fructose corn syrup, or any artificial sweeteners like sucralose or aspartame.
Stock Up on the Right Stuff
When you go shopping, go heavy on fruits, vegetables, beans, brown rice or quinoa, nuts, seeds, and fish. See the table for specific ingredients to purchase (and avoid) for this cleanse.
Rev Up Your Metabolism
Get in the habit of eating a nutritious breakfast within one hour of rising and then have a small meal or healthy snack every three to four hours.
Get Your Act Together
Start making your meal plan and shopping list for the first week of the cleanse. It's a good idea to have your pantry and refrigerator fully stocked and to do some advance recipe prep, like washing and chopping vegetables, making a salad, and cooking up some quinoa or rice. Spending a few hours upfront will make the cleanse that much more enjoyable, keeping you stress-free and on track.
Finally, get ready to feel amazing!
PHASE II: NOURISH AND CLEANSE
For the next twenty-one days, you will be completely eliminating the top eight allergens (wheat/gluten, dairy, soy, eggs, corn, peanuts, shellfish, and oranges) from your diet. In addition, you'll leave behind caffeine, sugar, and alcohol. If you suffer from joint pain or any overt inflammatory condition (e.g. inflammatory bowel disease), you may also choose to omit the nightshade vegetables (potatoes, tomatoes, eggplant, and peppers) to see if those alkaloid-containing foods are exacerbating your symptoms.
So what can you eat? What's left is a very robust and anti-inflammatory Mediterranean-style diet. You'll be nourishing yourself with vegetables, fruits, nuts, seeds, legumes, gluten-free whole grains, finfish, and small amounts of organic meat and poultry. You definitely won't go hungry if you plan well. This book will be an invaluable resource because every recipe that's included is cleanse-friendly!
As you review the table that follows of foods to include and exclude, be sure to take note of the "best bets" column for each category. The best-bet foods contain phytonutrients and/or essential fatty acids that reverse the inflammatory processes in your body. These are the foods that you should intentionally try to eat more regularly, with a goal of including at least five of them from any category in your diet every day.
Foods to Include and Exclude on the Cleanse
| FOODS TO INCLUDE | BEST BETS | FOODS TO EXCLUDE
---|---|---|---
Vegetables | Fresh raw, steamed, sautéed, juiced, or roasted vegetables | Beets, broccoli, brussels sprouts, carrots, chard, kale, onions, spinach, garnet or jewel yams | Corn, creamed vegetables, nightshades (eggplant, peppers, potatoes, tomatoes)
Fruits | Fresh or frozen fruits, unsweetened fruit juices | Apples, avocados, berries, grapes, kiwis, melons, pears | Oranges, orange juice
Grains | Amaranth, buckwheat, millet, oats (certified gluten-free), quinoa, rice, tapioca, teff | Buckwheat, quinoa, rice (brown or red) | Barley, corn, gluten-containing products, kamut, rye, spelt, wheat
Animal Proteins: Fish and Meat | Fresh, frozen, or canned (water-packed) fish; small amounts of 100% grass-fed beef, wild game, lamb, free-range organic poultry | Wild-caught finfish: black cod, cod, halibut, salmon, sardines | Canned meats, cold cuts, eggs, frankfurters, pork, sausage, shellfish
Vegetable Proteins: Legumes | All beans (except soy), peas, lentils | Adzuki beans, black-eyed peas, hummus, lentils, mung beans | All soy products, including edamame, soy milk, tempeh, tofu
Fats | Almond, coconut, flaxseed, grapeseed, extra-virgin olive, pumpkin, safflower, sunflower, sesame, and walnut oils | Coconut oil, grapeseed oil, extra-virgin olive oil, olives | Butter, margarine, mayonnaise, processed and hydrogenated oils, shortening, spreads
Nuts and Seeds | Almonds, cashews, walnuts; chia, pumpkin, sesame, and sunflower seeds; butters made from these nuts and seeds, tahini | Almonds, chia seeds, pumpkin seeds, walnuts | Peanuts, peanut butter
Spices, Condiments, and Confections | All spices, vinegar, mustard (grain-free, made from mustard seed and vinegar) | Cinnamon, garlic, ginger, oregano, rosemary, turmeric | Barbecue sauce, chocolate and chocolate sauce, chutney, cocoa, ketchup, relish, soy sauce, other condiments
Dairy and Milk Substitutes | Unsweetened almond, coconut, hemp, rice, and other nut or seed milks | Unsweetened almond, coconut, and hemp milks | Butter, cheese, cottage cheese, cow's milk, cream, frozen yogurt, ice cream, yogurt
Beverages | Filtered or distilled water, herbal tea, seltzer or mineral water | Herbal tea, water | Alcohol, coffee, all caffeinated and/or sweetened beverages
Sweeteners | Agave nectar, blackstrap molasses, coconut palm sugar, fruit sweetener, honey, pure maple syrup, stevia | Blackstrap molasses, coconut palm sugar, raw honey | Aspartame, corn syrup, evaporated cane juice, high-fructose corn syrup, refined sugar (white or brown), sucralose, other artificial sweeteners
Getting the Most From Your Cleanse
Here are some tips for a successful 21-day cleanse:
1. Eat breakfast within one hour of rising.
2. Eat a small meal or snack every three to four hours.
3. Put your eating emphasis on vegetables (focus on variety and lots of color!).
» Beets (raw, roasted, steamed)
» Cruciferous vegetables (broccoli, brussels sprouts, cabbage, cauliflower, kale)
» Garlic and onions
» Leafy greens (beet greens, chard, collard greens, dandelion greens, kale, spinach)
4. Eat anti-inflammatory sources of protein.
» Fish three to four times per week
» Vegetarian meals two to three times per week
» Other meat sources not more than one or two times per week (grass-fed beef, buffalo/bison, lamb, free-range organic poultry)
5. Eat plenty of good, healthy fats.
» Avocados
» Flaxseed oil
» Olives
» Raw nuts and seeds (lightly roasted is OK)
6. Eat at least five foods from the "best bets" column every day.
7. Stay well hydrated! Aim for sixty-four ounces of water and/or herbal tea every day (more if you're exercising).
8. Eat slowly and mindfully, focusing on the act of self-nourishment. Avoid watching TV, working on the computer, and other distractions while eating. Take three deep breaths before you start eating.
Menu Plans and Shopping Lists
The most challenging part of doing a cleanse (and eating healthy in general) is often the planning and prep. There are several ways you can approach this, and I always tell my clients to experiment until they find a routine that works within the context of their lives. The same goes for you. I've created a few different menus and accompanying shopping lists to accommodate a variety of eaters, including:
1. MENU PLAN 1: The Variety Lover—Perfect for the person who tires of leftovers quickly and enjoys experimenting with new recipes as part of the daily routine.
2. MENU PLAN 2: The Repeater—This menu is great for the person who says "I could eat the same thing every day, no problem!" It still adds some variety, but this menu requires far less cooking and fewer ingredients.
3. MENU PLAN 3: The Grain Avoider—If you feel like you tend to do better with no grains in your diet, this menu is for you. It's well-balanced yet completely devoid of all grains (even gluten-free ones).
You can choose the menu type that appeals most to you, or experiment with all three menus over the course of the 21-day cleanse. You can also use the menus to help guide you in your meal planning beyond the cleanse. You'll notice that balance and variety are key elements of menu planning. Snacks generally include a complex carbohydrate along with some type of protein and/or healthy fat. All meals have a healthy dose of fruits or vegetables along with some kind of protein. What follows are a few dos and don'ts to help make your planning and food prep easier:
DO
» Choose foods you enjoy
» Make a plan before you shop
» Shop once or twice a week
» Carefully read ingredient lists on packaged foods
» Wash, chop, and prep your foods as you're putting away the groceries
» Use shortcuts (e.g. canned beans or pre-washed and chopped veggies)
» Make a couple recipes on your day off and freeze extra portions
» Pack snacks to take to work or to have in the car
» Let servers know what foods you're avoiding when dining out and ask for guidance
» Eat when you're hungry
» Eat slowly and mindfully, savoring every bite
DON'T
» Feel like you need to cook something new at every meal (unless you love to cook!)
» Allow yourself to get overly hungry
» Get caught up in calorie counting and restricting
» Assume that foods labeled as "gluten-free" or "all natural" are healthy by default
» Rely too heavily on rice or rice-based products for convenience—the objective is to eat lots of colorful, fresh foods
» Ignore your body's natural cues
Storage Tips
» Use a salad spinner to dry lettuce, wrap it in paper towels, and store in the crisper (no bag necessary).
» Store chopped carrots and celery in an airtight container full of water.
» Store fresh herbs in a small vase or glass that is half full of water (like a little bouquet!) and refrigerate.
» Slice wet vegetables, such as cucumbers, as needed instead of in advance.
» Use Debbie Meyer GreenBags to extend the life of your produce by a few days. The bags can be rinsed and reused multiple times.
» Invest in glass storage containers and get rid of the plastic.
» Reuse glass jars to store and freeze portions of soups and stews.
MENU PLAN #1: THE VARIETY LOVER
| MONDAY | TUESDAY | WEDNESDAY
---|---|---|---
BREAKFAST | Berry Green Power Smoothie | Mango Muesli with Brazil Nut Topping | Berry Green Power Smoothie
SNACK | Apple and 1 tablespoon almond or cashew butter | ¼ cup pumpkin seeds | Pear and ¼ cup walnuts
LUNCH | Rainbow Quinoa with Roasted Asparagus and Adzuki Beans | Fish Taco Salad with Strawberry Avocado Salsa | Rainbow Quinoa with Roasted Asparagus and Adzuki Beans
SNACK | White Bean and Kalamata Olive Hummus and vegetables | Creamy Avocado Spinach Dip and vegetables | Anti-Inflammatory Trail Mix
DINNER | Poached White Fish with Mango Lime Chutney | Hearty Mushroom and Lentil Stew | Hearty Mushroom and Lentil Stew
MENU PLAN #1 CONTINUED: THE VARIETY LOVER
| THURSDAY | FRIDAY | SATURDAY | SUNDAY
---|---|---|---|---
BREAKFAST | Mango Muesli with Brazil Nut Topping | Berry Green Power Smoothie | Sweet or Savory Quinoa Crepes | Sweet Potato Hash with Lamb Sausage
SNACK | Apple and 2 tablespoons White Bean and Kalamata Olive Hummus | Anti-Inflammatory Trail Mix | Nutty Coconut Energy Truffles | Berry Green Power Smoothie
LUNCH | Wilted Kale Salad with Shredded Beets and Carrots | Super Greens Salad with Pomegranate and Toasted Hazelnuts and leftover salmon | Super Greens Salad with Pomegranate and Toasted Hazelnuts | Leftover kale and chard with canned skipjack tuna
SNACK | Artichoke and Basil Tapenade and celery | 1 cup grapes and ¼ cup pistachios | Artichoke and Basil Tapenade and celery | Crispy Curried Chickpeas
DINNER | Salmon en Papillote with Silky Celery Root Puree | Portobello Mushrooms with Samosa Filling and Wilted Kale Salad with Shredded Beets and Carrots | Pan-Fried Sardines with Sautéed Kale and Chard | Puttanesca-Style Beans and Greens
SHOPPING LIST FOR MENU PLAN #1
Fresh Produce
Apples, 2
Arugula, 5 cups
Asparagus, 1 pound
Avocados, 3
Banana, 1
Beets, 2
Carrots, 6
Celeriac (celery root), 1 medium
Celery, 1 head
Chard, 3 bunches
Cremini mushrooms, ½ pound
Garlic, 2 heads
Garnet yams, 3
Ginger, 1 (1-inch) piece
Grapes, 1 bunch
Green onions, 2 bunches
Kale, 4 bunches
Lemons, 3
Limes, 2
Mango, 1
Pear, 1
Pomegranate, 1
Portobello mushrooms, 3
Red onion, 1
Shallots, 3 small
Shiitake mushrooms, 1 pound
Spinach, 6 cups
Strawberries, 3
Sunchokes (Jerusalem artichokes), 4
Yellow onions, 3
Frozen Foods
Mango, 1 cup chopped
Mixed berries, 1 cup
Peas, ¾ cup
Fish, Meat, and Poultry
Anchovies, 4, or anchovy paste, 1 to 2 teaspoons
Lamb sausages, 2 (I recommend Uli's)
Light fish (cod, halibut, or red snapper), 1½ pounds
Salmon, 1½ pounds
Sardines, 1 pound fresh
White fish (black cod, cod, or halibut), 1½ pounds
Cereals, Grains, and Flours
Brown rice flour, ¾ cup
Oats, 1½ cups certified gluten-free
Quinoa flour, 2 cups
Rainbow quinoa, 1 cup
Nuts, Seeds, and Dried Fruits
Almond butter, ½ cup
Almonds, 1 cup raw
Blueberries, ½ cup freeze-dried
Brazil nuts, ½ cup
Cashews, ¾ cup raw
Cherries, ½ cup dried
Chia seeds, ¼ cup
Dates, ½ cup pitted
Hazelnuts, ¾ cup
Mangoes, ½ cup dried (unsweetened and unsulfured)
Pine nuts, ¼ cup
Pistachios, 1½ cups shelled
Pumpkin seeds (pepitas), 1½ cups shelled
Tahini (sesame seed paste), 1 tablespoon
Walnuts, 2 cups
Herbs and Spices
Basil, 4 to 5 large leaves or 2 teaspoons dried
Bay leaf, 1
Black peppercorns
Cardamom, ½ teaspoon ground
Cayenne pepper, ⅛ teaspoon
Celery seed, 1 teaspoon
Cilantro, 1 bunch
Cinnamon, 2 tablespoons ground
Cumin, 1 tablespoon ground
Curry powder, 2 teaspoons
Fenugreek, ½ teaspoon dried
Mint, 4 leaves
Nutmeg, ½ teaspoon ground
Oregano, 1 tablespoon fresh, or 2 teaspoons dried
Parsley, 1 bunch
Sage, 1 teaspoon dried
Sea salt
Thyme, 1 tablespoon fresh, or 2 teaspoons dried
Turmeric, 1 teaspoon ground
Oils and Vinegars
Balsamic vinegar, 2 tablespoons
Champagne vinegar, 1 tablespoon
Coconut oil, ¼ cup
Extra-virgin olive oil, 1 (16-ounce) bottle
Grapeseed oil, 1 (12-ounce) bottle
Sunflower oil (optional), 1 tablespoon
Sweeteners
Agave nectar, 3 teaspoons
Honey, 2 teaspoons
Maple syrup, 2 tablespoons
Other
Adzuki beans, 1 (15-ounce) can
Almond extract, ⅛ teaspoon
Almond milk, 1 cup unsweetened
Artichoke hearts marinated in olive oil, 1 (14-ounce) jar
Baby lima beans, 1 cup dried
Baking powder, 2 tablespoons
Baking soda, ½ teaspoon
Black beans, 1 (15-ounce) can
Cannellini or other white beans, 1 (15-ounce) can
Capers, 2 teaspoons
Chickpeas (garbanzo beans), 1 (15-ounce) can
Coconut, ½ cup unsweetened and shredded
Coconut milk, 1 cup unsweetened
Cooking sherry, 2 tablespoons
French lentils, ½ cup dried
Golden raisins, ¼ cup
Green olives, 1½ cups
Hemp milk (optional), ½ cup unsweetened
Kalamata olives, 3 cups pitted
Kombu (optional), 1 (2-ounce) package
Mushroom broth, 1 quart
Vegetable broth, 6 cups
NOTE: Add desired Sweet or Savory Quinoa Crepe fillings to shopping list.
MENU PLAN #2: THE REPEATER
| MONDAY | TUESDAY | WEDNESDAY
---|---|---|---
BREAKFAST | Berry Green Power Smoothie | Mango Muesli with Brazil Nut Topping | Berry Green Power Smoothie
SNACK | Apple and 1 tablespoon almond or cashew butter | Pear and ¼ cup walnuts | Apple and 1 tablespoon almond or cashew butter
LUNCH | Mediterranean White Bean Soup | Mediterranean White Bean Soup | Mediterranean White Bean Soup
SNACK | Artichoke and Basil Tapenade and vegetables | ¼ cup pumpkin seeds | Artichoke and Basil Tapenade and vegetables
DINNER | Hazelnut-Encrusted Halibut with Dipping Sauce | Toasted Pecan Quinoa Burgers | Toasted Pecan Quinoa Burgers
MENU PLAN #2 CONTINUED: THE REPEATER
| THURSDAY | FRIDAY | SATURDAY | SUNDAY
---|---|---|---|---
BREAKFAST | Mango Muesli with Brazil Nut Topping | Berry Green Power Smoothie | Power-Packed Granola with Currants and Chia Seeds and unsweetened almond milk | Power-Packed Granola with Currants and Chia Seeds and unsweetened almond milk
SNACK | Pear and ¼ cup walnuts | Apple and 1 tablespoon almond or cashew butter | Mixed berries and ¼ cup pumpkin seeds | Mixed berries and ¼ cup pumpkin seeds
LUNCH | Spring Pea and Jicama Salad | Spring Pea and Jicama Salad | Wilted Kale Salad with Shredded Beets and Carrots and Caramelized Carrot Soup | Wilted Kale Salad with Shredded Beets and Carrots and Caramelized Carrot Soup
SNACK | ¼ cup pumpkin seeds | Artichoke and Basil Tapenade and vegetables | Crispy Curried Chickpeas | Crispy Curried Chickpeas
DINNER | Pumpkin Coconut Curry with White Fish | Pumpkin Coconut Curry with White Fish | Spaghetti Squash Primavera with Basil Walnut Pesto | Spaghetti Squash Primavera with Basil Walnut Pesto
SHOPPING LIST FOR MENU PLAN #2
Fresh Produce
Apples, 2
Avocados, 2
Baby spinach or baby kale, 2 cups
Banana, 1
Beets, 2
Broccoli, 2 heads
Carrots, 14
Celery, 1 head
Chard, 1 bunch
Cremini mushrooms, 1 pound
Fresh spinach, ½ pound
Garlic, 1 head
Ginger, 1 (1-inch piece)
Jicama, 1 small
Kaffir lime leaves, 5 (or buy 2 limes for zest and juice if lime leaves are not available)
Kale, 2 bunches
Lemon, 1
Pears, 2
Radishes, 10
Red onion, 1 small
Scallions or green onions, 1 bunch
Spaghetti squash, 1 small
Yellow onions, 2 small
Frozen Foods
Mangoes, 1 (10-ounce) bag
Mixed berries, 1 (10-ounce) bag
Petite peas, 3 (10-ounce) bags
Fish, Meat, and Poultry
Halibut, 1½ pounds
White fish (black cod, cod, or halibut), 1 pound
Cereals, Grains, and Flours
Brown rice, ¼ cup
Brown rice flour, ¼ cup
Oats, 5½ cups certified gluten-free
Quinoa, 1 cup
Nuts, Seeds, and Dried Fruits
Almond butter, ½ cup
Brazil nuts, ½ cup
Cashews, ¼ cup raw
Chia seeds, ½ cup
Currants, ½ cup dried
Hazelnuts, 1 cup
Pecans, ¾ cup
Pine nuts, ¼ cup
Pumpkin seeds (pepitas), 1 cup shelled
Sesame seeds, ¼ raw
Sunflower seeds, ¾ cup
Walnuts, 1½ cups
Herbs and Spices
Basil, 4 to 5 leaves and 1 teaspoon dried
Black peppercorns
Cardamom, ½ teaspoon ground
Cayenne pepper, ¼ teaspoon
Cinnamon, 1 teaspoon ground
Cumin, 3 teaspoons ground
Curry powder, 2 teaspoons
Dill, 1 teaspoon dried
Fenugreek, ½ teaspoon dried
Italian herbs, ½ cup dried
Mint, 4 leaves
Nutmeg, ½ teaspoon ground
Oregano, 1 teaspoon dried
Sea salt
Turmeric, 1 teaspoon ground
Oils and Vinegars
Apple cider vinegar, 1 tablespoon
Balsamic vinegar, 2 tablespoons
Coconut oil, ¾ cup
Extra-virgin olive oil, 1 (16-ounce) bottle
Grapeseed oil, 3 tablespoons
Sweeteners
Agave nectar, ¼ cup
Coconut palm sugar, ½ cup
Honey, 1½ teaspoons
Other
Artichoke hearts marinated in olive oil, 1 (14-ounce) jar
Black beans, ½ cup
Cannellini beans, 1 (15-ounce) can
Chickpeas (garbanzo beans), 1 (15-ounce) can
Coconut milk, 2 cups
Dijon or stone-ground mustard, ½ teaspoon
Dill pickles, 1 (8-ounce) jar
Green olives, ½ cup
Hemp or almond milk, ½ cup unsweetened
Kalamata olives, ½ cup pitted
Pumpkin puree, 1 (15-ounce) can
Vegenaise, 1 cup soy-free
Vegetable broth, 4 quarts
MENU PLAN #3: THE GRAIN AVOIDER
| MONDAY | TUESDAY | WEDNESDAY
---|---|---|---
BREAKFAST | Berry Green Power Smoothie | Sweet Potato Hash with Lamb Sausage | Sweet Potato Hash with Lamb Sausage
SNACK | Apple and 1 tablespoon almond or cashew butter | Anti-Inflammatory Trail Mix | Pear and ¼ cup walnuts
LUNCH | Brussels Sprout Slaw and canned sardines | Black-Eyed Pea and Escarole Soup | Brussels Sprout Slaw and canned sardines
SNACK | Black Bean and Artichoke Hummus and vegetables | Creamy Avocado Spinach Dip and vegetables | Black Bean and Artichoke Hummus and vegetables
DINNER | Wilted Kale Salad with Shredded Beets and Carrots and grilled chicken breast | Salmon en Papillote with Silky Celery Root Puree and Wilted Kale Salad with Shredded Beets and Carrots | Zucchini Noodles with Pistachio Pesto and Black Lentils
MENU PLAN #3 CONTINUED: THE GRAIN AVOIDER
| THURSDAY | FRIDAY | SATURDAY | SUNDAY
---|---|---|---|---
BREAKFAST | Salmon lox and sautéed vegetables | Berry Green Power Smoothie | Fresh Berry Parfait with Coconut Cashew Cream | Salmon lox and sautéed vegetables
SNACK | ¼ cup pumpkin seeds | Anti-Inflammatory Trail Mix | Nutty Coconut Energy Truffles | Berry Green Power Smoothie
LUNCH | Black-Eyed Pea and Escarole Soup | Brussels Sprout Slaw and leftover white fish | Hearty Mushroom and Lentil Stew | Hearty Mushroom and Lentil Stew
SNACK | Apple and 1 tablespoon almond or cashew butter | Mixed berries and ¼ cup walnuts | Artichoke and Basil Tapenade and vegetables | Artichoke and Basil Tapenade and vegetables
DINNER | Poached White Fish with Mango Lime Chutney and steamed broccoli | Zucchini Noodles with Pistachio Pesto and Black Lentils | Oven-Roasted Black Cod with Smashed Sweet Peas | Grilled chicken breast with leftover zucchini noodles
SHOPPING LIST FOR MENU PLAN #3
Fresh Produce
Apple, 1
Avocados, 2
Banana, 1
Beets, 2
Brussels sprouts, ½ pound
Carrots, 12
Celeriac (celery root), 1 medium
Celery, 1 head
Chard, 1 bunch
Cremini mushrooms, ½ pound
Escarole, 1 head
Garlic, 1 head
Garnet yams, 2
Ginger, 1 (1-inch) piece
Kale, 4 bunches
Leeks, 3 medium
Lemons, 2
Lime, 1
Mango, 1
Mixed berries, 2 cups
Pear, 1
Portobello mushroom, 1
Purple cabbage, 1 small head
Red onions, 2
Shallots, 3
Shiitake mushrooms, 1½ pounds
Spinach, 4 cups
Sunchokes (Jerusalem artichokes), 4
White or yellow onions, 2 large
Zucchini, 3 medium
Frozen Foods
Butternut squash, 1 cup cubed
Mixed berries, 1 cup
Sweet peas, 2 cups
Fish, Meat, and Poultry
Black cod, 2 pounds
Chicken breasts, 2 (4-ounce) boneless and skinless (free-range organic)
Lamb sausages, 2 (I recommend Uli's)
Salmon, 1½ pounds
White fish (black cod, cod, or halibut), 1½ pounds
Nuts, Seeds, and Dried Fruits
Almond butter, 2 tablespoons
Almonds, 1 cup raw
Blueberries, ½ cup freeze-dried
Cashews, 1¼ cups raw
Cherries, ½ cup dried
Chia seeds, 2 tablespoons
Dates, ½ cup pitted
Golden raisins, ½ cup
Mangoes, ½ cup dried (unsweetened and unsulfured)
Pine nuts, ¼ cup
Pistachios, 1¼ cups shelled
Poppy seeds, 2 teaspoons
Pumpkin seeds (pepitas), 1½ cups shelled
Tahini (sesame seed paste), 1 tablespoon
Walnuts, 2½ cups
Herbs and Spices
Basil leaves, 4 to 5
Bay leaf, 1
Black peppercorns
Celery seed, 1 teaspoon
Cilantro, 1 bunch
Cinnamon, 2 tablespoons ground
Cumin, 1 teaspoon ground
Italian herbs, 1 tablespoon
Mint, 4 leaves
Oregano, 2 tablespoons fresh or 1 tablespoon dried
Parsley, 1 bunch
Sage, 1 teaspoon dried
Sea salt
Sweet paprika, 1 teaspoon
Thyme, 2 tablespoons fresh or 1 tablespoon dried
Oils and Vinegars
Balsamic vinegar, 2 tablespoons
Coconut oil, 3 tablespoons
Cooking sherry, 2 tablespoons
Extra-virgin olive oil, 1 (16-ounce) bottle
Flaxseed oil, 1 tablespoon
Rice wine vinegar, 3 tablespoons
Sunflower oil, 2 tablespoons
Toasted sesame oil, 2 tablespoons
Sweeteners
Agave nectar, 1 tablespoon
Honey, 3 teaspoons
Maple syrup (optional), 2 tablespoons
Other
Almond extract, ⅛ teaspoon
Black beans, 1 (15-ounce) can
Black lentils, ½ cup dried
Black-eyed peas, 1½ cups dried
Coconut milk, ½ cup unsweetened
Coconut, ½ cup unsweetened and shredded
French lentils, ½ cup dried
Green olives, ½ cup
Hemp or almond milk, ½ cup unsweetened
Kalamata olives, ½ cup pitted
Marinated artichoke hearts, 1 (14-ounce) jar
Mushroom broth, 1 quart
Vegetable broth, 3 quarts
Anti-Inflammatory Meal and Snack Ideas
BREAKFASTS
» Gluten-free oatmeal with nut butter, chia seeds, and coconut milk, sweetened with agave nectar or coconut palm sugar
» Sautéed vegetables over rice or quinoa with cashews and sunflower seeds
» Butternut Squash and White Bean Soup
» Hummus, avocado, and vegetables wrapped in a brown rice tortilla
» Smoked salmon with sautéed mushrooms and kale
» Power-Packed Granola with Currants and Chia Seeds with unsweetened almond or hemp milk and fresh fruit
SNACKS
» Hummus and assorted vegetables (carrots, celery, cucumbers, jicama, radishes, snap peas, zucchini)
» Celery spread with Artichoke and Basil Tapenade
» Fruit with nuts or seeds or nut or seed butter (almond, cashew, sesame, sunflower, walnut)
» Shelled pumpkin seeds
» Beans and quinoa with mango salsa
» Sardines
LUNCHES
» Lentil soup served over quinoa or rice
» Bean, greens, and grain bowl: black, adzuki, cannellini, or pinto beans with kale, chard, or spinach over quinoa or rice; add avocado and/or Karam's garlic sauce
» Lettuce wraps of canned skipjack tuna mixed with Soy-Free Vegenaise
» Mediterranean platter of vegetables, hummus, olives, salmon, and marinated mushrooms
» Salad with mixed vegetables, grilled chicken, avocado, and vinaigrette dressing
DINNERS
» Poached salmon with Wild Rice and Roasted Vegetables
» Brown rice tortillas with black or pinto beans, rice, shredded cabbage, shredded zucchini, and avocado
» Lamb, mushroom, and zucchini skewers and Roasted Cauliflower Soup with Gremolata
» Vegetarian stir-fry with toasted sesame oil and fresh ginger
» Bison burgers with oven-roasted sweet potato fries and mixed green salad
» Roasted free-range chicken with brussels sprouts and sweet potatoes
A note on portion control:
As a nutritionist who encourages mindful, intuitive eating, I am not a huge proponent of weighing and measuring every morsel of food that goes into your mouth. Generally speaking, when my clients start eating a variety of whole foods and limit or omit processed foods, portion control issues start to disappear. If you're obeying your hunger and satiety cues and eating slowly and mindfully that might be enough for you to self regulate. With that being said, it's quite possible to have too much of a good thing. There are some perfectly healthy whole foods that are calorically dense and portion awareness may be useful. Following are some serving size guidelines for some of those foods:
» Nuts and seeds = ¼ cup
» Nut and seed butters = 1 tablespoon
» Avocado = ¼ of a large avocado
» Grains (rice, quinoa, etc.) = ½ cup cooked
» Oatmeal = ¾ cup cooked
» Beans = ½ cup
» Hummus = ¼ cup
PHASE III: REINTRODUCTION
After enjoying three full weeks of clean eating and blissful nourishment, you might be feeling so good that you don't want to change a thing. While it's a good sign that you're still feeling enthusiastic about healthy eating as you approach the end of the cleanse, it's also important to remember that some of the foods you omitted during Phase II are perfectly great options. Let's take soy for example: If it's well tolerated and you plan to follow a more plant-based diet, whole soy foods like tofu, tempeh, and edamame can be a great way to get protein, calcium, and a host of other phytonutrients.
There are a couple options for reintroducing foods during this third phase. You can choose a less structured, slow reintroduction of the foods that you've been avoiding, or you may elect to do more formal food challenges. The latter is designed for those who are suspicious they might have specific food allergies or intolerances. When my clients have severe digestive disorders or skin issues that completely resolve during a cleanse, I typically suggest that they do formal food reintroductions to pinpoint which food or foods might be causing the symptoms.
Regardless of the option you choose, remember that you want to continue to build on your success and keep some of the healthy habits you've adopted. You also want to avoid shocking your squeaky clean system by bombarding your body with everything you've been avoiding in one fell swoop. It's generally not a good idea to celebrate the end of a cleanse with a large meat pizza, a pitcher of beer, and an ice cream sundae.
Option 1: Gentle Food Reintroduction (Less Structured)
1. Gradually ease foods back into your diet while holding strong to healthy eating habits (such as eating more vegetables and fresh, whole foods).
2. Try to limit food reintroductions to one new food group per day, and keep portion sizes reasonable (e.g., challenge dairy by having some plain yogurt with fruit and see how that goes).
3. Keep a journal to log any changes in how you're feeling as you introduce foods back into your diet. If there are any foods that aggravate your system in some way, this will help you connect the dots.
Option 2: Formal Food Reintroductions (More Structured)
1. Decide which food group to reintroduce first:
» Wheat/gluten
» Dairy
» Eggs
» Soy
» Corn
» Peanuts
» Citrus
» Nightshades (eggplant, peppers, potatoes, tomatoes)
» Optional: Caffeine, sugar, and alcohol (challenge these substances if you want to see how sensitive you are, but generally it's just good practice to enjoy them in moderation, if at all)
2. Eat two or three average-size portions of a pure form of foods from that group through the course of one day. A pure form would mean that the food does not have additives or other ingredients that you have been omitting from your diet (e.g., sugar). Some examples of pure foods from some of the groups:
» Wheat/gluten—Whole wheat tortilla (read ingredients), whole wheat pasta
» Dairy—Milk, cheese without added color or flavor, plain yogurt
» Soy—Edamame, tempeh, plain soy milk (Eden or other brand that contains only filtered soybeans and water)
3. After one day of eating from that food group, remove it from your diet again. You will keep this food group out of your diet through the end of the reintroduction phase, regardless of your reaction. Observe how you feel for two days, which gives you time to notice both immediate and delayed reactions. Use a journal or notebook to record the foods you're challenging and record any potential reactions—write down anything that is at all different from when you were in the full elimination phase of the diet. Examples of potentials reactions include:
» Skin irritations or break outs
» Gas, bloating, or abdominal pain
» Diarrhea or constipation
» Headache
» Fatigue, depression, or anxiety
» Muscle or joint pain
4. If you don't have any symptoms after two days, reintroduce the next food group. Remember that you are challenging each group individually, so be sure to remove the food group from your diet after challenging it even if you have no reaction until you've completed all food reintroductions. If you do have symptoms after challenging a food, stop eating that food and allow the symptoms to clear completely before starting the next challenge.
5. Repeat steps two through four for each food group.
PHASE IV: TRANSITION TO LONG-TERM ANTI-INFLAMMATORY EATING
Once you've completed your 21-day nutritional cleanse and made your way through the food reintroductions, you'll want to identify habits that are worth continuing, hopefully for a lifetime. I always tell my clients that this shouldn't be viewed as a short-term "Hollywood cleanse." The goal is set your body straight and stay on the path to wellness. If you find you're in a rush to return to some favorite inflammatory foods (processed, sugary treats or excess meat), just remind yourself of the many ways in which you benefited from a clean diet: more energy, a change in body composition, less achiness and joint pain, a glowing complexion, and the like.
Here are some tips to help you stay on track post-cleanse:
» Make a weekly or monthly food plan—healthy eating doesn't just happen!
» Follow Michael Pollan's advice: "Eat food. Mostly plants. Not too much."
» Eat twice as many vegetables as fruits.
» Tame that sugar beast (mostly by avoiding refined sugar).
» Choose whole grains over avoid highly processed flour products.
» Reduce your consumption of processed foods by packing your own lunches and snacks.
» Find joy in movement! Be creative with exercise.
» Take several moments each day to breathe, relax, and regroup.
» Create a bedtime ritual and aim for seven to nine hours of sleep every night.
» Be kind, patient, and loving toward yourself. Change is not easy!
# Breakfasts
Breakfast Burrito with Chickpeas and Avocado
Smoked Salmon and Avocado Tartine
Breakfast Rice with Crumbled Nori
Sweet Potato Hash with Lamb Sausage
Sweet or Savory Quinoa Crepes
Mango Muesli with Brazil Nut Topping
Power-Packed Granola with Currants and Chia Seeds
Fresh Berry Parfait with Coconut Cashew Cream
Berry Green Power Smoothie
What your mother always told you about breakfast is true. It really is the most important meal of the day. It brings you out of a fasting state, helps stabilize your blood sugar, and gives you the energy you need to face the day. Sadly, breakfast is still the most neglected meal. Busy schedules, sleep deprivation, lack of hunger cues, and a deficit of creative breakfast ideas are just a few of the reasons people give for skipping breakfast.
Well, friends, I'm happy to tell you that a little bit of planning and a few good recipes can help transform the chronic breakfast avoider into someone who springs out of bed in anticipation of that morning meal. If you're just getting used to the idea of having breakfast, start small. You might even find some of the snack ideas in the next chapter more appealing. If you just want to improve the quality of your breakfast, be sure to include some protein, complex carbs, and healthy fats in each morning meal. If you can start the day with a savory breakfast that includes vegetables, you'll be off to a winning start!
Breakfast Burrito with Chickpeas and Avocado
If you find yourself in a time crunch, and cooking in the morning is out of the question, this is the ideal savory breakfast. The chickpea-avocado combo makes a delicious burrito filling, and you can amp up the nutritional profile by adding your favorite greens. The sunflower seeds provide some crunch and a little extra protein. If you have leftover filling, you can have it for breakfast the next morning or use it as a dip. The avocado will discolor, but the taste will be just fine.
Makes 2 servings
1 (15-ounce) can of chickpeas, rinsed and drained
1 avocado
1 tablespoon freshly squeezed lemon juice
1 teaspoon sea salt
½ teaspoon ground cumin
½ teaspoon ground turmeric
2 tablespoons sunflower seeds
2 brown rice or teff tortillas
1 cup arugula, watercress, or microgreens
Put the chickpeas in a medium mixing bowl. Scoop the avocado into the bowl. Add the lemon juice, salt, cumin, and turmeric. Mash the mixture with a fork until the avocado is well incorporated. Allow some of the chickpeas to remain whole. Stir in the sunflower seeds.
Just before serving, sprinkle the tortillas with water and heat in the microwave for 20 seconds. Scoop half of the chickpea mixture into the center of each tortilla. Top with the arugula and roll up the tortillas like burritos.
The unique combination of healthy fats and phytosterols in avocados are what make them one of the top anti-inflammatory fruits. They are particularly useful in cooling down inflammation related to arthritis.
Smoked Salmon and Avocado Tartine
Smoked Salmon and Avocado Tartine
There's nothing like a savory open-faced sandwich after a long red-eye flight from coast to coast. I was nearly falling asleep in my latte at a French café in New York City, when I spotted a breakfast tartine amidst all the crepes and croissants on the menu. I felt myself come back to life as I polished off the tartine, and it inspired me to think up an endless combination of sandwich ingredients that would be suitable for breakfast.
If this breakfast can resurrect a person with severe jet lag, imagine what it will do for the average early riser. I refer to this tartine as the breakfast of champions because it's the perfect combination of carbohydrate, protein, and healthy fat.
Makes 4 servings
1 avocado
1 tablespoon freshly squeezed lime juice
1 teaspoon ground cumin
½ teaspoon sea salt
4 slices gluten-free bread, cut into quarters
8 ounces smoked salmon or lox
1 cup alfalfa sprouts or microgreens
4 radishes, thinly sliced
Scoop the avocado into a medium bowl. Add the lime juice, cumin, and salt and mash with a fork until the ingredients are well combined but the mixture is still chunky.
Arrange 4 bread pieces on each serving plate. Spread a layer of avocado mixture on each piece. Top with ½ ounce salmon. Stack some sprouts on the salmon and garnish with radish slices.
Salmon is an excellent source of anti-inflammatory omega-3s. Avocados are loaded with specific phytonutrients that block the inflammatory response and can decrease arthritic symptoms.
Breakfast Rice with Crumbled Nori
A recipe that naturopath Dr. Katherine Oldfield gives her patients inspired this quick and easy breakfast. It also happens to be a great way to clean out the refrigerator when you have leftover rice and soon-to-expire veggies in the produce bin. Anything goes in this breakfast sauté, so don't be afraid to add more vegetables and a different combination of nuts and seeds. It's also delightful when topped with a poached egg or some leftover salmon.
Makes 4 servings
1 tablespoon raw sesame seeds
2 teaspoons coconut oil
2 cloves garlic, crushed
1 shallot, chopped
6 cremini or button mushrooms, chopped
1 teaspoon sea salt
1 cup roughly chopped kale
1½ cups cooked brown rice (leftover rice works great!)
2 teaspoons toasted sesame oil
2 teaspoons mirin
¼ cup chopped raw cashews
2 tablespoons crumbled nori
Place a large sauté pan over medium heat. Add the sesame seeds to quickly toast until lightly browned, about 1 minute. Add the oil, garlic, and shallot and cook, stirring occasionally, until soft and fragrant, 3 to 4 minutes. Add the mushrooms and salt and sauté until tender, about 3 minutes. Fold in the kale and continue stirring until it starts to wilt, 3 to 4 minutes.
Stir in the rice, sesame oil, and mirin and cook until the rice is heated through, about 2 minutes. Top with the cashews and nori.
Sweet Potato Hash with Lamb Sausage
Brunch is one of my absolute favorite meals to host, and this dish is a hit every time I serve it. The key is thinly slicing the onions and leaving them in half rings so that they caramelize perfectly and create an amazing texture. You can use any type of sausage, but I much prefer using lamb. Not only does it have an impressive nutritional profile, but it also lends rich, complex flavors and blends perfectly with the sweetness of the garnet yams.
Makes 6 servings
2 lamb sausages (I recommend Uli's)
1 large white or yellow onion, thinly sliced into half rings
1 to 2 teaspoons sunflower oil (optional)
1 cup chopped cremini mushrooms
2 cloves garlic, minced
2 unpeeled garnet yams, cut into ¼-inch cubes
1 cup finely chopped kale
1 tablespoon fresh thyme
1 tablespoon fresh oregano
1 teaspoon ground sage
1 teaspoon sea salt
1 teaspoon freshly ground black pepper
Preheat the oven to 375 degrees F.
Squeeze the lamb out of its casing into a large cast-iron skillet over medium heat. As it cooks, break the lamb into small pieces with a spatula and sauté until it begins to brown, about 4 minutes. If using a leaner sausage, such as chicken, you may need to add some grapeseed or sunflower oil to the pan. With a slotted spoon, transfer the sausage to a bowl.
Add the onion to the skillet and sauté until it begins to caramelize, 4 to 5 minutes. Drizzle in sunflower oil as needed. Add the mushrooms and garlic and cook until softened, about three minutes. Stir in the yams, kale, thyme, oregano, sage, salt, and pepper and return the sausage to the skillet.
Transfer the skillet to the oven and roast for 20 to 25 minutes, or until the yams can be easily pierced with a fork.
Sweet or Savory Quinoa Crepes
One of my preferred pastimes is to chef it up with my good friend and mentor Barb Schiltz. Barb happens to be a masterful nutritionist and a longtime crepe lover, so we worked on this creation together and came up with a delicate but hearty crepe recipe. I prefer savory crepes and Barb prefers sweet, so I offer variations for both. Unless you have a special crepe pan, I find it easier to make smaller crepes (about three inches in diameter) and then stack the crepes with layers of filling in between.
Makes 6 servings
2 cups quinoa flour
2 tablespoons baking powder
½ teaspoon baking soda
¼ teaspoon sea salt
½ cup raw cashews
2 tablespoons chia seeds
1¾ cups water
1 cup unsweetened almond milk
2 tablespoons coconut oil
1 teaspoon freshly squeezed lemon juice
1 teaspoon maple syrup
Nonstick olive oil cooking spray
SWEET FILLING OPTIONS
2½ cups fresh or frozen berries (thawed if frozen)
2½ cups stewed apples or pears with cinnamon
2 cups cashew cream (1½ cups raw cashews blended with ½ cup water)
2½ cups mangoes blended with coconut milk
SAVORY FILLING OPTIONS
2½ cups sautéed onions, mushrooms, and spinach
2 cups hummus and 1 sliced avocado
2 ounces lox and 1 tablespoon capers
Whisk the flour, baking powder, baking soda, and salt in a medium mixing bowl and set aside. In a food processor or blender, grind the cashews and chia seeds until finely ground. Add the water, almond milk, oil, lemon juice, and maple syrup and blend for 2 to 3 minutes. Add the mixture to the dry ingredients, stirring until well blended. The batter should be the consistency of olive oil. Add more water to thin if necessary.
Spray a medium nonstick pan, cast-iron skillet, or crepe pan with oil. Put 2 to 3 tablespoons of the crepe batter in the pan and swirl around until there is a thin layer across the bottom. Cook each crepe for 1 minute per side. Fill with the desired filling.
Mango Muesli with Brazil Nut Topping
There's nothing better than waking up to find breakfast waiting for you in the fridge (except for maybe breakfast being served to you in bed!). I much prefer the texture of soaked oats to that of cooked porridge. The combination of mangoes with coconut milk gives this hearty cereal an exotic, tropical flair. And although Brazil nuts are commonly the ones that get left behind in the mixed nut dish, they're loaded with nutrients and good, healthy fats, making them a great topping for your muesli! Traditional muesli is served with yogurt; if you're avoiding dairy, you could try coconut, almond, or soy yogurt. It's also delicious with just the coconut milk.
Makes 4 servings
1½ cups certified gluten-free oats
2 cups water
1 cup coconut milk, plus more for serving
1 cup chopped mango (thawed frozen mango is OK)
1 teaspoon ground cinnamon
½ teaspoon ground nutmeg
½ teaspoon ground cardamom
½ cup roughly chopped Brazil nuts
In a large bowl with a lid, combine the oats, water, coconut milk, mango, cinnamon, nutmeg, and cardamom and stir well. Cover and place in the refrigerator for at least 4 hours or overnight. Serve with additional coconut milk or yogurt and sprinkle with Brazil nuts.
Brazil nuts are a rich source of anti-inflammatory unsaturated fats. They're also an excellent source of selenium, a powerful antioxidant that helps optimize thyroid function.
Power-Packed Granola with Currants and Chia Seeds
Power-Packed Granola with Currants and Chia Seeds
Part of my regular Sunday ritual is to make this granola and pop it into the oven first thing in the morning. I sit down with a cup of tea, watch the news, and take in the mouthwatering aroma that's reminiscent of oatmeal cookies. The timer dings, I salivate like I'm in a Pavlovian experiment, and then I enjoy a nice, warm bowl of granola. Store-bought granola is often loaded with sugar, but this recipe uses low-glycemic sweeteners and is high in protein. If you're a cereal lover, you can feel good about starting your day with this power-packed granola.
Makes 6 servings
4 cups certified gluten-free rolled oats
3 tablespoons chia seeds (I recommend Qia—a blend of chia, buckwheat, and hemp seeds)
¼ cup coconut palm sugar
¼ cup agave nectar
¼ cup coconut oil
2 tablespoons almond butter
½ cup dried currants
OPTIONAL ADDITIONS
½ cup chopped pecans
½ cup shelled pumpkin seeds (pepitas)
¾ cup unsweetened coconut flakes
½ cup raisins
½ cup dried cherries
Preheat the oven to 300 degrees F and line a baking pan with parchment paper.
In a large bowl, combine the oats, chia seeds, sugar, agave, oil, and almond butter and mix thoroughly. Spread the mixture evenly on the pan. Bake for 40 minutes, stirring after 20 minutes.
Allow the granola to cool completely before stirring in the currants and transferring to an airtight container.
Fresh Berry Parfait with Coconut Cashew Cream
Fresh Berry Parfait with Coconut Cashew Cream
I make coconut cashew cream as a fruit dip whenever I host brunch at my house, and people go crazy over it. They're always astonished when I tell them it's made from just cashews and coconut milk. Raw cashews have a wonderfully subtle sweetness and just the right amount of fat. They can easily be transformed into a perfect stand-in for whipped cream—particularly when you add coconut milk! Layering this creamy cashew goodness with fresh berries or mangoes creates a beautiful, satisfying breakfast that is loaded with antioxidants.
Makes 2 servings
1 cup raw cashews
½ cup unsweetened coconut milk
2 teaspoons honey
1 teaspoon ground cinnamon
2 cups berries (blackberries, blueberries, raspberries, strawberries—any combination works!)
Place the cashews, coconut milk, honey, and cinnamon in a food processor. Blend until smooth—the mixture should resemble creamy peanut butter. If it's too thick, slowly drizzle in some water and blend until it reaches the desired consistency.
Scoop two large spoonfuls of the cashew cream into the bottom of a small parfait glass. Add ½ cup of the berries and top with another layer of cashew cream. Finish with another ½ cup berries. Repeat in a second parfait glass.
Tip: The parfaits can be made ahead of time and stored in the refrigerator for 3 or 4 days for a ready-to-grab breakfast or a healthy-yet-decadent dessert.
Berry Green Power Smoothie
Smoothies can be a great way to get several servings of fruits and vegetables without lifting a fork. There's no shortage of smoothie recipes, and it can be fun to experiment with different combinations. Just be sure that you're including a variety of colorful foods, some form of protein, and a little healthy fat to keep you satiated. This smoothie has some anti-inflammatory fresh herbs that help tone down the greens and brighten up the rest of the ingredients. I think it's pure bliss in a glass!
Makes 1 serving
2 cups spinach and/or baby kale
1 cup frozen mixed berries
½ banana
¼ cup raw cashews
4 mint leaves
2 tablespoons chia seeds
¾ cup water
½ cup unsweetened hemp or almond milk
1 teaspoon honey
¼ teaspoon minced fresh ginger
Toss all the ingredients in a blender, blend thoroughly, and enjoy!
Ginger contains anti-inflammatory compounds called gingerols and also has immune-boosting properties, so it's a great safeguard during cold and flu season.
# Healthy Snacks
Black Bean and Artichoke Hummus
Crispy Curried Chickpeas
White Bean and Kalamata Olive Hummus
Shiitake Mushroom and Walnut Pâté
Creamy Avocado Spinach Dip
Artichoke and Basil Tapenade
Anti-Inflammatory Trail Mix
Nutty Coconut Energy Truffles
Tropical Quinoa Power Bars
Forget what you've heard about not snacking between meals. A well-planned snack in the afternoon might be just what the doctor ordered. Not only does it help keep your energy level up, but it also gives you another opportunity to work some nourishing anti-inflammatory foods into your daily diet. And if you avoid a blood sugar crash in the afternoon, you'll be much less likely to reach for the pro-inflammatory foods that are loaded with sugar.
A general rule of thumb is to avoid going more than four hours without eating. You should also be very intentional about fueling your body before workouts and replenishing it afterward. The snacks featured in this chapter are nourishing, well balanced, and can help reduce sugar cravings. I recommend preparing a few snack recipes on the weekend so you've got ready-made options to get you through the week. I also love the idea of setting out a dip with some veggies so the whole family can munch on something healthy while dinner is cooking.
Black Bean and Artichoke Hummus
I have to credit my sister for being the one who introduced me to the idea of adding artichoke hearts to a more traditional black bean dip, and it sure makes things more interesting! Use artichoke hearts marinated in olive oil to give this hummus a smoother finish and a richer flavor. I've also been known to stir in some fire-roasted peppers or sun-dried tomatoes for a slightly different twist on this dip.
Makes 6 servings
1 (15-ounce) can black beans, rinsed and drained
1 (6-ounce) jar artichoke hearts marinated in olive oil
2 cloves garlic, minced
1 tablespoon tahini
3 tablespoons extra-virgin olive oil
1 teaspoon sea salt
Place the beans, artichokes with oil, garlic, and tahini in a food processor. With the machine running, slowly drizzle in the oil and blend to the desired consistency. Season with the salt.
Serve with assorted vegetables, such as carrots, snap peas, cucumbers, zucchini, jicama, or daikon radishes.
Crispy Curried Chickpeas
Crispy Curried Chickpeas
This is such an easy and nutritious snack to make! Chickpeas (also called garbanzo beans) are a good source of protein, and they're rich in iron and potassium. But what you'll notice most is that these crispy chickpeas really hit the spot when you're in the mood for a crunchy, salty snack! You can get creative with the seasonings and experiment with different herbs and spices.
Makes 4 servings
1 (15-ounce) can chickpeas, rinsed and drained
1 tablespoon grapeseed oil
2 teaspoons ground cumin
1 teaspoon ground turmeric
1 teaspoon sea salt
½ teaspoon freshly ground black pepper
½ teaspoon fenugreek
Preheat the oven to 400 degrees F.
Pat the chickpeas dry with a paper towel, then place in a medium bowl. In a small bowl, combine the oil, cumin, turmeric, salt, pepper, and fenugreek and whisk with a fork. Pour the oil mixture over the chickpeas and stir until they're well coated.
Spread the chickpeas on a baking pan and bake for 40 minutes, or until they're golden brown and rattle around the pan. Serve immediately, or cool thoroughly before storing in an airtight container.
Tip: These chickpeas are actually best when eaten shortly after cooking. They tend to lose some of their crispiness after storing unless the container is completely airtight.
White Bean and Kalamata Olive Hummus
Hummus is a great snack, but it's easy to get burned out on the traditional chickpea variety. Using white beans is a simple way to change it up enough that hummus will once again become exciting and new. White beans also have a more neutral flavor than chickpeas, so even those who generally dislike hummus often enjoy this version. White bean hummus can also be a more nutritious substitute for mayo and works great as a spread on vegetarian sandwiches.
Makes 6 servings
1 (15-ounce can) cannellini or other white beans, rinsed and drained
2 cloves garlic
1 tablespoon tahini
3 tablespoons extra-virgin olive oil
¼ cup pitted kalamata olives
1 teaspoon sea salt
Place the beans, garlic, and tahini in a food processor. With the machine running, slowly drizzle in the oil and blend to the desired consistency. Add the olives and pulse until they are just chopped and incorporated. Season with the salt.
Serve with assorted vegetables, such as carrots, snap peas, cucumbers, zucchini, jicama, or daikon radishes. The hummus can be stored in an airtight container in the refrigerator for up to 5 days.
Specific compounds in olives, called polyphenols, have been shown to block inflammatory pathways and reduce specific markers that are used to measure inflammation in the body.
Shiitake Mushroom and Walnut Pâté
If you like the idea of pâté, but are not a fan of liver, this recipe is for you. Toasting the walnuts illuminates their sweetness and tones down the bitter quality of their skins. You can use any type of mushrooms or a combination of different types (such as chanterelle, oyster, or morel). Including butter beans helps create a silkier texture and also adds some fiber and nutrients.
Makes 6 servings
1 cup raw walnuts
1 tablespoon coconut oil
6 shiitake mushrooms, sliced
¾ teaspoon sea salt, divided
½ cup cooked butter beans or other white beans (canned is OK)
1 teaspoon maple syrup
Preheat the oven to 350 degrees F.
Spread the walnuts on a baking pan and bake for about 12 minutes, or until they start to brown and get fragrant. Set aside.
In a small sauté pan, heat the oil. Add the mushrooms, season with ¼ teaspoon of the salt, and cook, stirring occasionally, until the mushrooms are soft and juicy, about 5 minutes.
Combine the walnuts, mushrooms, beans, maple syrup, and remaining ½ teaspoon salt in a food processor and blend until smooth. The mixture will be somewhat thick and will resemble liver pâté.
Serve with assorted vegetables or rice crackers.
Walnuts are one of the few plant-based foods that contain an appreciable amount of omega-3s. They also contain some rare anti-inflammatory nutrients that are found in virtually no other commonly eaten foods.
Creamy Avocado Spinach Dip
Creamy Avocado Spinach Dip
It's hard to imagine what would make guacamole even more perfect than it already is. Enter spinach—a nutritional powerhouse and anti-inflammatory superstar that can happily join the party without interfering with the flavor. It's also kid-friendly, as evidenced by my two-year-old nephew who put his whole face in the bowl to lick it clean once he'd polished off the last of the dip. The kalamata olives are optional, but they add some nice saltiness that completely disguises the spinach.
Makes 6 servings
1 avocado
2 cups spinach
1 clove garlic, chopped
¼ cup olive oil, plus more as needed
2 tablespoons freshly squeezed lemon
1 teaspoon ground cumin
1 teaspoon sea salt
½ cup pitted kalamata olives (optional)
Scoop the avocado into a food processor. Add the spinach, garlic, and oil and blend. Additional oil can be added for a smoother texture. Add the lemon juice, cumin, and salt and blend until smooth. Add the olives and pulse until they are just chopped and incorporated.
Serve with assorted vegetables, such as carrots, cucumbers, cauliflower, or jicama. This dip also works great as a sandwich spread or a topping for bean and rice dishes.
Spinach contains high amounts of unique carotenoids, which are plant compounds that have been shown to calm inflammation, particularly in the digestive tract.
Artichoke and Basil Tapenade
Behold the beautiful olive! While technically a fruit, olives are better known for their fatty, salty, briny characteristics. They also happen to be one of the main components of a Mediterranean diet. I absolutely love the fact that we can get healthy fats and other anti-inflammatory compounds via such a remarkably tasty delivery system. The artichoke hearts add a different texture to the tapenade while the basil really helps balance the saltiness and elevates the flavor. I like to bring this snack to Super Bowl parties with the secret intention of unclogging the arteries of those who are overindulging in deep-fried chicken wings.
Makes 6 servings
1 (14-ounce) jar artichoke hearts marinated in olive oil
½ cup pitted green olives
½ cup pitted kalamata olives
4 to 5 large basil leaves or
2 teaspoons dried
Place all the ingredients in a blender or food processor and pulse until finely chopped but not pureed. Serve with assorted vegetables, such as carrots, cauliflower, or celery.
Tip: For a slightly fancier spin on this recipe, scoop the tapenade into endive leaves and arrange in a circular pattern on a large plate. This is a hit at parties, and it couldn't be easier for the host!
Anti-Inflammatory Trail Mix
Let's face it, trail mix has all the elements of a perfect snack. It's portable, it's satisfying, and it can be fun to eat. Unfortunately, the stuff available in the bulk section can be loaded with bits of candy, chocolate, and sugar-sweetened dried fruit. Sure, that might be what makes trail mix so much "fun," but if you have to mine out the candy and leave the nuts behind, you're really not doing yourself any favors. This homemade version is a delicious and nutritious solution. I encourage families to have a trail-mix-making party where you set out small bowls of nuts, seeds, and freeze-dried or unsweetened, unsulfured dried fruits, and have everyone create their own baggies of trail mix to take to school, work, or soccer practice.
Makes 16 (¼-cup) servings
1 cup raw almonds
1 cup shelled pistachios
1 cup shelled pumpkin seeds (pepitas)
½ cup freeze-dried blueberries
½ cup chopped unsweetened, unsulfured dried mangoes
OPTIONAL ADDITIONS
1 cup walnuts
1 cup pecans 1 cup Brazil nuts
1 cup macadamia nuts
1 cup sunflower seeds
1 cup unsweetened coconut flakes
½ cup raisins
½ cup dried figs
½ cup dried apricots
1 cup freeze-dried strawberries
1 cup freeze-dried raspberries
In a large bowl, combine the almonds, pistachios, pumpkin seeds, blueberries, and mangoes and toss well. Place any optional ingredients in small bowls and set them out to choose from. Combine a little of each desired item in a ¼-cup measuring cup and pour the contents from the cup into snack bags for preportioned, power-packed snacks that are ready to grab and go!
Tip: Active adults and older children (8-plus years) may want to use ½ cup as a serving. Adults watching their weight and younger children should stick with the ¼-cup portion.
Nutty Coconut Energy Truffle
Nutty Coconut Energy Truffles
This recipe is definitely one of my top five greatest hits. The delectable combination of nuts, dried fruit, and coconut make it a universal favorite. I knew I was on to something when I took these yummy little treats to a campout with friends and they were gone within seconds of opening the container. My friends still talk about them today. They're a great high-energy snack to take along when hiking or biking, and they can be enjoyed as a much healthier stand-in for a candy bar. I recommend using Medjool dates because they're always fresh and moist with just the right amount of stickiness to bind these truffles together.
Makes 12 servings
2 cups raw walnuts
1 tablespoon ground cinnamon
⅛ teaspoon sea salt
½ cup pitted dates
½ cup dried cherries
2 tablespoons coconut oil
2 tablespoons almond butter
⅛ teaspoon almond extract
2 tablespoons maple syrup (optional)
½ cup unsweetened shredded coconut
Place the walnuts, cinnamon, and salt in a food processor. Process until the nuts are finely ground, about 1 minute. Add the dates, cherries, oil, almond butter, and extract. Process until well combined; the mixture should have a thick, sticky consistency. Check to see if you can form a truffle by rolling some of the mixture in your hands—if it falls apart easily, blend in the maple syrup.
Spread the coconut on a plate. Scoop the nut mixture with a large spoon and roll into 1-inch balls. Roll in the coconut until the balls are generously coated. Store the truffles in an airtight container in the refrigerator for up to 1 week and store extras in the freezer for up to 6 months.
Cinnamon inhibits the release of pro-inflammatory signals in the body and helps regulate blood sugar.
Tropical Quinoa Power Bars
Clients often ask me for ideas for portable, whole food snacks. While I think of quinoa as a super food, it's not generally known for portability. However, it's possible to take advantage of the sticky quality of freshly cooked quinoa, along with a few other key ingredients, to create a bar that travels well. I love the tropical trio of apricots, coconut, and macadamia nuts (which always seem like a treat!). I choose to go fairly light on the honey because I prefer a more subtle sweetness, but that can certainly be adjusted to taste.
Makes 12 servings
2 cups water
1 cup quinoa, rinsed and drained
¼ teaspoon sea salt
½ cup raw macadamia nuts
½ cup unsweetened shredded coconut
½ cup chopped dried apricots
2 tablespoons honey
2 tablespoons tahini
2 teaspoons ground cinnamon
¼ cup coconut flour
Place the water, quinoa, and salt in a medium saucepan over medium-high heat. Bring to a boil, cover, and reduce the heat to low. Simmer for 20 minutes without stirring. The quinoa will look like it sprouted and grew a tiny tail, and all of the water should be absorbed. Fluff with a fork and allow to cool for about 5 minutes.
In a large bowl, combine 2 cups of the quinoa (you'll have some leftover) with the macadamia nuts, coconut, and apricots. Stir in the honey, tahini, and cinnamon. Cover with the coconut flour and use your hands to knead everything together until well incorporated.
Shape the mixture into bars approximately 3 inches long and 1 inch thick. Place them in a shallow baking dish lined with parchment paper, cover, and refrigerate for 4 hours or overnight.
Tip: I like to wrap these bars in parchment paper, then double wrap in foil, and store them in the freezer. I'll pull one out in the morning and toss it in my workbag, so I can look forward to a satisfying afternoon snack.
# Soups and Stews
Mediterranean White Bean Soup
Roasted Cauliflower Soup with Gremolata
Creamy Asparagus and Sunchoke Soup
Butternut Squash and White Bean Soup
Slow-Cooked Black Bean and Broccoli Stew
Three-Bean Stew with Red Quinoa
Caramelized Carrot and Ginger Soup
Black-Eyed Pea and Escarole Soup
Lentil and Spinach Stew with Roasted Garlic
Vegetable and Chicken Pho
When I think of soups and stews, the first word that comes to mind is nourishing. A steaming bowl of soup can take the chill out of your bones and right wrongs of the world. In traditional Chinese medicine, soups are often described as tonics that have specific healing properties.
Many of the ingredients that are featured in this chapter have particular nutrients with known medicinal properties. So while these soups and stews are warming you up, they're cooling down systemic inflammation and strengthening your immune system.
Most of the recipes in this section make six to eight servings, so I recommend investing in some mason jars or glass storage containers and freezing a few servings for later. This helps prevent same-food fatigue, and you get to stock your freezer with some nourishing options for those nights that you just don't feel like cooking.
Mediterranean White Bean Soup
This Tuscan-inspired white bean soup is my go- to soup for clients who are just starting to practice anti-inflammatory eating. It's easy to make; contains beans and rice, which make a complete protein; and it freezes well, so leftovers can be spread out. I like to serve a small cup of this soup as a starter when I have a lamb dish on the menu, but it's definitely satisfying enough to be a meal all on its own.
Makes 8 servings
1 tablespoon extra-virgin olive oil
1 small red onion, chopped
1 medium green onion, chopped
1 cup sliced mushrooms
2 cloves garlic, minced
¼ cup dried Italian herbs
1 teaspoon sea salt, plus more for seasoning
2 quarts vegetable broth
1½ to 2 cups water
1 (15-ounce) can cannellini beans, rinsed and drained
¼ cup brown rice
3 cups chopped chard or spinach
Freshly ground black pepper
Heat the oil in a large stockpot over medium heat. Add the red and green onions and sauté until they start to sweat and become soft. Add the mushrooms, garlic, Italian herbs, and salt and cook, stirring occasionally, for 2 to 3 minutes, or until the mushrooms start to soften and release liquid.
Add the broth, water, beans, and rice; increase the heat to high; and bring the soup to a boil. Reduce the heat to medium-low and simmer for 45 minutes. Stir in the chard and continue simmering for another 10 to 15 minutes. Season to taste with salt and pepper and serve.
Roasted Cauliflower Soup with Gremolata
I created this soup for a cooking class on healthy hormone balance and was pleasantly surprised at how popular it was with the students. Roasting the cauliflower and leeks with turmeric and cumin really showcases their sweeter characteristics and makes the soup incredibly rich. As delicious as it tastes, the soup is lacking in color and tends to look pretty lonely in bowl without a colorful garnish. The gremolata adds a vibrant punch, and the parsley and lemon zest brighten up the flavor of the soup. If desired, you can use less broth to create more of a puree than a soup.
Makes 4 servings
FOR THE SOUP:
1 leek (white and light green parts only), rinsed well and roughly chopped
1 large head cauliflower, separated into florets
2 tablespoons grapeseed oil
1 teaspoon ground turmeric
1 teaspoon ground cumin
1 teaspoon coarse sea salt
3 cups vegetable broth
½ cup coconut milk
FOR THE GREMOLATA:
½ cup finely chopped parsley
1 clove garlic, minced
1 tablespoon extra-virgin olive oil
1 teaspoon freshly grated lemon zest
Preheat the oven to 400 degrees F.
In a large bowl, toss the leek and cauliflower with the grapeseed oil, turmeric, and cumin. Spread on a baking pan, sprinkle with the salt, and roast for 15 to 20 minutes, or until the leeks start to brown.
Transfer the vegetables to a food processor. Add the broth and coconut milk and blend well, adding more broth or water as needed to achieve the desired consistency. Pour the soup into a saucepan and place over low heat until ready to serve.
To make the gremolata, combine the parsley, garlic, olive oil, and lemon zest in a small bowl and mix well.
Ladle the soup into bowls and top with a spoonful of gremolata.
Creamy Asparagus and Sunchoke Soup
When asparagus makes its debut at my local farmers' market, I know that spring has officially sprung. Time to trade in the winter squash and hearty root vegetables and make way for the lighter, greener foods that help us detox and recharge. Sunchokes, also known as Jerusalem artichokes, are rather unfortunate looking, but it's worth getting past their homely, knobby exterior. They have a slightly nutty, artichoke-like flavor that is subtle when combined with the stronger flavor of asparagus. They can also be shredded into a salad or roasted, so save a few for another day. You'll be pleasantly surprised at the creaminess of this soup without the use of any dairy products—the sunchokes and oats help smooth out the texture and give the illusion of a cream-based soup.
Makes 6 servings
1 tablespoon extra-virgin olive oil
1 leek (white and light green parts only), rinsed well and chopped
3 sunchokes, scrubbed and chopped into small pieces
1 bunch asparagus, ends trimmed, chopped into 1-inch pieces
1 quart mushroom broth
1 cup water
⅓ cup certified gluten-free rolled oats
1 teaspoon sea salt
½ teaspoon freshly ground black pepper
Drizzle the oil into a large saucepan or stockpot. Add the leek, stir until well coated with oil, and sauté over medium heat until soft. Add the sunchokes and asparagus and sauté for 2 to 3 minutes. Add the broth, water, oats, and salt. Bring to a boil over high heat, reduce the heat to medium-low, and simmer for 15 to 20 minutes.
Carefully transfer the soup in batches to a food processor and blend until smooth and creamy. Reheat on the stove top as needed and season with the pepper.
Both asparagus and sunchokes contain inulin, an indigestible form of fiber that acts as a prebiotic and may help with inflammation in the gut.
Butternut Squash and White Bean Soup
Butternut squash soup must rank in the top three favorite soups among adults and kids alike. I'd always found it to be a bit too sweet for my liking, which is why I decided to cut some of the sweetness by adding white beans. And then of course the nutritionist side of me loves the fact that adding beans really increases the fiber, adds essential minerals, and makes the soup heartier and more satiating. Now I'm proud to report that I, too, am a butternut squash soup lover! For dinner parties, this soup is fun to serve in hollowed-out gourds or mini pumpkins. It's festive and looks beautiful on the table.
Makes 6 servings
1 tablespoon extra-virgin olive oil
1 small yellow onion, chopped
1 medium butternut squash, peeled, seeded, and cubed
2 cloves garlic, minced
6 cups vegetable broth
1 (15-ounce) can cannellini or other white beans, rinsed and drained
2 teaspoons sea salt
1 teaspoon ground cumin
½ teaspoon ground coriander
¼ teaspoon cayenne pepper
1 teaspoon freshly ground black pepper
Drizzle the oil into a large saucepan and add the onions, stirring until well coated with oil. Sauté over medium heat until the onions soften and start to become translucent, about 3 to 4 minutes. Add the squash, garlic, broth, beans, salt, cumin, coriander, and cayenne and bring to a boil. Reduce the heat to low, cover, and simmer for 35 to 40 minutes.
Carefully transfer small batches of the soup to a food processor or blender, blend until smooth, and then return to the pan to keep warm. Season with the pepper.
Tip: This soup can be stored in an airtight container in the refrigerator for 4 to 5 days and also freezes well. With leftovers, I like to add fresh spinach or chopped kale and serve it over quinoa, just to change it up.
Slow-Cooked Black Bean and Broccoli Stew
Finding good vegetarian slow cooker recipes can be challenging, which is what inspired this creation. Black beans tend to require a longer cooking time than most beans, so they're perfectly suited for slow cooking. The blend of herbs and spices gives this stew a Moroccan feel and creates a heavenly aroma that will welcome you home after a long day of work. If you want another shortcut, you can look for a North African spice blend called ras el hanout. It has many of the same anti-inflammatory spices called for here (plus a few extras!). I usually make some red rice or quinoa for serving during the stew's last half hour of cooking.
Makes 6 servings
1 cup dried black beans, soaked overnight
4 cups water
1 small white onion, diced
1 teaspoon ground black mustard seed
1 teaspoon garlic powder
1 teaspoon ground cumin
1 teaspoon ground turmeric
1 teaspoon ground fennel seed
1 teaspoon chili powder
½ teaspoon ground coriander
¼ teaspoon ground cinnamon
¼ teaspoon ground ginger
2 teaspoons sea salt
1 head broccoli, separated into florets
Rice or quinoa, for serving
Rinse and drain the beans and put them in a slow cooker with the water, onion, and all of the spices except for salt. Cover and cook on low for 8 hours.
Add the salt and broccoli, cover, and cook for another 30 minutes. Serve over rice or quinoa.
Phytonutrients in broccoli have been found to suppress the inflammatory signaling system and may also decrease allergic responses that lead to inflammation.
Three-Bean Stew with Red Quinoa
This vegetarian stew has a colorful array of beans and features the lesser-known adzuki bean, a small red bean that's a particular favorite in macrobiotic diets. It's a nutritional powerhouse, loaded with minerals and easier to digest than some other beans. Kombu is a sea vegetable (aka seaweed) that helps tenderize the beans and imparts minerals during cooking. I like to think of it as the bay leaf for beans. Adding chunky vegetables and quinoa to this stew gives it a hearty enough texture to satisfy any meat-eating friends or family. Note that the black beans should be soaked separately as they are cooked longer than the others.
Makes 6 servings
⅓ cup dried black beans, soaked overnight
⅓ cup dried adzuki beans, soaked overnight
⅓ cup dried cannellini beans, soaked overnight
3 cups vegetable broth
1 (6-inch) strip kombu
1 tablespoon coconut oil
1 small yellow onion, diced
2 small zucchini, chopped
1 unpeeled sweet potato, scrubbed and chopped into ½-inch cubes
2 cloves garlic, minced
¼ cup red quinoa
1 tablespoon chili powder
2 teaspoons ground cumin
1 teaspoon ground ginger
1 teaspoon sea salt
½ teaspoon ground cinnamon
OPTIONAL ADDITIONS
1 cup corn kernels
¼ cup diced green chilies
1 cup fire-roasted peppers
Rinse the beans thoroughly and drain. Put the black beans in a large stockpot with the broth and kombu. Bring to a boil over medium-high heat, then reduce the heat to medium-low and simmer for 20 minutes. Add the adzuki and cannellini beans and simmer for another 60 to 90 minutes, or until all the beans are tender.
Meanwhile, heat the oil in a medium sauté pan over medium heat. Add the onion and sauté until translucent. Add the zucchini and sauté for 3 to 4 minutes. Set aside.
When the beans are tender, discard the kombu. Add the onions and zucchini, sweet potato, garlic, quinoa, chili powder, cumin, ginger, salt, and cinnamon to the stockpot. Stir in any optional items at this time. Reduce the heat to low and simmer for 30 to 40 minutes to allow the flavors to mingle and the sweet potato to soften.
Quinoa is higher in healthy unsaturated fats than most other grains and even contains a small amount of omega-3s. It's also loaded with a unique blend of phytonutrients known to be anti-inflammatory.
Caramelized Carrot and Ginger Soup
Caramelized Carrot and Ginger Soup
Carrots are a great example of a vegetable with multiple personalities. As a child, I would only eat them raw, straight out of my grandmother's garden. I had no use for peeled carrots in general, and I detested cooked ones (thankfully I'm over that now). Roasted carrots, slightly caramelized with a little bit of sweetener, are an entirely different experience than steamed or sautéed carrots. The buttery sweetness of the caramelized carrots makes this recipe unique compared to other carrot-ginger soups. Coconut palm sugar is great here because it's a low-glycemic sweetener that is reminiscent of brown sugar.
Makes 6 servings
8 unpeeled carrots, scrubbed well and cut into 1-inch chunks
1 small yellow onion, sliced into thick rings
1 tablespoon coconut oil, melted
2 tablespoons coconut palm sugar
2 cups vegetable broth
½ cup water
1½ teaspoons freshly grated peeled ginger
½ teaspoon sea salt
Preheat the oven to 400 degrees F.
Place the carrots and onion in a large bowl and toss with the coconut oil and palm sugar until well coated. Spread the vegetables evenly on a baking pan. Roast in the oven for 30 minutes, or until the onions start to caramelize and the carrots begin to turn golden brown.
Transfer the vegetables to a food processor. Add the broth, water, ginger, and salt and blend until smooth. Reheat the soup in a saucepan over medium heat and serve warm.
Black-Eyed Pea and Escarole Soup
Black-Eyed Pea and Escarole Soup
I remember the first time I saw someone add escarole to soup, I thought to myself, "Why is that person putting leaf lettuce into soup?" As I learned when I tasted the soup, escarole is perfectly suited for light cooking and is an extremely versatile leafy green. It has a slightly bitter note that mellows when cooked, and it's perfectly balanced in this recipe by the sweetness from the butternut squash. Save yourself some time and head to the frozen foods section to find squash that's already cubed and ready to use.
Makes 6 servings
1½ cups dried black-eyed peas, soaked overnight
6 cups vegetable broth
2 cloves garlic, minced
1 tablespoon fresh thyme or
2 teaspoons dried
1 tablespoon fresh oregano or
2 teaspoons dried
1 tablespoon coconut oil
1 large or 2 small leeks (white and light green parts only), rinsed well and sliced into half moons
2 stalks celery, chopped
1 cup frozen cubed butternut squash
1 head escarole, coarsely chopped (about 2 cups)
1 teaspoon sea salt
½ teaspoon freshly ground black pepper
Rinse and drain the black-eyed peas and put in a large stockpot. Add the broth, garlic, thyme, and oregano. Bring to a boil, and then reduce the heat to medium-low to maintain a simmer.
Meanwhile, heat the oil in a medium skillet over medium heat. Add the leeks and celery and sauté for 5 minutes, or until the leeks sweat and the celery starts to soften. Transfer the vegetables to the stockpot, cover, and simmer for 40 minutes.
Stir the squash and escarole into the soup and simmer for another 20 minutes. Season with the salt and pepper and serve. Leftovers can be stored in the freezer for up to 6 months.
Escarole is a good source of vitamin K, which is an important nutrient that helps regulate the body's inflammatory processes. One cup provides more than 100 percent of the recommended daily intake.
Lentil and Spinach Stew with Roasted Garlic
Lentils are so delightfully simple. There's no soaking required, and the cooking time is shorter than most legumes. Red lentils are often the first choice for dal, an Indian dish that's the consistency of mashed potatoes. This is primarily because red lentils don't hold their shape when you cook them, which makes them desirable for thicker soups and stews. In this recipe, you'll be simmering them for ninety minutes over low heat so they will soften and break apart, blending into the stew. The addition of balsamic vinegar and roasted garlic really brings the flavors together and makes this the ultimate anti-inflammatory comfort food!
Makes 6 servings
1 head garlic
1 tablespoon extra-virgin olive oil, divided
2 leeks (white and light green parts only), rinsed well and sliced into half moons
1 carrot, diced
1 stalk celery, diced
3 cups vegetable broth
1½ cups water
1½ cups dried red lentils, rinsed and drained
2 bay leaves
1 tablespoon fresh thyme or
2 teaspoons dried
1½ cups chopped spinach
3 tablespoons balsamic vinegar
1 teaspoon sea salt
Freshly ground black pepper (or crushed red pepper flakes for more heat)
Preheat the oven to 350 degrees F.
Slice the top off the head of garlic to expose the cloves. Drizzle with 1 teaspoon of the oil and wrap the head in foil. Place in the oven and roast for 50 minutes.
Meanwhile, heat the remaining 2 teaspoons oil in a large saucepan or stockpot over medium heat. Add the leeks, carrot, and celery and sauté for 5 minutes, or until the leeks begin to sweat and the carrots and celery soften. Add the broth, water, lentils, bay leaves, and thyme. Reduce the heat to low and simmer, stirring occasionally, for 50 to 60 minutes.
Discard the bay leaves. Gently squeeze the roasted garlic cloves into the stew. Stir in the spinach and balsamic vinegar. Season with the salt and pepper and serve.
Vegetable and Chicken Pho
When pho mania hit Seattle, I was late to the party. I'm not sure what the holdup was, because pho has all of my prerequisites for the perfect comfort food. It's warm, it has noodles, and the aroma just feels like coming home. After making up for lost time with countless pho outings, I decided to try to create a slow cooker version with less meat, more veggies, and plenty of flavor. If you can't find Kaffir lime leaves, just substitute a teaspoon of lime zest and a generous squeeze of lime juice. When serving the pho, choose a selection of suggested toppings from the list to pass at the table.
Makes 6 servings
1 (8-ounce) bone-in free-range organic chicken breast, skin removed
3 green onions, chopped
5 cloves garlic
1 tablespoon chopped peeled fresh ginger
1 ounce dried porcini mushrooms
1 quart mushroom broth
1 quart water
2 tablespoons Coconut Aminos (tamari or Bragg Liquid Aminos if avoiding soy)
2 tablespoons agave nectar
2 tablespoons five-spice powder
2 teaspoons sea salt
4 ounces wide rice noodles
1 cup chopped bok choy
1 cup chopped broccoli florets
5 Kaffir lime leaves
FOR SERVING:
1 cup bean sprouts
1 cup shredded carrots
1 cup chopped fresh basil leaves
1 cup chopped fresh cilantro
Sriracha, hot peppers, or crushed red pepper flakes
Place the chicken in a slow cooker. Add the green onions, garlic, ginger, porcinis, broth, water, aminos, agave, five-spice powder, and salt. Cover and cook on low for 8 hours.
Remove the chicken and use a fork to shred the meat from the bone. Return the shredded chicken to the slow cooker. Add the noodles, bok choy, broccoli, and lime leaves. Add more water to cover as needed. Cover, increase the temperature to high, and cook for an additional 30 minutes. Discard the lime leaves.
Ladle the soup into bowls and offer the desired selection of ingredients for serving on the side.
# Salads and Sides
Wilted Kale Salad with Shredded Beets and Carrots
Spring Pea and Jicama Salad
Rainbow Quinoa with Roasted Asparagus and Adzuki Beans
Wild Rice and Roasted Vegetables
Brussels Sprout Slaw
Bhutanese Rice and Flageolet Bean Salad
Warm Brussels Sprout Salad with Pecans and Currants
Kale and Kohlrabi Salad with Creamy Avocado Vinaigrette
Oven-Roasted Beets with Sautéed Greens
Braised Greens and Roasted Fennel with Silky Walnut Sauce
Super Greens Salad with Pomegranate and Toasted Hazelnuts
Shredded Cabbage and Snow Pea Sauté
The goal of an anti-inflammatory Mediterranean-style diet is to make sure that vegetables occupy at least half of the real estate on your plate. If that feels like a substantial shift from what your plate looks like now, please don't panic. This chapter of salads and sides is designed to make veggie eating second nature.
There are a wide variety of vegetables featured in the recipes that follow, and some may be outside of your usual repertoire. Challenge yourself to be an explorer. Why limit yourself to romaine lettuce and carrots when you can tantalize your taste buds with fennel, kohlrabi, and chard? If you look beyond your usual produce picks and expand your vegetable horizon, you'll exponentially increase the variety of anti-inflammatory nutrients in your diet, and what's more, you'll actually enjoy it!
Remember that nightshades (tomatoes, potatoes, peppers, and eggplant) are not featured in these recipes to accommodate those who might have an inflammatory reaction to those foods. However, if you are not one of those people, I encourage you to add them for more color, flavor, and phytonutrients.
Wilted Kale Salad with Shredded Beets and Carrots
This salad gives you all the benefits that raw kale, beets, and carrots have to offer. No nutrients were destroyed in the making of this dish. You'll wilt the kale with the heat of your own hands when you thoroughly massage the dressing into the leaves. This is a great way to break down kale and make it more tender and enjoyable to eat raw. And you get the added benefit of silky smooth hands from the avocado and olive oil.
Makes 6 servings
2 bunches kale, stemmed and coarsely chopped
1 avocado
2 tablespoons balsamic vinegar
1 tablespoon extra-virgin olive oil
½ teaspoon sea salt
4 carrots, shredded
2 unpeeled beets, scrubbed and shredded
¼ cup pine nuts, toasted
Put the kale in a large bowl. Scoop the avocado into the bowl and drizzle with the balsamic vinegar, oil, and salt. Using both hands, massage the avocado and dressing into the kale. It will start to take on a wilted appearance as it gets tender.
Add the carrots, beets, and pine nuts. Toss well and serve.
Beets contain phytonutrients that support healthy detoxification and reduce inflammation in the body.
Spring Pea and Jicama Salad
Spring Pea and Jicama Salad
There's a relatively short window of opportunity to get your hands on some fresh spring peas, but it's a window you don't want to miss. When you open the pod and see the vibrant, perfect green row of plump little peas, you'll know right away that you're in for a treat. Don't fret if you miss the spring pea harvest—frozen peas are a perfectly suitable option. This light and creamy salad will dress up any barbecue or picnic. The firm yet creamy texture of the peas is offset by the crunch from the jicama and celery. A tangy lemon-dill dressing brings this beautifully balanced salad together.
Makes 6 servings
1½ cups shelled spring peas, or 1 (10-ounce) bag frozen petite peas, thawed
10 radishes, quartered
2 stalks celery, chopped
1 small jicama, peeled and shredded
¼ cup sunflower seeds
½ cup Soy-Free Vegenaise
2 tablespoons freshly squeezed lemon juice
1 tablespoon apple cider vinegar
1 clove garlic, minced
1 teaspoon dried dill
1 teaspoon dried basil
1 teaspoon sea salt
In a large bowl, combine the peas, radishes, celery, jicama, and sunflower seeds.
In a small bowl, whisk together the Vegenaise, lemon juice, vinegar, garlic, dill, basil, and salt.
Drizzle the dressing over the salad and toss until all ingredients are well coated.
Rainbow Quinoa with Roasted Asparagus and Adzuki Beans
This colorful bean and grain salad is packed with nutrients and makes a great side dish or a tasty stand-alone lunch. Quinoa is one of the most nutritious whole grains, and adzuki beans outshine many other beans because of their mineral content. If you can't find rainbow quinoa, traditional or red quinoa will work just fine. You can play with variations of this recipe by substituting different beans (such as black, chickpea, or pinto) and switch up the veggies based on what's in season.
Makes 4 servings
1 pound asparagus, ends trimmed, cut into 1-inch pieces
6 tablespoons extra-virgin olive oil, divided
2 cups vegetable broth
1 cup rainbow quinoa
2 cups stemmed and chopped chard
1 (15-ounce) can adzuki beans, rinsed and drained
½ cup chopped kalamata olives
2 tablespoons freshly squeezed lime juice
1 teaspoon agave nectar or honey
1 teaspoon sea salt
Freshly ground black pepper
Preheat the oven to 375 degrees F.
In a large bowl, toss the asparagus with 2 tablespoons of the oil. Spread the asparagus on a baking pan and roast for 15 to 20 minutes, or until tender.
Meanwhile, in a large saucepan over high heat, combine the broth and quinoa. Bring to a boil, reduce the heat to low, cover, and simmer for 20 minutes without stirring. Measure 1 cup of the quinoa to use in the salad, reserving the extra for another use.
In a large bowl, combine the asparagus, 1 cup quinoa, chard, beans, and olives.
In a small bowl, whisk together the remaining ¼ cup oil, lime juice, and agave. Drizzle over the salad and toss until well coated. Season with the salt and pepper and serve.
Wild Rice and Roasted Vegetables
This dish is a mainstay in my repertoire of sides because it's simple, versatile, and a crowd-pleaser. The types of veggies I roast correspond with what's in season, so sometimes it includes more root vegetables, like rutabaga, turnips, and beets, and other times it includes zucchini, mushrooms, and sweet potatoes. Regardless of the combination I use, dinner guests almost always ask for the recipe.
Makes 6 servings
2 cups vegetable or mushroom broth
1 cup wild rice, rinsed well and drained
3 medium carrots, chopped into 1-inch chunks
1 zucchini, chopped into 1-inch chunks
1 small yellow onion, chopped
2 cups broccoli florets
1½ cups cremini mushrooms, quartered
2 tablespoons grapeseed or sunflower oil
1½ teaspoons coarse sea salt, divided
3 tablespoons extra-virgin olive oil
2 tablespoons balsamic vinegar
2 cloves garlic, minced
1 tablespoon dried Italian herbs
Preheat the oven to 425 degrees F.
Put the broth and rice in a large saucepan and bring to a boil over high heat. Reduce the heat to low, cover, and simmer for approximately 50 minutes, or until the broth is absorbed and the rice is tender but not mushy.
Meanwhile, combine the carrots, zucchini, onion, broccoli, and mushrooms in a large bowl. Drizzle with the grapeseed oil, sprinkle with 1 teaspoon of the salt, and toss until well coated.
Spread the vegetables in a single layer on a large baking pan. Roast for 20 minutes, stirring after 10 minutes. Transfer the vegetables back to the large bowl. Stir in the rice.
In a small bowl, whisk together the olive oil, balsamic vinegar, garlic, remaining ½ teaspoon salt, and Italian herbs. Drizzle over the vegetables and rice and toss until well combined.
Brussels Sprout Slaw
Brussels Sprout Slaw
Eating brussels sprouts raw may seem like absolute madness to most. Until you think about the fact that it's just baby cabbage, so it's actually genius to use it for slaw. If you object to mayonnaise-laden salads, this will be a welcome replacement. The sprouts and purple cabbage are nicely dressed in a light vinaigrette with a bit of an Asian flair. The sesame oil offsets the grassy notes of the flaxseed oil and balances perfectly with the tangy sweet rice wine vinegar. I use flaxseed oil because it's a good vegetarian source of omega-3s, but you can substitute extra-virgin olive oil if you don't have flaxseed on hand.
Makes 6 servings
3 tablespoons rice wine vinegar
2 tablespoons toasted sesame oil
1 tablespoon flaxseed oil
1 tablespoon agave nectar
½ pound brussels sprouts, ends trimmed
½ small head purple cabbage
¼ cup golden raisins
2 teaspoons poppy seeds
In a small bowl, whisk together the rice wine vinegar, sesame oil, flaxseed oil, and agave. Set aside.
Chop the brussels sprouts into fine strips or use a food processor to shred them. Chop the purple cabbage into thin strips or shred in a food processor.
Combine the brussels sprouts and cabbage in a large bowl with the raisins and poppy seeds. Drizzle the vinaigrette over the slaw and toss thoroughly to coat.
Tip: This slaw gets better after marinating in the dressing for at least a couple hours in the refrigerator. I love to pair leftovers with grilled halibut or cod for fish tacos.
Bhutanese Rice and Flageolet Bean Salad
If you've ever been served red rice when you ordered brown at a Thai restaurant, chances are it was Bhutanese rice. This nutty, earthy, red-toned rice is as nutritious as brown rice but cooks in half the time and is a whole lot more interesting. Flageolet beans are a favorite in French cuisine, and their pale green hue makes them a beautiful companion to the red rice. If you can't find flageolets, you can substitute cannellini beans or baby limas. Kombu is a type of seaweed that imparts minerals and helps break down the fiber in the beans.
Makes 6 servings
1 cup dried flageolet beans, soaked overnight
3 cups water
1 (6-inch) strip kombu (optional)
1 cup Bhutanese red rice
1½ cups vegetable broth
1 bunch chard, stemmed and chopped
1 small fennel bulb, chopped
½ cup chopped fresh parsley
½ cup sunflower seeds
¼ cup extra-virgin olive oil
2 tablespoons sherry vinegar
1 clove garlic, minced
¼ teaspoon sea salt
Rinse and drain the beans and transfer to a large saucepan. Add the water and kombu and bring to a boil over high heat. Reduce the heat to low and simmer, uncovered, for 45 to 50 minutes, or until the beans are tender. Discard the kombu and drain any excess water from the beans. Measure 1 cup of the beans for use in the salad, reserving any extra for another use.
Meanwhile, combine the rice and broth in a medium saucepan and bring to a boil over high heat. Cover, reduce the heat to low, and simmer for 20 minutes. Measure 1 cup of the rice for use in the salad, reserving any extra for another use.
In a large bowl, combine the 1 cup beans, 1 cup rice, chard, fennel, parsley, and sunflower seeds.
In a small bowl, whisk together the oil, vinegar, garlic, and salt. Drizzle over the salad and toss until well coated. Serve warm or cold.
Warm Brussels Sprout Salad with Pecans and Currants
I have converted more brussels sprout haters with this recipe than any other. Sautéing the shredded sprouts gives them an almost creamy texture. Adding the toasted pecans and currants brings in just enough sweetness to disguise any lingering bitterness from the sprouts. I've used this as a side dish for Thanksgiving meals, and it pairs very well with poultry or white fish.
Makes 6 servings
½ cup pecans
2 tablespoons extra-virgin olive oil
1 pound brussels sprouts, ends trimmed, thinly sliced
¼ cup dried currants
1 teaspoon sea salt
¼ cup vegetable broth
Preheat the oven to 350 degrees F.
Spread the pecans on a baking pan and roast for 8 to 10 minutes, or until fragrant. Transfer to a food processor and pulse until roughly chopped. (Alternatively, you can coarsely chop the pecans by hand.) Set aside.
In a large sauté pan or cast-iron skillet, heat the oil over medium heat. Add the brussels sprouts and toss until thoroughly coated with oil. Add the currants and salt and sauté, stirring occasionally for 2 to 3 minutes. Deglaze the pan with the broth, scraping up any brown bits. Reduce the heat to medium-low, cover, and simmer for 10 to 15 minutes, until the brussels sprouts are tender. Remove from the heat, stir in the pecans, and serve.
Brussels sprouts help reduce inflammation in a number of ways. Compounds in this member of the cabbage family help the body process toxins, fight off free radicals, and block pro-inflammatory pathways.
Kale and Kohlrabi Salad with Creamy Avocado Vinaigrette
I like to dress up a classic kale salad with some interesting shredded veggies. Kohlrabi is one of the lesser-known members of the cabbage family and often gets neglected because of its unusual appearance. It has the texture of a turnip but tastes more like a cross between a radish and cauliflower. The avocado-based dressing mimics a creamy balsamic vinaigrette, and it makes this kale salad (or any salad!) irresistible. This salad will hold up for a couple days in the refrigerator, and the flavor and texture get even better with time.
Makes 4 servings
1 small bunch kale, stemmed and finely shredded
2 small or 1 medium kohlrabi, peeled and shredded
2 carrots, shredded
1 avocado
3 tablespoons balsamic vinegar
2 tablespoons extra-virgin olive oil
½ teaspoon sea salt, plus more for seasoning
1 cup water
¼ cup hemp or sunflower seeds
Combine the kale, kohlrabi, and carrots in a large bowl and toss well.
Scoop the avocado into a food processor or blender along with the balsamic vinegar, oil, and salt. With the machine running, slowly add the water.
Drizzle the dressing over the salad, add the sunflower seeds, and toss until all ingredients are well coated. Season to taste with salt.
Kale easily tops the list of anti-inflammatory vegetables. It's a good source of omega-3s, and it's rich in vitamin K, which is an inflammation regulator.
Oven-Roasted Beets with Sautéed Greens
Whenever I encounter someone who claims to dislike beets, I always ask whether they've ever tried them roasted. It's such a different experience than eating them canned or steamed beyond recognition. Oven-roasted beets have an earthy sweetness that I find irresistible. And not only are the beet greens loaded with vitamin K, an anti-inflammatory nutrient, they are also delicious when lightly sautéed. Early-season beets often have greens that are delicate enough to use raw in a salad. However, in the winter, when the greens are heartier, I prefer to sauté them or add them to soups.
Makes 4 servings
3 medium red or gold beets with greens attached
2 tablespoons grapeseed or sunflower oil
1 tablespoon coarse sea or kosher salt
2 teaspoons extra-virgin olive oil
1 tablespoon balsamic vinegar
Preheat the oven to 375 degrees F.
Trim the beet greens from the beets and set aside. Thoroughly scrub the unpeeled beets and trim off any stems or roots. Chop the beets into 1-inch chunks and toss with the grapeseed oil until well coated. Spread the beets on a baking pan, sprinkle with the salt, and roast for about 25 minutes, or until the beets are tender.
Meanwhile, wash the beet greens thoroughly, trim the stems, and cut the leaves into thin strips. Heat the olive oil in a large sauté pan over medium heat, add the greens, and sauté until wilted, 3 to 4 minutes.
Toss the beets and greens in a large bowl with the balsamic vinegar and serve.
Many of the unique phytonutrients in beets have anti-inflammatory effects. The mechanism of action is similar to some of the nonsteroidal anti-inflammatory drugs but without the side effects!
Braised Greens and Roasted Fennel with Silky Walnut Sauce
Covering half your plate with braised greens is one of the best ways to start building an anti-inflammatory plate. Braising, which really just means sautéing in oil and then steaming with liquid, is a quick and easy way to soften up hearty greens. And just to make it even more interesting, roasted fennel joins the party to share its sweet anise flavor and its own unique anti-inflammatory benefits. You'll top it all off with a silky, savory walnut sauce that not only makes the greens irresistibly delicious, but it also gives you an extra boost of omega-3s. Roasting the walnuts and adding a touch of maple syrup help to mellow any lingering bitterness of the greens.
Makes 4 servings
FOR THE WALNUT SAUCE:
1 cup raw walnuts
½ cup canned butter beans, rinsed and drained
1 tablespoon tahini
1 tablespoon maple syrup
1 teaspoon sea salt
¾ cup water
FOR THE GREENS AND FENNEL:
2 fennel bulbs, cut into 1-inch chunks
2 teaspoons grapeseed oil
½ teaspoon coarse sea or kosher salt
2 tablespoons extra-virgin olive oil
1 shallot, diced
1 cup sliced shiitake mushrooms
1 bunch kale or chard, stemmed and cut into ribbons
2 tablespoons mirin
1 tablespoon balsamic vinegar
Preheat the oven to 350 degrees F.
To make the walnut sauce, first spread the walnuts on a baking pan and roast for 10 minutes. Allow to cool slightly, then transfer to a food processor along with the beans, tahini, maple syrup, and salt. With the machine running, slowly add the water until the mixture is thin enough to drizzle. Set aside.
Increase the oven temperature to 400 degrees F.
In a medium bowl, toss the fennel and grapeseed oil to coat. Spread the fennel on a baking pan, sprinkle with the salt, and roast for 40 minutes.
Put the olive oil and shallots in a large sauté pan over medium heat and cook for 2 minutes. Add the mushrooms and sauté for 5 minutes. Add the kale and sauté until it begins to wilt. Add the fennel to the pan along with the mirin and balsamic vinegar. Reduce the heat to low, cover, and simmer for 5 minutes.
To serve, divide the braised vegetables among four plates and drizzle each with the walnut sauce.
Tip: The Silky Walnut Sauce also works well as a creamy salad dressing. You can even drizzle it over fresh baked pears or apples for a delectable sweet and savory dessert.
Super Greens Salad with Pomegranate and Toasted Hazelnut
Super Greens Salad with Pomegranate and Toasted Hazelnuts
I was tempted to call this the Ultimate Antioxidant Salad, but it just didn't seem quite as sexy. All of the featured ingredients are loaded with immune-boosting, cancer-fighting, heart-protecting antioxidants. But that's not the best part. The wooden-spoon method of getting the seeds out of the pomegranate is a great stress reliever and just downright fun. I taught this technique to a friend's two young sons at a Thanksgiving gathering, and they're still talking about it (and they now love pomegranates).
Makes 6 servings
¾ cup raw hazelnuts
1 pomegranate
¼ cup extra-virgin olive oil
1 tablespoon champagne vinegar
1 teaspoon honey
¼ teaspoon sea salt
2 cups spinach
1 cup arugula
1 cup stemmed and chopped chard
Preheat the oven to 350 degrees F.
Spread the hazelnuts on a baking pan and roast for 10 minutes. Transfer the nuts to a clean dish towel while they're still warm. Wrap the pile of nuts in the towel and aggressively massage them to remove the skins. Coarsely chop the hazelnuts and set aside.
Cut the pomegranate in half. Over a medium bowl, hold one half of the pomegranate, cut side down. Use a wooden spoon to repeatedly strike the sides of the pomegranate while gently squeezing with the hand holding it. Seeds will start to fall into the bowl. If no seeds are falling, put some muscle into it. Empty the seeds from both halves of the pomegranate into the bowl and pick out any of the white membrane that may have also fallen in. Pour the seeds through a strainer over a small bowl to capture the pomegranate juice. Measure ½ cup of the pomegranate seeds to use in the salad, reserving any extras for another use.
Add the oil to the pomegranate juice and whisk in the vinegar, honey, and salt until well blended.
In a large bowl, combine the spinach, arugula, and chard. Add the hazelnuts and ½ cup pomegranate seeds. Drizzle the dressing over the salad and toss until the greens are well coated.
Antioxidants help destroy free radicals in the body and prevent oxidation, both of which can cause inflammation. They also help support the immune system.
Shredded Cabbage and Snow Pea Sauté
I have such an appreciation for vegetarian Indian dishes that bring vegetables to life with interesting combinations of herbs and spices. That's what makes this recipe one of my all-time favorite anti-inflammatory side dishes. Heating the mustard seeds, turmeric, and asafetida in oil unlocks the powerful flavors that welcome the cabbage and snow peas with open arms. Asafetida is an extremely pungent spice (with some unflattering nicknames) that is used almost exclusively in Indian cuisine. It's commonly used in curry and dal and is thought of as a replacement for onions or garlic. A little bit goes a long way!
Makes 4 servings
2 teaspoons black mustard seeds
1 teaspoon coarse sea salt
½ teaspoon ground turmeric
¼ teaspoon asafetida
2 tablespoons coconut oil
½ large head green cabbage, shredded (about 2 cups)
1 cup snow peas
¼ cup water
In a small bowl, combine the mustard seeds, salt, turmeric, and asafetida.
Heat the oil in a cast-iron skillet or large sauté pan over medium-high heat. Add the spices and stir to coat with the oil. When the mustard seeds begin to pop, add the cabbage and snow peas. Toss the vegetables until they're well coated with the spice-infused oil. Sauté for 2 to 3 minutes, or until the cabbage begins to soften. Add the water to deglaze the pan and continue cooking until the water is absorbed. Serve immediately.
Turmeric contains curcumin, which has been shown to have similar anti-inflammatory effects as prescription medications.
# Vegetarian Main Dishes
Spaghetti Squash Primavera with Basil Walnut Pesto
Black-Eyed Peas and Forbidden Rice with Crispy Kale
Zucchini Noodles with Pistachio Pesto and Black Lentils
Portobello Mushrooms with Samosa Filling
Puttanesca-Style Beans and Greens
Toasted Pecan Quinoa Burgers
Hearty Mushroom and Lentil Stew
Southwestern-Style Buckwheat Polenta Stacks
Quinoa-Stuffed Collard Rolls
Golden Beet and Mushroom Faux Gratin
Veggie Pizza with Cauliflower-Yam Crust
When I'm counseling clients on anti-inflammatory eating and reviewing their food journals, there are three words that I say over, and over, and over again... eat more vegetables! In fact, I'm convinced that we would resolve the vast majority of our diet-related health issues if everyone would just take that simple advice. And I would probably be out of a job.
As an advocate for a more plant-based diet, I feel it's my responsibility to make vegetable eating not just tolerable, but downright enjoyable. I often encourage clients to experiment with just one or two meatless meals per week, so they can discover how satisfying a vegetarian or vegan dish can be. Whether you're a vegan or a meat-loving omnivore, this chapter will help you think outside the produce box. Have you ever made noodles from squash? Or used collards like a tortilla? Or created a pizza crust from cauliflower and sweet potatoes? Well hold on to your apron strings because things are about to get pretty wild in the kitchen!
Spaghetti Squash Primavera with Basil Walnut Pesto
This is the perfect recipe for the squash skeptic and is usually a hit with the kids. When you scrape out those spaghetti-like strands, it seems like something magical is happening. Using walnuts in the pesto provides a more sophisticated flavor and amps up the anti-inflammatory benefits of this dish.
Makes 4 servings
½ cup water
1 small spaghetti squash, halved lengthwise and seeded
¾ cup raw walnuts
2 cups packed fresh basil leaves
2 cloves garlic
½ cup plus 1 tablespoon extra-virgin olive oil, divided
½ teaspoon sea salt
1 cup sliced cremini mushrooms
1½ cups coarsely chopped broccoli
2 medium carrots, chopped
½ cup vegetable broth
1 cup fresh or frozen peas
½ pound fresh spinach, thoroughly washed
Freshly ground black pepper
Preheat the oven to 350 degrees F.
Pour the water into a shallow baking dish and place the squash in the water face down. Cover with foil and bake for 45 to 50 minutes, or until the squash is tender. Meanwhile, lightly toast the walnuts in a dry sauté pan over medium-low heat for about 5 minutes. Combine the walnuts, basil, garlic, ½ cup of the olive oil, and salt in a food processor and blend until smooth. Set aside.
In a large skillet over medium heat, add the remaining 1 tablespoon olive oil and mushrooms. Cover and cook until the mushrooms are tender, 2 to 3 minutes. Add the broccoli, carrots, and broth and sauté for 5 to 7 minutes, or until broccoli is just starting to get tender (do not overcook). Reduce the heat to low and add the peas and spinach.
Use a fork to scrape the squash flesh directly into the skillet. Add the pesto and stir well to combine all ingredients. Season to taste with pepper.
Black-Eyed Peas and Forbidden Rice with Crispy Kale
This is one of my favorite variations of the "beans, greens, and grains" dishes. Delicata is a versatile winter squash with lots of personality. Not only is it perfectly sweet, but the skin is thinner than most winter squashes, so you don't have to peel it before roasting. The combination of sweetness from the squash and savory notes from the roasted mushrooms create the perfect party when they're nestled into forbidden rice and mingling with black-eyed peas. The crunchy kale is the crowning jewel, and it's a great excuse to make up a tray of kale chips to snack on while you're waiting for dinner.
Makes 6 servings
1 head garlic
3½ tablespoons extra-virgin olive oil, divided
1 bunch curly kale, stemmed and torn into 2-inch pieces
4 teaspoons coarse sea salt, divided
1 delicata squash, ends trimmed and halved lengthwise
6 cremini mushrooms, quartered
2 tablespoons grapeseed oil
1 cup vegetable broth
½ cup forbidden rice
2 tablespoons balsamic vinegar
1 cup canned black-eyed peas, rinsed and drained
Preheat the oven to 350 degrees F.
Slice the top off the head of garlic to expose the cloves. Drizzle with ½ teaspoon of the olive oil and wrap the head in foil. Place in the oven to one side of the middle rack.
Toss the kale with a scant 1½ tablespoons olive oil, spread on a baking pan, and sprinkle with 1 teaspoon of the salt. Bake for about 12 minutes, or until the kale starts to brown and become crispy. Set aside (feel free to do some snacking!). Leave the garlic in the oven.
Increase the oven temperature to 400 degrees F.
Scoop the seeds from the squash with a spoon and cut the flesh into 1-inch chunks. Toss the squash and mushrooms with the grapeseed oil and 2 teaspoons of the salt until well coated. Spread the vegetables on a baking pan and bake for 20 minutes, or until the squash is tender and begins to caramelize. Remove the vegetables and garlic from the oven and set aside.
Meanwhile, bring the broth and rice to a boil in a medium saucepan over medium-high heat. Reduce the heat to low, cover, and simmer for 30 minutes.
Squeeze the garlic cloves into a small bowl. Stir in the remaining 2 tablespoons olive oil, balsamic vinegar, and remaining 1 teaspoon salt.
In a large bowl, combine the roasted vegetables, rice, and black-eyed peas. Drizzle with the garlic mixture and toss well to coat. Crumble the crispy kale over the top right before serving.
Delicata squash contains anti-inflammatory phytonutrients, and it's also rich in antioxidants like beta-carotene and vitamin C.
Zucchini Noodles with Pistachio Pesto and Black Lentils
Zucchini Noodles with Pistachio Pesto and Black Lentils
It took many years of cooking for me to truly appreciate the versatility of zucchini, and I wasn't that big of a fan until I discovered zucchini noodles. It makes a pasta-inspired dish that's gluten-free, and it's just a fun way to transform an overused vegetable. This no-pasta noodle dish also features the oh-so-adorable black lentil, also known as the Beluga lentil for it's caviar-like appearance. If you cook the lentils al dente, they will retain their shape, and this will look like the most expensive faux pasta dish ever served. And you'll be dishing up one more surprise when you use pistachios in place of pine nuts in the pesto. Delicious!
Makes 6 servings
1½ cups vegetable broth, divided
½ cup dried black lentils, rinsed and drained
1 cup packed basil leaves
¼ cup shelled pistachio nuts
1 small clove garlic
1 teaspoon sea salt, divided
¼ cup plus 2 teaspoons extra-virgin olive oil, divided
¼ cup water
3 medium zucchini, ends trimmed
8 shiitake mushrooms, thinly sliced
Place 1¼ cups of the broth and lentils in a medium saucepan and bring to a boil over medium-high heat. Reduce the heat to low, cover, and simmer for 15 minutes for al dente.
Meanwhile, put the basil, pistachios, garlic, and ½ teaspoon of the salt in a food processor. With the machine running, drizzle in ¼ cup of the oil, then slowly add the water until the pesto is slightly thinner than olive oil. Set aside.
Use a spiral vegetable slicer with a 3-millimeter blade to shred the zucchini into noodles. (Alternatively, you can hand slice the zucchini or use a peeler. Cut the zucchini in half or in thirds, then slice into very thin strips.)
Drizzle the remaining 2 teaspoons oil in a large skillet or sauté pan over medium heat and add the mushrooms and remaining ½ teaspoon salt. Sauté for 3 to 4 minutes, or until the mushrooms start to soften. Add the remaining ¼ cup broth and zucchini noodles to the skillet. Use tongs to gently toss the noodles with the mushrooms. Sauté for 5 minutes, or until the zucchini just begins to soften. Remove from the heat and stir in the lentils and pesto until well combined. Serve warm.
Pistachios contain high amounts of antioxidants that support the immune system, as well as specific phytonutrients called proanthocyanidins that help reduce inflammation.
Portobello Mushrooms with Samosa Filling
Not only are portobello mushrooms juicy, delicious, and nutritious, they are also the perfect vessel for a hearty samosa-style filling made from garnet yams and green peas. This can be a satisfying vegan meal, or you can use cremini mushrooms and serve it as an appetizer. It's pleasing to the eye and the palate!
Makes 2 servings
2 portobello mushrooms, stemmed and cleaned
3 tablespoons grapeseed or sunflower oil, divided
1 teaspoon sea salt
1 shallot, minced
1 garnet yam, peeled and chopped into 1-inch chunks
1 cup vegetable or mushroom broth, divided
¾ cup fresh or frozen peas
Preheat the oven to 350 degrees F.
With a basting brush, coat the portobellos with 2 tablespoons of the oil, seasons with the salt, and place on a baking pan. Bake for about 20 minutes, turning the mushrooms halfway through cooking.
Meanwhile, heat the remaining 1 tablespoon oil in a sauté pan over medium heat. Add the shallot and sauté for 2 to 3 minutes, or until it starts to caramelize. Add the yams and ½ cup of the broth. Cover and simmer, stirring occasionally, until the yams are soft, about 10 minutes.
Transfer the yam mixture to a food processor, add the remaining ½ cup broth, and blend until smooth. (Alternatively, you can fork-mash the mixture.) Transfer to a medium bowl.
In a small saucepan over medium heat, add the peas and enough water to cover. Bring to a gentle boil, then drain the peas and stir into the yams. Spoon the mixture into the mushroom caps and serve immediately.
Puttanesca-Style Beans and Greens
Puttanesca-Style Beans and Greens
This recipe was inspired by my good friend Greg Janssen, who tried a similar dish at a local restaurant and became obsessed with trying to recreate it at home. While I never got to taste the original, I am assured that this comes very close. A secret ingredient that I added when I wanted to put my own spin on it is sun-dried tomatoes, which really elevates the flavor with an acidic sweetness. The anchovies are optional, but I really encourage adding them for the lovely saltiness they offer and also for their delightful oiliness that make them a good anti-inflammatory food. You could also top this dish with a few sardines for good measure.
Makes 6 servings
3 cups water
1 cup dried baby lima beans, soaked overnight
1 (6-inch) strip kombu (optional)
1 cup pitted kalamata olives
¾ cup pitted green olives
½ cup sun-dried tomatoes in olive oil (optional)
½ large yellow onion 2 teaspoons capers
6 tablespoons extra-virgin olive oil, divided
2 cups shredded greens (kale, chard, dandelion greens, or beet greens all work well)
4 anchovies, or 1 to 2 teaspoons anchovy paste (optional)
1 to 2 cloves garlic
In a large pot, combine the water, lima beans, and kombu and bring to a boil over high heat. Reduce the heat to medium-low, cover, and simmer for about 40 minutes, or until the beans are tender. Drain the beans and discard the kombu.
Meanwhile, in a food processor or blender, combine the olives, sun-dried tomatoes, onion, and capers and pulse until the mixture is coarsely chopped.
In a large skillet, heat 2 tablespoons of the oil over medium heat. Add the olive mixture and sauté for about 5 minutes, or until the onions start to become translucent. Add the greens, anchovies, and garlic and sauté for 3 to 4 minutes. Stir in the beans until well incorporated. Serve warm or at room temperature.
Toasted Pecan Quinoa Burgers
Toasted pecans have a sweet, earthy flavor that helps to make these meat-free "burgers" unique. The quinoa lends itself to the nutty flavor of the pecans and rounds out the nutritional profile of these tasty burgers. When I want to have a more "burger-like" experience, I put this on a gluten-free bun with lettuce, tomatoes, and onion, or sometimes I just place the patty between two large lettuce leaves. Serve these to your vegan friends and watch them savor every bite!
Makes 8 servings
¾ cup pecans
2¼ cups vegetable broth, divided
1 cup quinoa, rinsed and drained
1 teaspoon sea salt, plus more for seasoning
½ cup sunflower seeds
¼ cup sesame seeds
1 teaspoon ground cumin
1 teaspoon dried oregano
1 carrot, shredded
½ cup canned black beans, rinsed and drained
Freshly ground black pepper
1 teaspoon coconut or sunflower oil
1 avocado, thinly sliced
Preheat the oven to 375 degrees F.
Spread the pecans on a baking pan and roast for 5 to 7 minutes.
Combine 2 cups of the broth, quinoa, and salt in a large saucepan and bring to a boil over medium-high heat. Reduce the heat to low, cover, and simmer for 20 minutes without stirring. Measure 1 cup of the quinoa for use in the burgers, reserving any extra for another use.
Combine the pecans, sunflower seeds, sesame seeds, cumin, and oregano in a food processor and grind to a medium-coarse texture.
In a large bowl, stir together the 1 cup quinoa, nut mixture, carrot, and beans. While stirring, slowly add the remaining ¼ cup broth until mixture becomes tacky. Season to taste with salt and pepper.
Form the mixture into 8 patties about ½-inch thick and cook, refrigerate, or freeze immediately.
Heat the coconut oil in a large skillet over medium-high heat. Add half of the patties and cook for about 2 minutes on each side (cook longer from frozen). Repeat with the remaining patties. Top the burgers with avocado slices.
Quinoa is the only grain that is considered a complete protein, and it also contains a small amount of omega-3s.
Hearty Mushroom and Lentil Stew
The combination of mushrooms and lentils makes for a hearty, satisfying stew that even meat lovers enjoy. I like to make a big batch when I have company for the weekend, so we can heat it up for a quick lunch or serve it over quinoa with some sautéed greens for dinner. Any type of black, brown, or green lentil will work in this stew, and you can also play around with different types of mushrooms. I've used chanterelles and porcinis and had good results.
Makes 8 servings
2 teaspoons sunflower oil
2 cups diced red onion
1 cup chopped celery
1 cup chopped carrots
2 cloves garlic, minced
1 bay leaf
5 cups sliced shiitake mushrooms
1½ cups sliced portobello mushrooms
½ cup dried French lentils
1 quart mushroom broth
2 tablespoons cooking sherry Sea salt and freshly ground black pepper
¼ cup chopped fresh parsley, for garnish
Heat the oil in large soup pot or Dutch oven over medium heat. Add the onion and sauté for 5 minutes, or until tender. Add the celery, carrots, garlic, and bay leaf and sauté, stirring frequently, for 10 minutes, or until the onion is golden brown. Stir in the mushrooms and sauté for 10 minutes, or until most of the liquid has evaporated. Stir in the lentils, broth, and sherry. Season to taste with salt and pepper. Bring the mixture to a boil, then reduce the heat to medium-low, cover, and simmer for 30 to 40 minutes, or until the lentils are tender.
Discard the bay leaf. Ladle the stew into individual bowls, sprinkle with the parsley, and serve.
Southwestern-Style Buckwheat Polenta Stacks
Buckwheat fascinates me. Even though it has "wheat" in its name, it's actually not a grain at all. It's related to rhubarb and is safe for those who are avoiding gluten. It also happens to be a good source of tryptophan, so you might just find that you sleep like a baby after this nutritious meal! Buckwheat has a characteristically strong flavor, but I think it works nicely with the spicy bean mixture in this recipe. It also makes a great alternative to corn polenta, which takes much longer to cook and requires a lot of stirring. You can get creative with the bean mixture: I love to add roasted red peppers.
Makes 9 buckwheat stacks
2 cups water
1 cup toasted buckwheat groats
¼ teaspoon sea salt
2 teaspoons extra-virgin olive oil
1 small onion, diced
1 (15-ounce) can pinto beans with liquid
2 teaspoons maple syrup
1 teaspoon ground cumin
1 teaspoon garlic powder
1 teaspoon dried oregano
½ teaspoon paprika
¼ teaspoon cayenne pepper
2 cups chopped spinach
1 avocado, thinly sliced
In a large saucepan, bring the water to a boil. Add the groats and salt, reduce the heat to low, cover, and simmer for 10 minutes.
Transfer the groats to a food processor and blend. Texture will be somewhat rough and thick, like polenta. Allow the mixture to cool slightly for a couple minutes.
Take a small handful of the buckwheat mixture—the size of a large meatball—roll it, and then flatten onto a baking pan. Press each disk to a ¼-inch thickness. Repeat until you have used all of the buckwheat mixture.
Drizzle the oil in a large skillet, add the onion, and stir until well coated. Place the skillet over medium heat and sauté the onion until translucent. Add the beans, maple syrup, cumin, garlic powder, oregano, paprika, and cayenne. Stir well and sauté for 3 to 5 minutes. Use potato masher or fork to smash some of the beans, allowing some to remain whole. The mixture should be the consistency of chunky refried beans.
Place a small handful of spinach on a buckwheat cake. Scoop a generous amount of the bean mixture over the spinach. Top with avocado slices. Repeat with the rest of the buckwheat cakes.
Tip: For a crispy cake, before stacking, place the buckwheat rounds under the broiler for 3 minutes on each side.
Quinoa-Stuffed Collard Rolls
Collard greens are most commonly used in southern dishes. They're generally cooked with bacon, ham, or some form of pork that lends saltiness and cuts the bitterness that's inherent to this member of the cabbage family. The savory quinoa filling and rich, buttery flavor of the toasted pecans also helps disguise the bitterness of the greens—no pork required. You can save some time if you have large, pliable collard leaves that are flexible enough to roll without blanching. You can also choose to skip the step of baking the finished rolls if you want the benefit of eating the greens raw. Steaming in the oven just helps soften the collards a bit more and may make them more desirable for those who are still on the fence about these mysterious greens.
Makes 6 servings
2½ cups vegetable broth, divided
1 cup quinoa
1 bunch collard greens, washed with stems removed
½ cup pecans
1 tablespoon grapeseed oil
1 small onion, diced
¾ cup chopped mushrooms
2 carrots, shredded
1 clove garlic
1 teaspoon ground sage
1 teaspoon dried oregano
2 teaspoons sea salt
In a medium saucepan, add 2 cups of the broth and quinoa and bring to a boil over medium-high heat. Reduce the heat to low, cover, and simmer for 20 minutes without stirring. Fluff with a fork and let cool for about 5 minutes.
Preheat the oven to 350 degrees F.
Bring a large pot of water to a boil. Add the collard greens and blanch for 2 minutes. Drain and rinse with cold water. Set aside.
Place the pecans in a small dry sauté pan and toast over medium heat until they become aromatic and start to brown slightly. Coarsely chop the nuts.
Heat the oil in a large sauté pan over medium heat. Add the onion and sauté until translucent. Add the mushrooms and sauté for 2 minutes. Add the carrots, garlic, sage, and oregano and sauté for 3 minutes. Stir in the quinoa, pecans, and salt.
Place a generous scoop of filling onto a collard leaf. Roll the leaf burrito-style and place seam down in a square baking dish. Repeat until all the collards and filling have been used. Pour the remaining ½ cup broth over the rolls, cover with foil, and bake for 25 minutes.
Collard greens are right up there with kale when it comes to outstanding anti-inflammatory foods. They're an excellent source of vitamin K, which directly regulates the body's inflammatory response, as well as a good source of omega-3s.
Golden Beet and Mushroom Faux Gratin
I don't like to play favorites, but this recipe is certainly a contender for first place in my book. It really is the whole package: the sweet earthiness of the roasted beets, the savory succulence of the portobellos, and the tangy, nutty creaminess of the layers in between. The gratin is extremely rich and surprisingly filling, but it's still hard for me to put down my fork. I'm hoping this dish will result in a slew of beet-loving converts!
Makes 6 servings
1 cup raw cashews
½ cup unsweetened almond milk
2 tablespoons tahini
2 tablespoons nutritional yeast
1 tablespoon freshly squeezed lemon juice
1½ teaspoons coarse sea salt, divided
1 teaspoon garlic powder
4 medium unpeeled golden beets, well scrubbed and ends trimmed
2 large portobello mushrooms
2 tablespoons extra-virgin olive oil, divided
Preheat the oven to 425 degrees F.
In a food processor, finely chop the cashews. Add the almond milk, tahini, nutritional yeast, lemon juice, ½ teaspoon of the salt, and garlic powder. Blend until the mixture is the consistency of creamy peanut butter, adding water as needed. Set aside.
Slice the beets into ¼-inch-thick slices (a mandolin works well for this). Cut the portobellos into large ¼-inch-thick slices.
Drizzle 1 tablespoon of the oil in a 9-by-12-inch baking dish. Arrange one layer of beets along the bottom of the dish. Arrange a layer of mushrooms over the beets. Spread half of the cashew mixture over the mushrooms. Arrange another layer of beets on top of the nut spread and repeat the layering ingredients in this order (it's best to end with a layer of mushrooms). Drizzle the remaining 1 tablespoon oil over the top layer and sprinkle with the remaining 1 teaspoon salt. Cover and bake for 55 minutes. Serve warm.
Veggie Pizza with Cauliflower-Yam Crust
Veggie Pizza with Cauliflower-Yam Crust
If I were to take a poll and ask my gluten-free, dairy-free clients what food they miss the most, I'm certain that pizza would easily top the list. I've played around with dozens of variations of a gluten-free crust and topped them with some pretty pathetic cheese wannabes, but nothing comes close to the real thing. What I love about this veggie pizza is that it's not trying to be a big, doughy, cheesy pizza when it grows up. It's entirely unique, with a vegetable-based crust, pesto in place of cheese, and succulent sautéed vegetables on top. It's quite tasty the next day, and who wouldn't want to eat cold pizza for breakfast and have it be healthy!
Makes 4 servings
½ medium head cauliflower, broken into small florets
½ medium garnet yam, peeled and chopped into ½-inch chunks
1 tablespoon dried Italian herbs
¾ teaspoon sea salt, divided
1 cup brown rice flour
1 tablespoon coconut oil, plus more for greasing pizza stone
1 small red onion, sliced into thin rings
½ cup sliced cremini mushrooms
1 yellow summer squash or zucchini
2 cups spinach
¼ to ½ cup vegan pesto (store-bought is OK)
Preheat the oven to 400 degrees F. If you have a pizza stone, put it in the oven.
Place a steamer basket in a large pot with 1 inch of water. Put the cauliflower and yam in the basket and steam until both are easily pierced with a fork, about 15 minutes. Do not overcook or the vegetables will be too wet.
Transfer the vegetables to a food processor. Add the Italian herbs and ½ teaspoon of the salt and blend until smooth. Transfer the mixture to a large bowl. Gradually add the flour, stirring until the mixture is well combined.
Grease the pizza stone or a pizza pan with coconut oil. Pile the cauliflower mixture in the center of the pizza stone. Use a spatula to carefully spread the "dough" evenly in a circular pattern (much like spreading frosting) until the crust is about ⅛ inch thick. Bake for 40 to 45 minutes. Turn on the broiler and broil the crust for 2 minutes to make the top crispy (watch carefully to avoid burning).
Meanwhile, heat the coconut oil in a medium skillet over medium heat. Add the onion and sauté for 2 minutes. Add the mushrooms, squash, and remaining ¼ teaspoon salt and sauté for 3 to 4 minutes. Stir in the spinach and remove from the heat just as it begins to wilt.
Spread the pesto evenly over the pizza crust. Spread the sautéed vegetables over the pesto. Slice the pizza, it's ready to serve!
Tip: The cauliflower crust serves as a great stand-in for flatbread and can even be used to make sandwiches.
# Pescatarian Main Dishes
Hazelnut-Encrusted Halibut with Dipping Sauce
Poached White Fish with Mango Lime Chutney
Pan-Fried Sardines with Sautéed Kale and Chard
Salmon en Papillote with Silky Celery Root Puree
Mediterranean Salmon Skewers
Pumpkin Coconut Curry with White Fish
Sizzling Salmon and Quinoa Skillet
Nori-Wrapped Mackerel with Wasabi "Mayo"
Fish Taco Salad with Strawberry Avocado Salsa
Oven-Roasted Black Cod with Smashed Sweet Peas
One of the hallmarks of a Mediterranean diet is that the primary source of animal protein is fish. As we start to gain a better understanding of the role inflammation plays in the development of various diseases, it becomes more obvious why populations eating more fish and vegetables tend to be healthier. We know that oily finfish, like salmon, halibut, mackerel, and sardines, are rich sources of inflammation-blocking omega-3s.
When I give clients the recommendation to eat fish three to four times per week, they often tell me that they're just not confident about cooking it. So it should come as a relief that fish is actually one of the easiest, quick-cooking forms of protein out there. It takes less time to prepare than chicken. It's more versatile than beef. And there are a variety of cooking methods that can guarantee perfect results every time. It just takes some practice. The recipes in this chapter will encourage you to try poaching, roasting, pan-frying, and steaming in parchment. Be fearless in your attempts, and any insecurities about being able to cook fish correctly will soon be overcome.
Another concern that's often raised about seafood is sustainability and accumulation of heavy metals (like mercury) and other contaminants. That's why it's important to choose your fish wisely. One of the best resources I've found is the Monterey Bay Aquarium's Seafood Watch Program. You can visit their website or download the app and see which types of fish are the best bets.
Hazelnut-Encrusted Halibut with Dipping Sauce
I like to think of this as a healthier, anti-inflammatory version of fish and chips. Hazelnuts have a buttery sweetness when roasted, and they make a delicious bread-free crust. You won't get the golden-brown coloration and crispy texture of battered and fried fish, but you'll still get good flavor and a lot more anti-inflammatory unsaturated fats. The dipping sauce is reminiscent of tartar sauce. Just throw in some oven-roasted sweet potatoes fries and the package will be complete!
Makes 4 servings
FOR THE HALIBUT:
1 cup raw hazelnuts
¼ cup brown rice flour
1 tablespoon Italian herbs
½ teaspoon sea salt
1½ pounds halibut or any other firm white fish, skin removed and cut into 4 fillets
3 tablespoons extra-virgin olive oil
FOR THE SAUCE:
½ cup Soy-Free Vegenaise
½ teaspoon Dijon or stone-ground mustard
1 small dill pickle, finely chopped
1 teaspoon pickle juice
½ teaspoon honey
¼ teaspoon freshly grated lemon zest
Preheat the oven to 350 degrees F.
Spread the hazelnuts on a baking sheet and roast for 10 minutes. Transfer the nuts to a clean dish towel while they're still warm. Wrap the pile of nuts in the towel and aggressively massage them to remove the bitter skins. When the nuts are cool to the touch, transfer to a food processor and pulse until finely chopped. (Alternatively, put the hazelnuts in a plastic bag, seal, and use a meat tenderizer to pound them into fine crumbs.
Combine the hazelnuts, flour, Italian herbs, and salt in a shallow dish. Use a basting brush to coat both sides of the halibut fillets with the oil and then carefully press each fillet in the hazelnut mixture, making sure both sides are thoroughly coated.
Place the halibut in a baking dish and bake for 15 to 20 minutes. The fish should be just opaque but not dry.
Meanwhile to make the sauce, whisk together the Vegenaise, mustard, pickle, pickle juice, honey, and lemon zest.
Place a small spoonful of sauce on each fillet or serve on the side for dipping.
Halibut and other types of oily white fish are great sources of omega-3s and magnesium, a winning combination for reducing inflammation and protecting the heart.
Poached White Fish with Mango Lime Chutney
Poaching fish is perhaps the easiest of all cooking methods. It also guarantees that the fish will stay moist because you're immersing it in liquid. And it saves you the trouble of removing the skin from the fish because it will just slide right off after cooking. The mango lime chutney helps brighten up the dish and makes you feel like you're dining on the beach in the tropics (or at least you can pretend).
Makes 4 servings
FOR THE CHUTNEY:
1 mango, peeled, pitted, and diced
1 small shallot, minced
1 teaspoon minced peeled ginger
¼ cup finely chopped fresh cilantro
2 tablespoons freshly squeezed lime juice
½ teaspoon sea salt
FOR THE FISH:
1 cup vegetable broth
1 cup water
1½ pounds white fish (halibut, cod, or black cod), cut into 4 fillets
1 teaspoon sea salt
½ teaspoon freshly ground black pepper
To make the chutney, in medium bowl, stir together the mango, shallot, ginger, cilantro, lime juice, and salt. Set aside to allow the flavors to mingle.
In a deep cast-iron skillet or saucepan, bring the broth and water to a gentle boil over medium-high heat. Place fillets in the skillet and season with the salt and pepper. Cover and reduce the heat to medium. Poach for 8 to 9 minutes. The fish should be opaque in the center but still moist.
Carefully transfer the fillets from the skillet to plates, leaving the skin behind. Top with the chutney and serve.
Tip: Prepare an extra fillet and reserve some chutney for a fish taco the next day. I like to serve mine with Brussels Sprout Slaw.
Pan-Fried Sardines with Sautéed Kale and Chard
Pan-Fried Sardines with Sautéed Kale and Chard
Sardines. You either love 'em or you hate 'em. I've been on a crusade to try to convert the masses, but I have to admit, it's slowgoing. For me, sardines have always been an easy sell. I used to sit on my grandpa's lap and we'd eat canned sardines with a row of Ritz crackers. As an adult (and a nutritionist), I've come to appreciate the fabulous anti-inflammatory qualities of sardines. They're one of the most concentrated sources of omega-3 fatty acids, and they're also low on the food chain, so they don't accumulate mercury or other contaminants. This meal, featuring sardines and kale, would rank at the top of all the anti-inflammatory dishes in this book. Have I converted you yet?
Makes 4 servings
¾ cup brown rice flour
2 teaspoons curry powder
1 teaspoon ground cumin
1 pound fresh sardines, cleaned, scaled, and heads removed
1 tablespoon grapeseed or sunflower oil
1 small yellow onion, diced
1 bunch kale, stemmed and chopped
1 bunch chard, stemmed and chopped
1 clove garlic, chopped
¼ cup vegetable broth
¼ cup golden raisins
In a shallow baking dish, whisk together the flour, curry powder, and cumin. Dredge the sardines in the flour mixture until well coated.
Heat the oil in a cast-iron skillet over medium-high heat. When hot, add the sardines to the skillet in batches and cook for 1½ to 2 minutes per side. Transfer the sardines to a plate lined with paper towels. Repeat with the remaining sardines, adding more oil to the skillet as needed.
Reduce the heat to medium and add the onion to the skillet. Sauté for 4 to 5 minutes, or until the onion starts to sweat and soften. Add kale, chard, and garlic. Deglaze the pan with the broth and cook the greens until they begin to wilt, 2 to 3 minutes. Stir in the raisins. Divide the greens among each plate and top with the sardines.
Salmon en Papillote with Silky Celery Root Puree
Cooking fish in parchment is a foolproof way to ensure perfectly cooked fish that's moist. The celery root puree is smooth and silky, resembling expertly blended mashed potatoes but with a completely different flavor profile. Adding sunchokes (also called Jerusalem artichokes) to the puree gives it a bit of a nutty essence. If you can't find sunchokes, try parsnips instead. The salmon fillets look beautiful resting on pillows of pureed goodness. All you need to do is add some greens to create the ideal anti-inflammatory plate!
Makes 4 servings
FOR THE SALMON:
1½ pounds salmon, bones removed and cut into 4 fillets
1 lemon, halved
1 teaspoon sea salt
1 teaspoon celery seed
¼ cup chopped Italian parsley or celery leaves, for garnish (optional)
FOR THE PUREE:
1 medium celeriac (celery root), peeled and cut into 1-inch chunks
4 sunchokes, thoroughly scrubbed and cut into 1-inch chunks
1½ cups vegetable broth
1 teaspoon sea salt
Freshly ground black pepper
Preheat the oven to 350 degrees F.
Cut sheets of parchment paper into 4 large heart-shaped pieces that can cover each fillet. Place a fillet on each parchment heart, slightly off center. Season each fillet with a squeeze of lemon juice, ¼ teaspoon salt, and ¼ teaspoon celery seed. Fold the parchment over the fillet so all edges of the paper meet. Working your way around parchment, fold the edges over twice and end with a twist at the bottom of the heart. The fillets should be well sealed inside the paper so no steam escapes.
Place the packages on a baking pan and bake for 15 to 20 minutes.
Meanwhile, to make the puree, place a steamer basket in a pot with 1 inch of water. Put the celeriac and sunchokes in the basket and steam until tender, about 15 minutes. Transfer the vegetables to a food processor, add the broth and salt, and blend until smooth. Season to taste with pepper.
Carefully remove the fish from the parchment paper. Put a generous scoop of the puree on a plate. Place a fillet on top of the puree. Garnish with the parsley and serve with a side of sautéed greens.
In addition to being a great source of omega-3s, salmon contains unique bioactive proteins that may be particularly effective in reducing joint inflammation.
Mediterranean Salmon Skewers
Mediterranean Salmon Skewers
There's something about putting food on skewers that almost immediately makes it more interesting to eat. I love using salmon when making these skewers in the spring and summer months, and I'm more inclined to use lamb in fall and winter. You can choose which type of protein you'd like to use (the marinade works well on most types of fish, meat, or poultry) and then switch up the veggies for a whole new experience. I find that it helps to put your picky eaters to work on the skewer assembly line so they can custom-make a couple skewers with their favorite ingredients.
Makes 8 servings
½ cup extra-virgin olive oil
1 tablespoon freshly squeezed lime juice
1 teaspoon freshly grated lime zest
1 clove garlic, minced
2 tablespoons chopped fresh oregano
1 tablespoon chopped fresh mint
1 teaspoon sea salt
¾ pound salmon, skin and bones removed, cut into 1½-inch cubes
12 cremini mushrooms, quartered
1 small head broccoli, broken into florets
1 summer squash or zucchini, cut into 1-inch chunks
½ cup pitted kalamata olives
8 wooden skewers, soaked in water for at least 30 minutes
Preheat the oven to 375 degrees F.
In a medium bowl, combine the oil, lime juice and zest, garlic, oregano, mint, and salt.
Combine the salmon, mushrooms, broccoli, and squash in a baking dish. Drizzle with the marinade and gently toss until the salmon and vegetables are well coated.
Layer the salmon, vegetables, and olives on the skewers, alternating the order on each skewer.
Place the skewers on a broiler pan and bake for 20 to 25 minutes, turning them halfway through cooking.
Pumpkin Coconut Curry with White Fish
Just saying "pumpkin coconut curry" out loud makes my mouth water in anticipation. I created this recipe for a cooking class I was teaching in the fall, and it quickly became a favorite among the class participants and the cooking assistants. If you like pumpkin and you are a fan of curry, you'll love this dish. The rich, complex flavors of the pumpkin and coconut unite with the earthy heat of the curry and cayenne to create the perfect balance of flavors. If you're still working on becoming a fish lover, this is an exquisite way to enjoy a mild white fish that will blend right in without any hopes of overpowering the sauce. I've also recreated this dish as a vegetarian meal by replacing the fish with chickpeas and adding more veggies (like mushrooms, broccoli, and cauliflower).
Makes 6 servings
2 tablespoons coconut oil
1 small yellow onion, diced
2 cloves garlic, minced
1 teaspoon minced peeled ginger
1 (15-ounce) can pumpkin puree
1 cup vegetable broth
1 cup coconut milk
1½ cups frozen peas
5 Kaffir lime leaves
2 teaspoons curry powder
¼ teaspoon cayenne pepper
1 pound white fish (halibut, cod, or black cod), skin and bones removed, cut into 1-inch cubes
2 teaspoons sea salt
Forbidden black rice, for serving
Heat the oil in a deep cast-iron skillet or saucepan over medium heat. Add the onion and sauté until translucent, about 5 minutes. Add garlic and ginger and sauté for 2 to 3 minutes. Stir in the pumpkin puree, broth, coconut milk, peas, lime leaves, curry powder, and cayenne. Reduce the heat to medium-low, add the fish and salt to the skillet, cover, and simmer for 10 to 15 minutes.
Serve the curry over forbidden black rice for a beautiful contrast of color.
Pumpkin is rich in antioxidants like beta-carotene and vitamin C, so it helps keep the immune system tuned up and prevents oxidative damage that leads to inflammation.
Sizzling Salmon and Quinoa Skillet
One-pot meals are the norm at my house. I've always enjoyed comingling the food on my plate, so why not just throw all of my favorite ingredients into a sizzling skillet and call it dinner? This dish looks like a work of art too, with its earthy tones from the mushrooms and quinoa, various shades of green, and of course the beautiful pink salmon. I make this often for impromptu dinner parties when I'm in the mood for a more rustic, casual theme, but I also love to serve it as a brunch entrée. In fact, I encourage you to eat any leftovers for breakfast. I can't think of a more balanced meal to help you kick off the day.
Makes 4 servings
1 head garlic
½ teaspoon extra-virgin olive oil
2½ cups mushroom broth, divided
1 cup quinoa, rinsed and drained
1 tablespoon coconut oil
½ pound chanterelle mushrooms, sliced
1 cup shredded brussels sprouts
1 cup frozen petite peas
2 tablespoons nutritional yeast
2 tablespoons chopped fresh basil
1 tablespoon dried oregano
½ pound salmon, skin and bones removed, cut into 1-inch cubes
Sea salt and freshly ground black pepper
Preheat the oven to 350 degrees F.
Slice the top off the head of garlic to expose the cloves. Drizzle with the olive oil and wrap the head in foil. Place in the oven and roast for 50 minutes.
Meanwhile, combine 2 cups of the broth and quinoa in large saucepan. Bring to boil over high heat, then reduce the heat to low, cover, and cook for 20 minutes without stirring. Measure 1 cup of the quinoa to use in this recipe, reserving any extra for another use.
In a large skillet, heat the coconut oil over medium heat. Add the mushrooms and sauté for 5 minutes, or until they release liquid and become soft. Add the brussels sprouts and sauté for 3 minutes, adding up to ¼ cup broth as needed to prevent the mushrooms and sprouts from sticking to the skillet. Add the peas, nutritional yeast, basil, and oregano and sauté, stirring occasionally, for 5 minutes. Add the salmon to the skillet and toss gently to combine. Gently squeeze the garlic cloves into the skillet. Cover and cook, stirring occasionally, for 4 to 5 minutes.
Add the 1 cup quinoa and remaining ¼ cup broth to the skillet and stir until well combined. Season to taste with salt and pepper and serve.
Tip: A perfectly poached egg can be the crowning jewel on this skillet mix if you're heating it up for breakfast the next day.
Nori-Wrapped Mackerel with Wasabi "Mayo"
Mackerel is one of the top picks for anti-inflammatory eating. It's an oilier fish, which means it's ultra high in omega-3s. When I ask my cooking class if they ever eat mackerel, the typical response is "only in sushi." So I was inspired to create a cooked version that would be easy for the home cook. I bake fish in nori for the same reason I use parchment—it steams the fish and prevents it from drying out.
Makes 4 servings
2 cups water
1 cup brown basmati rice, rinsed and drained
¾ teaspoon sea salt, divided
2 tablespoons toasted sesame oil
1 tablespoon rice wine vinegar
½ teaspoon minced peeled ginger
1 pound mackerel, skin and bones removed
1 (1-ounce) package nori sheets
¼ cup Soy-Free Vegenaise
2 teaspoons wasabi powder
Preheat the oven to 350 degrees F.
Put the water, rice, and ¼ teaspoon of the salt in a large saucepan and bring to a boil over medium-high heat. Reduce the heat to low, cover, and simmer for 45 to 50 minutes, or until all of the water is absorbed.
Meanwhile, in a small bowl, combine the oil, vinegar, ginger, and remaining ½ teaspoon salt.
Slice the mackerel into strips about 1 inch wide by 3 inches long. Place a strip of mackerel on a sheet nori. Top with about 3 tablespoons of the rice. Drizzle the oil mixture over the rice and carefully roll the nori sheet like a burrito, folding in the ends as you go. Place the rolls in a baking dish, cover with foil, and bake for 15 minutes.
Meanwhile, in a small bowl, blend the Vegenaise and wasabi powder.
Serve the wrapped mackerel with a generous spoonful of the wasabi mayo for dipping.
Tip: Spritzing the nori sheets with a little bit of water minutes before filling makes them much more pliable and easier to roll without tearing.
Fish Taco Salad with Strawberry Avocado Salsa
Fish Taco Salad with Strawberry Avocado Salsa
The first time I made this salsa, I had an audience of dinner guests looking on as I tossed the strawberries into a bowl with the rest of the salsa ingredients. "That's interesting" was one of the comments. "I never would have thought to put strawberries in salsa" was another. I sensed some skepticism. After a few tentative bites of the fish salad with the salsa, they were all scooping generous amounts onto their plates. The fruity salsa works well with the peppery arugula and creates a fish taco salad that surprises the skeptics. Don't forget the margaritas!
Makes 4 servings
FOR THE SALSA:
½ avocado, diced
3 strawberries, hulled and diced
¼ cup canned black beans, rinsed and drained
1 small shallot, diced
2 green onions, thinly sliced
¼ cup finely chopped fresh cilantro
1 teaspoon finely chopped peeled ginger
3 tablespoons freshly squeezed lime juice
½ teaspoon sea salt
⅛ teaspoon cayenne pepper
FOR THE FISH:
2 tablespoons extra-virgin olive oil or avocado oil
2 teaspoons agave nectar
1 tablespoon freshly squeezed lime juice
4 cups arugula
1½ pounds light fish (halibut, cod, or red snapper), cut into 4 fillets
1 teaspoon sea salt
½ teaspoon freshly ground black pepper
Preheat a gas or charcoal grill.
To make the salsa, in a medium bowl, combine the avocado, strawberries, beans, shallot, green onions, cilantro, ginger, lime juice, salt, and cayenne. Mix until the ingredients are well blended and set aside.
To make the salad, in a small bowl, whisk together the oil, agave, and lime juice. Put the arugula in a large bowl and toss with the vinaigrette.
Season the fish fillets with the salt and pepper. Grill over direct high heat for 7 to 9 minutes, turning the fish once during cooking. The fish should be opaque and flake easily.
To serve, pile 1 cup of the arugula salad on each plate. Place a fillet on the salad and top with a heaping spoonful of salsa.
The unique combination of healthy fats and phytosterols in avocados are what make them particularly useful in reducing inflammation related to arthritis.
Oven-Roasted Black Cod with Smashed Sweet Peas
My first introduction to black cod was a rather unsuccessful grilling experience, but I'm so glad that I tried it again. Black cod, also called sablefish, is loaded with omega-3s, and it ranks high on the sustainability list. I prefer to roast or poach it, and I'm in love with the subtle, buttery flavor. Even those who are finicky about fish generally appreciate black cod. The smashed peas are such a simple side dish and they add so much to the plate. And the best part? This gourmet-looking dinner only takes twenty minutes to make.
Makes 4 servings
2 leeks (white and light green parts only), sliced into thick rings
1 pound shiitake mushrooms, sliced
2 tablespoons grapeseed oil
2 teaspoons coarse sea salt, divided, plus more for seasoning
2 pounds black cod, skin and bones removed, cut into 4 fillets
1 teaspoon freshly ground black pepper
1 teaspoon sweet paprika
1 tablespoon extra-virgin olive oil, divided
2 shallots, chopped
2 cloves garlic, chopped
2 cups fresh or frozen sweet peas
¾ cup vegetable broth
Preheat the oven to 400 degrees F.
Combine the leeks, mushrooms, grapeseed oil, and 1 teaspoon of the salt in a large bowl and toss until well coated. Spread the vegetables in a 9-by-14-inch baking dish and roast for 10 minutes.
Remove the dish from the oven and place the fish fillets on top of the vegetables. Season with the remaining 1 teaspoon salt, pepper, and paprika. Bake for 8 to 10 minutes, or until the fish flakes easily and is opaque in the center.
Meanwhile, heat 2 teaspoons of the olive oil in a medium skillet over medium heat. Add the shallots and sauté for 2 to 3 minutes. Add the garlic and sauté for 2 minutes. Stir in the peas, add the remaining 1 teaspoon olive oil, and sauté for about 5 minutes.
Transfer half of the pea mixture to a blender or food processor. Drizzle in the broth and pulse until the peas begin to resemble a thick soup. Return the blended peas to the skillet, season with the salt and pepper, and stir well.
To serve, put a generous scoop of pea mash to one side of each plate; place a fish fillet in the center of the plate, slightly overlapping the smashed peas; and top with the roasted vegetables.
Not only do shiitake mushrooms contain more than a hundred compounds that may help prevent cancer, they also keep our blood vessels healthy and block inflammation that can lead to heart disease.
# Hint-of-Meat Main Dishes
Moroccan Lamb Tagine with Chickpeas and Apricots
Bison Lettuce Cups with Garnet Yam Home Fries
Spring Lamb Stew
Steak Salad with Massaged Kale
Veggie Beef Burger with Rocket Salad
Sweet Potato Shepherd's Pie
While eating copious amounts of meat and poultry can lead to more inflammation, there's definitely a way to include them as part of the Mediterranean-style diet. It's called moderation. I like to think of meat as a condiment on the plate rather than the main event.
Meat adds flavor and texture, and it's a great source of protein. It's also the best delivery system for iron and vitamin B12, and it contains other nutrients that can be more challenging to get in appreciable amounts from plant sources. So there are advantages to being an omnivore.
With that said, it's also important to consider the type and source of the meat you choose to eat. There's plenty of research to confirm that animals raised on their native diets in environments that are closer to their natural habitats will produce meat that is more nutrient-rich and less inflammatory. Assays on grass-fed beef confirm that there are more omega-3 fatty acids in their flesh than in that of grain-fed beef. The meats that are featured in this chapter were selected because they tend to be less inflammatory than other options. Lamb, for example, has a very unique fatty acid profile that mimics the ideal ratio of omega-6s to omega-3s for an anti-inflammatory diet.
My general words of wisdom for meat selection are:
» Look for 100 percent grass-fed beef and lamb (the labels "all natural" and even "organic" do not always mean 100 percent grass-fed)
» Take a walk on the wild side and try game meats or bison (buffalo)
» Choose free-range organic poultry
Moroccan Lamb Tagine with Chickpeas and Apricots
A traditional tagine would be made in an unusually shaped earthenware dish that resembles an upside down funnel. This version can be made right in your slow cooker. I recommend searing the meat and sautéing the onion the night before if you want to throw this together in the morning before work. Leave it to simmer while you're away and you'll be greeted by the mouthwatering aroma of a Moroccan-inspired mix of spices.
Makes 6 servings
¾ cup dried chickpeas, soaked overnight
2 teaspoons grapeseed oil
1 pound boneless lamb shoulder or leg, trimmed of fat and cut into 1-inch cubes
1 teaspoon sea salt, divided
Freshly ground black pepper
1 small yellow onion, diced
3½ cups beef broth
¾ cup dried apricots, chopped
5 cloves garlic, chopped
1 teaspoon minced peeled ginger
1 teaspoon ground turmeric
1 teaspoon ground cumin
½ teaspoon sweet paprika
½ teaspoon ground cinnamon
¼ teaspoon ground nutmeg
¼ cup chopped flat-leaf parsley
Rinse and drain the chickpeas and place in a slow cooker.
Heat the oil in a large skillet over medium-high heat. Add the lamb, season with ¼ teaspoon of the salt and pepper, and allow the pieces to brown on one side, 1 to 2 minutes. Turn the lamb pieces, season with another ¼ teaspoon of the salt and pepper, and allow the other sides to brown. Remove from the skillet with a slotted spoon and transfer to the slow cooker.
Add the onion to the skillet and sauté until it begins to brown. Transfer to the slow cooker.
Add the broth, apricots, garlic, ginger, turmeric, cumin, paprika, cinnamon, and nutmeg to the slow cooker and stir well. Cover and cook on low for 4 hours. Season with the remaining ½ teaspoon salt and stir in the parsley right before serving.
Bison Lettuce Cups with Garnet Yam Home Fries
Bison Lettuce Cups with Garnet Yam Home Fries
Bison, or buffalo, is a naturally lean meat with a slightly stronger flavor than beef. Bison graze on their native diet of grass, so their meat is higher in omega-3s than red meat from animals that spend more time in the feedlots. I actually prefer the taste of bison to beef, and I think that sweet paprika, garlic, and pepper really elevate the savory flavors of the meat. Be sure to look for lettuce leaves that are large and somewhat pliable for best results. Or you can create dainty hors d'oeuvres using endive. The garnet yam fries are a universal hit and provide the perfect punch of color on the plate.
Makes 4 servings
2 unpeeled garnet yams, well scrubbed
2 teaspoons sunflower oil
1½ teaspoons coarse sea salt, divided
1 teaspoon five-spice powder
1 teaspoon coconut oil
1 pound ground bison (buffalo)
1 small white onion, chopped
2 carrots, shredded
1 yellow summer squash or zucchini, shredded
2 tablespoons chopped fresh oregano
2 teaspoons garlic powder
1 teaspoon sweet paprika
½ teaspoon freshly ground black pepper
½ teaspoon cayenne pepper
1 large head lettuce (I like Bibb or leaf lettuce)
¼ cup chopped green onions
Preheat the oven to 400 degrees F.
Cut the yams into strips the size of small steak fries (thinner if you prefer crispier fries). Toss with the sunflower oil, 1 teaspoon of the salt, and five-spice powder until well coated. Spread the fries evenly on a baking pan. Bake for 25 to 30 minutes, or until the fries start to brown and edges are becoming crispy. Turn off the oven, leaving the fries in there until ready to serve.
Heat the coconut oil in a large skillet over medium-high heat. Crumble the bison into the skillet and continue to break it up with a spatula as it cooks for 3 to 4 minutes. Use a slotted spoon to transfer the bison to a bowl. Add the onion and carrots to the skillet and sauté for 5 minutes. Add the squash and return the bison to the skillet. Stir in the oregano, garlic powder, paprika, remaining ½ teaspoon salt, pepper, and cayenne and sauté for 3 to 4 minutes.
To serve, scoop a generous helping of the bison mixture into a lettuce leaf and top with green onions. Repeat until you run out of filling or lettuce leaves. Place a couple lettuce cups on each plate and stack some yam fries beside them.
Tip: Use any leftover bison mixture as a topping on a green salad for a punch of flavor and boost of protein.
Spring Lamb Stew
This hearty stew is made lighter with fresh spring peas and mustard greens, and the flavors are spectacular! If parsnips are difficult to find, you can substitute sweet potatoes or regular potatoes if you're not avoiding nightshades. Lamb is an extremely healthy meat that has a better omega-6 to omega-3 ratio than most red meat, so it fits well into an anti-inflammatory diet. This stew could also be made in a slow cooker: just cook on low for six hours and add the kale thirty minutes before serving.
Makes 6 servings
1 tablespoon sunflower oil
2 pounds boneless lamb shoulder or leg, trimmed of fat and cut into 1-inch cubes
Sea salt and freshly ground black pepper
1 yellow onion, chopped
6 large carrots, cut into 1-inch chunks
6 parsnips, cut into 1-inch chunks
3 cloves garlic, minced
½ cup dry white wine
2 teaspoons Dijon mustard
1 quart mushroom broth
2 cups chopped mustard greens or kale
1 cup fresh or frozen peas
1 cup chopped fresh parsley
Preheat the oven to 325 degrees F.
Heat the oil in large, deep cast-iron skillet or Dutch oven. Season the lamb to taste with salt and pepper. Add the lamb to the pan in a single layer and brown the meat on all sides. Use a slotted spoon to transfer the lamb to a bowl.
Add the onion to the pan and sauté until it begins to caramelize. Add the carrots, parsnips, and garlic and sauté for 5 minutes. Stir in the wine, mustard, and broth. Return the lamb to the pan and place it in the oven. Cook, uncovered, for about 1 hour, adding water as needed.
Stir in the greens, peas, and parsley and return the pan to the oven to cook for 15 to 20 minutes. Season to taste with salt and pepper and serve.
Steak Salad with Massaged Kale
Steak Salad with Massaged Kale
This is a pretty interesting twist on steak salad and a great example of allowing vegetables to take center stage while the meat makes a flavorful cameo appearance. If you're just not sure about eating raw kale, do yourself a favor and try the massaged version. Playing rough with the kale helps break down the fibers and definitely makes it more enjoyable to eat raw. It also helps when the kale is marinating in a balsamic vinaigrette that's made creamy by our good friend Avocado. You can practice teamwork in the kitchen with your significant other if one of you loves to grill and the other prefers to manhandle the veggies.
Makes 4 servings
1 pound grass-fed flank or flatiron steak
2½ teaspoons coarse sea salt, divided
1 teaspoon dried Italian herbs
½ teaspoon freshly ground black pepper
¼ teaspoon freshly grated lemon zest
2 bunches Tuscan (Dino) kale, stemmed and roughly chopped
1 avocado
3 tablespoons balsamic vinegar
2 tablespoons extra-virgin olive oil
2 tablespoons agave nectar
2 carrots, shredded
Preheat a gas or charcoal grill.
Lay the steak out on a cutting board or baking pan. In a small bowl, combine 1 teaspoon of the salt, Italian herb blend, pepper, and lemon zest. Rub each side of the steak with the seasoning mixture.
Grill the steak over direct high heat for 4 minutes per side for medium-rare. Transfer to a plate, cover with foil, and allow the steak to rest for 5 minutes.
Meanwhile, put the kale in a large bowl. Scoop the avocado into the bowl. Drizzle in the balsamic vinegar, olive oil, agave, and remaining 1½ teaspoons salt. Using both hands, massage the avocado and dressing into the kale. It will start to take on a wilted appearance as it gets tender. Add the carrots and toss well.
Cut the steak across the grain into thin strips. Place a mound of salad on each plate and top with 4 or 5 steak strips.
Veggie Beef Burger with Rocket Salad
I'm a sucker for a good, juicy burger, and I absolutely love the idea of embellishing it with some sautéed veggies mixed right into the meat. This whole concept was inspired by my turkey meatloaf, which gets its moisture from an array of vegetables that are discretely integrated. I discovered this translates well to beef burgers, and it's a good example of how to enjoy a bit of meat with a lot of veggies and still feel like you're eating a hearty, all-American meal. Rocket salad is the more descriptive British name for arugula. Although it's an adaptation of roquette, the French name for arugula, I like to think of it instead as a nod to the peppery flavor that blasts off in your mouth after the first bite.
Makes 4 servings
FOR THE VEGGIE BEEF BURGER:
1 tablespoon sunflower oil
1 small yellow onion, diced
1 cup diced cremini mushrooms
1½ teaspoons sea salt, divided
1 carrot, shredded
1 small zucchini, shredded
1 pound ground grass-fed beef
¼ cup sunflower seeds
2 tablespoons dried parsley
2 teaspoons dried tarragon
1 teaspoon dried sage
½ teaspoon freshly ground black pepper
4 gluten-free hamburger buns (optional)
FOR THE SALAD:
2 tablespoons extra-virgin olive oil
1 tablespoon white balsamic vinegar
1 teaspoon stone-ground mustard
1 teaspoon honey
½ teaspoon sea salt
3 cups arugula
1 cup shredded purple cabbage
1 apple, cored and diced
1 English cucumber, chopped
To make the burgers, heat the sunflower oil in a large skillet or sauté pan over medium heat. Add the onion and sauté until it starts to become translucent. Add the mushrooms and 1 teaspoon of the salt and sauté for 5 minutes, or until the mushrooms start to release liquid and become soft. Add the carrot and zucchini and sauté for 2 to 3 minutes. Remove the skillet from the heat and let cool for 10 minutes.
Preheat a gas or charcoal grill.
Place the beef in a large mixing bowl. Add the sautéed vegetables, sunflower seeds, parsley, tarragon, sage, remaining ½ teaspoon salt, and pepper. Use both hands to knead the ingredients together until well combined. Form the mixture into 4 patties approximately ¾-inch thick.
Grill the patties over direct high heat for 8 to 10 minutes, turning once. Transfer to a plate, cover with foil, and allow the burgers to rest for 2 minutes.
To make the salad, whisk together the olive oil, vinegar, mustard, honey, and salt in a small bowl. Combine the arugula, cabbage, apple, and cucumber in a large bowl. Drizzle the vinaigrette over the salad and toss well to coat.
Place the burgers and a generous helping of salad on the gluten-free buns, or go bun-free and serve the salad in bowls with the burgers on top.
Arugula is a great source of folic acid and rich in immune-boosting vitamins like A, C, and K. It's also a good plant-based source of calcium.
Sweet Potato Shepherd's Pie
Shepherd's pie is my rainy day, stay-in-my-PJs-and-watch-movies supper. This version of a classic comfort food foregoes the cream but features high-quality meat and a garlicky sweet potato topping. Even though this pie is made mostly from vegetables and is dotted with green peas, I still feel compelled to serve some greens on the side, like the Wilted Kale Salad with Shredded Beets and Carrots or just some simple sautéed broccoli.
Makes 8 servings
1 head garlic
½ teaspoon extra-virgin olive oil
2 pounds sweet potatoes, peeled and cut into 1-inch cubes
¾ cup vegetable broth
1 pound ground bison, lamb, or grass-fed beef
1 teaspoon sea salt, plus more for seasoning
½ teaspoon freshly ground black pepper, plus more for seasoning
½ teaspoon ground nutmeg
1 small yellow onion, chopped
1 cup chopped cremini mushrooms
4 carrots, finely chopped
1 cup frozen peas
Preheat the oven to 400 degrees F.
Slice the top off the head of garlic to expose the cloves. Drizzle with the olive oil and wrap the head in foil. Place in the oven and roast for 50 minutes.
Meanwhile, place a steamer basket in a large saucepan with 1 inch of water. Put the sweet potatoes in the basket, cover the pan tightly, and steam for 15 to 20 minutes, or until the potatoes are tender and pierce easily with a fork. Drain and transfer the potatoes to the bowl of a food processor or stand mixer. Carefully squeeze the garlic cloves into the bowl. Add the broth and blend until the potatoes are thick and creamy. Season to taste with salt and pepper.
Turn on the broiler.
Place the bison in a large skillet over medium heat. Add the salt, pepper, and nutmeg and sauté the meat until brown. Use a slotted spoon to transfer the bison to a bowl.
Add the onion to the skillet and sauté until tender. Add the mushrooms and carrots, along with a little olive oil if needed, and sauté for 3 to 4 minutes. Add the peas and return the meat to the skillet. Cover and cook for 5 minutes.
Transfer the meat mixture to a 9-by-12-inch baking dish or deep pie pan and spread the sweet potatoes evenly over the top and to the edge of the pan. Place the dish in the oven and broil until the sweet potatoes begin to brown, 5 to 7 minutes.
Sweet potatoes, which are not a member of the nightshade family, are an excellent source of complex carbohydrates and fiber.
# Desserts
Pumpkin Coconut Pie with Almond Crust
Mixed Berry Walnut Crumble
Rustic Pear and Fig Crostatas
So-Easy Coconut Mango Sorbet
Baked Pears or Apples with Cashew Cream
Strawberry Rhubarb Crumble
Vanilla Wafer Pudding
No-Bake Peach Pie
Following an anti-inflammatory diet doesn't mean you don't get to have any treats. While refined sugar is high on the list of pro-inflammatory foods, there are plenty of great alternatives to satisfy a sweet tooth. Nature provides us with the best variety of desserts available in the form of fruits.
Sugar addicts may be rolling their eyes right now, doubting whether an apple could ever take the place of a chocolate frosted cupcake. The key is to get creative and learn what combinations of naturally decadent foods bring the same satisfaction and a lot less guilt (and pain!). Over time, your palate will even adjust to a lower level of sweetness, and fruits like mangoes and figs will give you as much of a sugar rush as you can handle.
In this chapter, you'll discover winning dessert combinations—like cashews and coconuts, dates and pecans—as well as lots of imaginative ways to use fruit. There's no chocolate on the menu, but that's only to make these recipes suitable for anyone on the anti-inflammatory cleanse; it's not explicitly forbidden otherwise. Cocoa is loaded with antioxidants and can also be used to create some amazing desserts. In fact, sometimes all you really need is a nice piece of dark chocolate.
Pumpkin Coconut Pie with Almond Crust
Pumpkin Coconut Pie with Almond Crust
This pie is gluten-free, dairy-free, and absolutely delicious! Combining mildly sweet pumpkin with thick, rich coconut milk makes for a decadent filling that's sure to impress your dairy-loving friends. And since the crust is made from almonds and the filling is made from a vegetable in the squash family, it's a low-glycemic, guilt-free dessert. This recipe makes a double crust, but you'll only need half for a single pie, so the second portion can be frozen for up to three months for future use.
Makes 8 servings
FOR THE CRUST:
2 cups almond meal
⅓ cup arrowroot powder
1 teaspoon baking powder
1 teaspoon sea salt
½ teaspoon xanthan gum
⅓ cup coconut oil
¼ cup cold water
FOR THE FILLING:
⅓ cup hot water
2 tablespoons ground flaxseed
2 cups pumpkin puree (canned is OK)
1 cup coconut milk
½ cup agave nectar or maple syrup
1 teaspoon ground cinnamon
1 teaspoon ground ginger
½ teaspoon sea salt
½ teaspoon freshly grated or ground nutmeg
¼ teaspoon ground cloves or allspice
To make the crust, in a large bowl, combine the almond meal, arrowroot powder, baking powder, salt, and xanthan gum and mix well. Add the coconut oil and blend with a fork or pastry cutter until the mixture is crumbly. Slowly stir in the water until the dough forms a ball. Divide into 2 equal balls, wrap in plastic wrap, and refrigerate for about 1 hour.
Preheat the oven to 375 degrees F.
To make the filling, in a small bowl, stir together the hot water and flaxseed. Allow the mixture to sit for 10 minutes, then transfer to a large bowl. Add the pumpkin puree, coconut milk, agave, cinnamon, ginger, salt, nutmeg, and cloves and mix thoroughly.
Place 1 dough ball between 2 sheets of waxed paper and gently roll it out to about 10 inches in diameter. (The dough tears easily.) Carefully transfer the dough to a 9-inch pie pan and press it gently in place. Bake for 8 to 10 minutes to warm the crust. The crust may start to brown slightly. Pour the filling into the crust and bake for 35 to 45 minutes, or until the filling no longer jiggles. You may need to place foil around the edges if the crust starts to brown too much.
Let the pie cool completely and refrigerate for 2 to 3 hours before serving.
Tip: If you're not avoiding eggs, you can substitute 2 eggs in place of the hot water and flaxseed mixture.
Mixed Berry Walnut Crumble
Mixed Berry Walnut Crumble
I love to make this dessert when my family comes to visit because everyone likes it, and it can double as breakfast the next morning. Trust me, no one ever complains about eating dessert for breakfast. When I'm serving it for dessert, I like to serve it warm and top it with some Coconut Bliss, a delicious nondairy ice cream substitute.
Makes 10 servings
FOR THE FILLING:
8 cups fresh or frozen mixed berries
¼ cup agave nectar
2 tablespoons arrowroot powder
1 teaspoon ground cinnamon
½ teaspoon ground nutmeg
FOR THE CRUMBLE:
2 cups certified gluten-free oats
1 cup chopped walnuts
½ cup brown rice flour
⅓ cup agave nectar
2 teaspoons ground cinnamon
½ teaspoon ground allspice
½ cup coconut oil, plus more for greasing dish
Preheat the oven to 350 degrees F. Lightly grease a 9-by-13-inch baking dish with coconut oil.
To make the filling, in a large bowl, combine the berries, agave, arrowroot powder, cinnamon, and nutmeg and toss until the berries are well coated. Transfer to the baking dish, cover with foil, and bake for 35 minutes.
Meanwhile, to make the topping, combine the oats, walnuts, flour, agave, cinnamon, and allspice in a medium bowl. Add the coconut oil and blend with a fork or pastry cutter until the mixture is crumbly.
Spread the topping over the fruit filling. Bake, uncovered, for 20 minutes, or until the topping is browned. Allow the crumble to cool slightly before serving.
Rustic Pear and Fig Crostatas
When I prepare this dessert in the cooking classes I teach, I tell students that the beauty of making a "rustic" dessert means that the crust doesn't have to look picture perfect. The amazingly easy gluten-free crust was inspired by a local gluten-free bakery called Flying Apron, and it can be a bit crumbly but it still works extremely well as a rustic wrap for fruit. The rich flavor of the figs blends beautifully with several varieties of pears, including Bartlett, Bosc, and Comice. This is sure to be a hit at any dinner party. Top it with cashew cream or some vanilla Coconut Bliss nondairy frozen dessert and you will think you've died and gone to heaven.
Makes 6 servings
7 to 8 fresh figs, cut into ½-inch chunks
1 large unpeeled pear, thinly sliced
1½ cups brown rice flour, plus more for dusting
¼ teaspoon sea salt
½ cup plus 3 tablespoons coconut oil
3 tablespoons agave nectar or maple syrup
1 to 2 tablespoons cold water
¼ cup honey
Preheat the oven to 350 degrees F.
Combine the figs and pear in a medium bowl and set aside.
In a small bowl, combine the flour and salt. In the bowl of a stand mixer fitted with the paddle attachment, beat the coconut oil until softened, about 1 minute. With the mixer on low speed, slowly add the flour mixture until incorporated. Add the agave and cold water and continue beating until a soft dough has formed. (Alternatively, you can mix the dough by hand, kneading the ingredients to incorporate.)
Dust a large piece of parchment paper (placed on top of a cutting board for ease) and your hands well with brown rice flour. Divide the dough into 6 equal balls. Flatten 1 ball and sprinkle with more flour. Roll the dough into a 4-inch disk. Spread 2 large spoonfuls of the fruit mixture into the center, leaving ½-inch of dough around the edges. Pull the dough up around the edges of the fruit, leaving the very center exposed. Repeat with the remaining dough balls and fruit.
Carefully transfer the parchment with the crostatas to a baking pan. Drizzle honey over the top of each crostata and bake for 25 minutes, or until the crusts are golden brown and the pears are soft.
Pears contain some unique phytonutrients called flavonols, which help improve blood sugar balance, support the immune system, and decrease inflammation.
So-Easy Coconut Mango Sorbet
So-Easy Coconut Mango Sorbet
Rich, creamy coconut milk and naturally sweet mangoes make this dessert seem absolutely sinful—but it's not! You can enjoy this quick and easy treat without a care in the world. For my clients who complain of having uncontrollable ice cream cravings when family members are enjoying theirs in front of them, this sorbet does the trick. You can experiment with different types of fruit, but there's something about the tropical combination of mangoes and coconut milk that provides just the right amount of sweetness and the perfect silky texture.
Makes 4 servings
1 (10-ounce) bag frozen mangoes
1 cup coconut milk
½ cup hazelnut or almond milk
½ teaspoon vanilla extract
Place all the ingredients in food processor or blender and blend until smooth. Serve immediately or freeze until the sorbet is the desired consistency.
Baked Pears or Apples with Cashew Cream
Whenever I make cashew cream, I marvel at the simplicity of it! It's such a naturally delicious topping, and it pairs extremely well with baked pears or apples. Your house will smell like a freshly baked pie, and you'll have a healthy, satisfying dessert! The cashew cream is also great as a dip for raw fruit.
Makes 6 servings
3 large unpeeled pears or apples, cored and halved
¾ cup water, divided
1 to 2 tablespoons ground cinnamon
2 tablespoons agave nectar
1 cup raw cashews
Preheat the oven to 375 degrees F.
Place the pears or apples face down in a medium baking dish. Add ½ cup of the water to the dish, cover with foil, and bake for 45 minutes. Uncover, sprinkle the fruit with cinnamon, drizzle with agave, and bake for another 5 minutes, or until the fruit is soft but not mushy.
Meanwhile, place the cashews in a food processor. With the machine running, slowly drizzle in the remaining ¼ cup water until the cream is the desired consistency.
Scoop the fruit into bowls and top with a heaping spoonful of cashew cream. Serve warm.
Strawberry Rhubarb Crumble
Strawberry rhubarb pie is definitely the way to my husband's heart. It reminds him of spending time in the kitchen with Grandma Babb on the farm. So I knew this recipe was a winner when my strawberry-rhubarb-pie connoisseur gave it the seal of approval and went back for seconds... and thirds. It doesn't last long in our house, but that's exactly what I expect of a good dessert.
Makes 8 servings
FOR THE FILLING:
⅓ cup coconut palm sugar
1½ tablespoons arrowroot powder
⅛ teaspoon sea salt
1 pound trimmed rhubarb, cut into ¼-inch-thick pieces (about 4 cups)
3 cups strawberries, hulled and quartered
FOR THE CRUMBLE:
½ cup coconut palm sugar
½ cup coconut flour
¼ cup white rice flour
1 teaspoon baking powder
½ teaspoon ground cinnamon
¼ teaspoon sea salt
½ cup certified gluten-free rolled oats
3 tablespoons hot water
1 tablespoon ground flaxseed
2 tablespoons agave nectar or maple syrup
¼ cup coconut oil, melted
Preheat the oven to 375 degrees F.
To make the filling, in a medium bowl, combine the coconut palm sugar, arrowroot powder, and salt. Add the rhubarb and strawberries, and gently toss until coated. Spread the fruit in a 10-inch pie pan and set aside.
To make the topping, in another medium bowl, combine the coconut palm sugar, flours, baking powder, cinnamon, and salt. Stir in the oats.
In a small bowl, stir together the hot water and flaxseed. Allow the mixture to sit for 10 minutes, then add it to the dry ingredients. Stir in the agave. Using your hands, knead the ingredients until well combined. Sprinkle the topping evenly over the fruit. Drizzle the coconut oil evenly all over the topping.
Bake the crumble for 45 to 50 minutes, or until the topping turns a golden-brown color and the fruit is bubbling. Allow to cool slightly before serving.
Rhubarb is a good source of powerful antioxidants like lutein and lycopene, so it helps rid the body of free radicals that can cause inflammation and it keeps the immune system in check.
Vanilla Wafer Pudding
This recipe was a happy accident. I was trying to make macadamia nut cream, thinking it would have similar properties to cashew cream. Turns out that's not the case, but I kept adding ingredients to smooth out the texture and then put it the refrigerator and forgot about until the next day. When I pulled it out and had a couple bites, I kept feeling like it reminded me of something very familiar. A couple more bites and it dawned on me... it tastes exactly like vanilla wafers!
Makes 2 servings
1 cup raw macadamia nuts
½ ripe banana
¾ cup water
¼ cup coconut milk
2 teaspoons chia seeds
1 tablespoon agave nectar
Place all the ingredients in a food processor and blend until smooth. Transfer to an airtight container and refrigerate for at least 2 hours or overnight.
No-Bake Peach Pie
As a person who enjoys cooking much more than baking, there are a few qualities that make some desserts more desirable than others for me and this dessert has them all. It's simple, doesn't require baking, and it's absolutely decadent! I've made this pie using a variety of fruits, including mangoes, mixed berries, pears, and even fresh figs. You can't go wrong topping it with whatever is in season. Be sure to cut thin slices because the coconut cashew cream is extraordinarily rich and filling.
Makes 10 to 12 servings
FOR THE CRUST:
3 cups raw walnuts
¾ cup pitted dates (I prefer Medjool dates)
1 tablespoon ground cinnamon
¼ teaspoon sea salt
FOR THE FILLING:
¾ cup hot water
½ cup creamed coconut, chopped into chunks
2 cups raw cashews
2 tablespoons maple syrup
Seeds from 1 vanilla bean
3 ripe peaches, peeled and thinly sliced
To make the crust, combine the walnuts, dates, cinnamon, and salt in a food processor and blend until the ingredients are finely chopped and well combined.
Scrape the mixture into a 9-inch pie pan greased with coconut oil. Press into the bottom of the pan, pushing the crust up to reach the top edge all around. Wash the food processor before proceeding.
To make the filling, put the hot water and creamed coconut in the food processor and let sit for 5 minutes so the coconut can soften. Add the cashews, maple syrup, and vanilla bean seeds and blend until smooth.
Spread the cream evenly into the crust. Arrange the peaches in a circular pattern over the cream filling. Can be served immediately.
Walnuts contain higher amounts of omega-3s than any other nut and are a rich source of unique anti-inflammatory phytonutrients.
# Acknowledgments
It takes a village to write a cookbook, and I couldn't have done it without the constant encouragement and support from my incredibly patient husband along with my family, friends, and colleagues. Barb Schiltz put me on the path to becoming a nutritionist and I will be forever indebted to her for helping me find my true passion. She's my mentor, sage (in the non-herbal sense), and favorite cooking companion.
I developed my love for cooking later in life, when I was a grad student at Bastyr University. I took my first whole foods cooking class from Cynthia Lair and I was hooked. She made healthy cooking accessible and exciting and she didn't even ridicule me when I showed up on the first day with an herb knife instead of a chef's knife.
Four years later, I was developing and teaching my own cooking classes with a nutritional bent. I will be infinitely grateful to the incredibly organized team at Puget Consumers Co-op (PCC), Marilyn McCormick, Jackie DeCicco, and Alicia Guy, to name a few. It's because of them that the anti-inflammatory cooking classes I teach get noticed and are well attended. I'm proud to be a part of a co-op that can boast being one of the largest cooking schools in the nation.
The recipes in this book are a labor of love and a reflection of my passion. They also went through several rounds of tasting and tweaking. Heartfelt gratitude to my fabulous recipe testers, Bill Babb, Maribeth Evezich, Kim Campbell, Jenny Harris, Rob Sweet, and Vicki Duffy. You've made these delicious dishes infinitely better with your thoughtful comments and food-savvy suggestions. And a big thank you to my mom, who didn't make the cut for recipe tester because of her mysterious aversion to all herbs and spices, but she's always there to cheer me on and celebrate my successes.
Now let's eat!
# Index
Note: Photographs are indicated by italics.
## A
anti-inflammation, about
anti-inflammatory pantry
anti-inflammatory plate
Artichoke and Basil Tapenade
Artichoke and Black Bean Hummus
Asparagus and Adzuki Beans, Rainbow Quinoa with Roasted
Asparagus and Sunchoke Soup, Creamy
Avocado and Chickpeas, Breakfast Burrito with
Avocado Spinach Dip, Creamy
Avocado Strawberry Salsa, Fish Taco Salad with, 6.1, 6.2
Avocado Tartine, Smoked Salmon and, 1.2
Avocado Vinaigrette, Kale and Kohlrabi Salad with Creamy
## B
Bars, Tropical Quinoa Power
beans. See legumes beef
Steak Salad with Massaged Kale, 7.2
Sweet Potato Shepherd's Pie
Veggie Beef Burger with Rocket Salad
beets
Golden Beet and Mushroom Faux Gratin
Oven-Roasted Beets with Sautéed Greens
Wilted Kale Salad with Shredded Beets and Carrots
Berry Green Power Smoothie
Berry Parfait with Coconut Cashew Cream, Fresh, 1.5
Berry Walnut Crumble, Mixed, 8.1, 8.2
Bison Lettuce Cups with Garnet Yam Home Fries, 7.5
Brazil Nut Topping, Mango Muesli with
breakfasts
broccoli
Mediterranean Salmon Skewers, 6.3
Slow-Cooked Black Bean and Broccoli Stew
Spaghetti Squash Primavera with Basil Walnut Pesto
Vegetable and Chicken Pho
Wild Rice and Roasted Vegetables
brussels sprouts
Brussels Sprout Slaw, 4.5
Sizzling Salmon and Quinoa Skillet
Warm Brussels Sprout Salad with Pecans and Currants
Buckwheat Polenta Stacks, Southwestern-Style
Burgers, Toasted Pecan Quinoa
Burger with Rocket Salad, Veggie Beef
Burrito with Chickpeas and Avocado, Breakfast
## C
Cabbage and Snow Pea Sauté, Shredded
Carrot and Ginger Soup, Caramelized, 3.5
Cashew Cream, Baked Pears or Apples with
Cashew Cream, Fresh Berry Parfait with Coconut, 1.6
Cauliflower Soup with Gremolata, Roasted
Cauliflower-Yam Crust, Veggie Pizza with, 5.1, 5.2
Celery Root Puree, Salmon en Papillote with Silky
chard. See greens Chia Seeds, Power-Packed Granola with Currants and, 1.1, 1.2
Chicken and Vegetable Pho
cleanse, fm4.1-day
foods to include/exclude
meal and snack ideas
menu plan for grain avoider
menu plan for repeater
menu plan for variety lover
menu plans, overview of
overview, fm4.1, fm4.2
phase I (preparation)
phase II (nourish and cleanse)
phase III (reintroduction)
phase IV (transition to long-term plan)
portion control
tips for, fm4.1, fm4.2
Coconut Energy Truffles, Nutty, 2.1, 2.2
Coconut Mango Sorbet, So-Easy, 8.5
Collard Rolls, Quinoa-Stuffed
Crepes, Sweet or Savory Quinoa
Crostatas, Rustic Pear and Fig
Crumble, Mixed Berry Walnut, 8.1, 8.2
Crumble, Strawberry Rhubarb
curry
Crispy Curried Chickpeas, 2.1, 2.2
Pan-Fried Sardines with Sautéed Kale and Chard, 6.1, 6.2
Pumpkin Coconut Curry with White Fish
## D
desserts
Dip, Creamy Avocado Spinach, 2.6
## E
Escarole and Black-Eyed Pea Soup, 3.1, 3.2
## F
Fennel and Braised Greens with Silky Walnut Sauce, Roasted
Fig and Pear Crostatas, Rustic
fish
Fish Taco Salad with Strawberry Avocado Salsa, 6.1, 6.2
Hazelnut-Encrusted Halibut with Dipping Sauce
Oven-Roasted Black Cod with Smashed Sweet Peas
Poached White Fish with Mango Lime Chutney
Pumpkin Coconut Curry with White Fish
Salmon and Avocado Tartine, Smoked, 1.8
Salmon and Quinoa Skillet, Sizzling
Salmon en Papillote with Silky Celery Root Puree
Salmon Skewers, Mediterranean, 6.6
Sweet or Savory Quinoa Crepes
## G
Ginger Soup, Caramelized Carrot and, 3.6
Granola with Currants and Chia Seeds, Power-Packed, 1.1, 1.2
Gratin, Golden Beet and Mushroom Faux
greens
Berry Green Power Smoothie
Black-Eyed Pea and Escarole Soup, 3.1, 3.2
Black-Eyed Peas and Forbidden Rice with Crispy Kale
Breakfast Burrito with Chickpeas and Avocado
Breakfast Rice with Crumbled Nori
Creamy Avocado Spinach Dip, 2.8
Lentil and Spinach Stew with Roasted Garlic
Mediterranean White Bean Soup
Pan-Fried Sardines with Sautéed Kale and Chard, 6.1, 6.2
Puttanesca-Style Beans and Greens, 5.8
Quinoa-Stuffed Collard Rolls
Southwestern-Style Buckwheat Polenta Stacks
Spaghetti Squash Primavera with Basil Walnut Pesto
Spring Lamb Stew
Veggie Pizza with Cauliflower-Yam Crust, 5.1, 5.2
See also salads, main dish; salads and sides
## H
Hummus, Black Bean and Artichoke
Hummus, White Bean and Kalamata Olive
## I
inflammation, about
## J
Jicama and Spring Pea Salad, 4.1, 4.2
## K
kale. See greens
Kohlrabi and Kale Salad with Creamy Avocado Vinaigrette
## L
lamb
Moroccan Lamb Tagine with Chickpeas and Apricots
Spring Lamb Stew
Sweet Potato Hash with Lamb Sausage
Sweet Potato Shepherd's Pie
legumes, fm3.1, fm4.1
Bhutanese Rice and Flageolet Bean Salad
Black Bean and Artichoke Hummus
Black-Eyed Pea and Escarole Soup, 3.1, 3.2
Braised Greens and Roasted Fennel with Silky Walnut Sauce
Breakfast Burrito with Chickpeas and Avocado
Butternut Squash and White Bean Soup
Crispy Curried Chickpeas, 2.1, 2.2
Fish Taco Salad with Strawberry Avocado Salsa, 6.1, 6.2
Hearty Mushroom and Lentil Stew
Lentil and Spinach Stew with Roasted Garlic
Mediterranean White Bean Soup
Moroccan Lamb Tagine with Chickpeas and Apricots
Puttanesca-Style Beans and Greens, 5.2
Rainbow Quinoa with Roasted Asparagus and Adzuki Beans
Slow-Cooked Black Bean and Broccoli Stew
Southwestern-Style Buckwheat Polenta Stacks
Three-Bean Stew with Red Quinoa
Toasted Pecan Quinoa Burgers
White Bean and Kalamata Olive Hummus
Zucchini Noodles with Pistachio Pesto and Black Lentils, 5.1, 5.2
lentils. See legumes
Lettuce Cups with Garnet Yam Home Fries, Bison, 7.9
## M
Mackerel with Wasabi "Mayo," Nori-Wrapped
main dishes, hint-of-meat
main dishes, pescatarian
main dishes, vegetarian
Mango Lime Chutney, Poached White Fish with
Mango Muesli with Brazil Nut Topping
Mango Sorbet, So-Easy Coconut, 8.9
Mediterranean-style diet, about
menu plans and shopping lists, fm4.1, fm4.2
Muesli with Brazil Nut Topping, Mango
Mushroom and Golden Beet Faux Gratin
Mushroom and Lentil Stew, Hearty
Mushroom and Walnut Pâté, Shiitake
Mushrooms with Samosa Filling, Portobello
## N
Nori, Breakfast Rice with Crumbled
Nori-Wrapped Mackerel with Wasabi "Mayo,"
## O
oats
Creamy Asparagus and Sunchoke Soup
Mango Muesli with Brazil Nut Topping
Mixed Berry Walnut Crumble, 8.1, 8.2
Power-Packed Granola with Currants and Chia Seeds, 1.1, 1.2
Strawberry Rhubarb Crumble
olives
Artichoke and Basil Tapenade
Creamy Avocado Spinach Dip, 2.6
Mediterranean Salmon Skewers, 6.8
Puttanesca-Style Beans and Greens, 5.5
Rainbow Quinoa with Roasted Asparagus and Adzuki Beans
White Bean and Kalamata Olive Hummus
## P
pantry, anti-inflammatory
Parfait with Coconut Cashew Cream, Fresh Berry, 1.6
Pâté, Shiitake Mushroom and Walnut
Pea and Jicama Salad, Spring, 4.1, 4.2
Peach Pie, No-Bake
Pear and Fig Crostatas, Rustic
Pears or Apples with Cashew Cream, Baked
pescatarian diet, about
Pho, Vegetable and Chicken
Pistachio Pesto and Black Lentils, Zucchini Noodles with, 5.1, 5.2
Pizza with Cauliflower-Yam Crust, Veggie, 5.1, 5.2
Pomegranate and Toasted Hazelnuts, Super Greens Salad with, 4.1, 4.2
Pudding, Vanilla Wafer
Pumpkin Coconut Curry with White Fish
Pumpkin Coconut Pie with Almond Crust, 8.7
## Q
quinoa
Quinoa-Stuffed Collard Rolls
Rainbow Quinoa with Roasted Asparagus and Adzuki Beans
Sizzling Salmon and Quinoa Skillet
Sweet or Savory Quinoa Crepes
Three-Bean Stew with Red Quinoa
Toasted Pecan Quinoa Burgers
Tropical Quinoa Power Bars
## R
Rhubarb Crumble, Strawberry
rice
Bhutanese Rice and Flageolet Bean Salad
Black-Eyed Peas and Forbidden Rice with Crispy Kale
Breakfast Rice with Crumbled Nori
Mediterranean White Bean Soup
Nori-Wrapped Mackerel with Wasabi "Mayo,"
Wild Rice and Roasted Vegetables
## S
salads, main dish
Fish Taco Salad with Strawberry Avocado Salsa, 6.1, 6.2
Steak Salad with Massaged Kale, 7.4
Veggie Beef Burger with Rocket Salad
salads and sides
Salsa, Fish Taco Salad with Strawberry Avocado, 6.1, 6.2
Sardines with Sautéed Kale and Chard, Pan-Fried, 6.1, 6.2
Shepherd's Pie, Sweet Potato
shopping lists and menu plans, fm4.1, fm4.2
Slaw, Brussels Sprout, 4.8
Smoothie, Berry Green Power
snack and meal ideas
snacks, 2.1
Snow Pea and Cabbage Sauté, Shredded
Sorbet, So-Easy Coconut Mango, 8.8
soups and stews, 3.1, 5.1, 7.1
spinach. See greens
squash and zucchini
Bison Lettuce Cups with Garnet Yam Home Fries, 7.4
Black-Eyed Pea and Escarole Soup, 3.1, 3.2
Black-Eyed Peas and Forbidden Rice with Crispy Kale
Butternut Squash and White Bean Soup
Mediterranean Salmon Skewers, 6.2
Pumpkin Coconut Curry with White Fish
Pumpkin Coconut Pie with Almond Crust, 8.7
Spaghetti Squash Primavera with Basil Walnut Pesto
Three-Bean Stew with Red Quinoa
Veggie Beef Burger with Rocket Salad
Veggie Pizza with Cauliflower-Yam Crust, 5.1, 5.2
Wild Rice and Roasted Vegetables
Zucchini Noodles with Pistachio Pesto and Black Lentils, 5.1, 5.2
Strawberry Avocado Salsa, Fish Taco Salad with, 6.1, 6.2
Strawberry Rhubarb Crumble
sunchokes
Creamy Asparagus and Sunchoke Soup
Salmon en Papillote with Silky Celery Root Puree
sweet potatoes
Bison Lettuce Cups with Garnet Yam Home Fries, 7.1
Portobello Mushrooms with Samosa Filling
Sweet Potato Hash with Lamb Sausage
Sweet Potato Shepherd's Pie
Three-Bean Stew with Red Quinoa
Veggie Pizza with Cauliflower-Yam Crust, 5.1, 5.2
## T
Tagine with Chickpeas and Apricots, Moroccan Lamb
Tapenade, Artichoke and Basil
Tartine, Smoked Salmon and Avocado, 1.2
Trail Mix, Anti-Inflammatory
Truffles, Nutty Coconut Energy, 2.1, 2.2
turmeric
Breakfast Burrito with Chickpeas and Avocado
Crispy Curried Chickpeas, 2.1, 2.2
Moroccan Lamb Tagine with Chickpeas and Apricots
Roasted Cauliflower Soup with Gremolata
Shredded Cabbage and Snow Pea Sauté
Slow-Cooked Black Bean and Broccoli Stew
## W
walnuts
Braised Greens and Roasted Fennel with Silky Walnut Sauce
Mixed Berry Walnut Crumble, 8.1, 8.2
No-Bake Peach Pie
Nutty Coconut Energy Truffles, 2.1, 2.2
Shiitake Mushroom and Walnut Pâté
Spaghetti Squash Primavera with Basil Walnut Pesto
## Y
yams. See sweet potatoes
## Z
zucchini. See squash and zucchini
# Conversions
VOLUME
---
US | METRIC | IMPERIAL
¼ tsp. | 1.25 ml |
½ tsp. | 2.5 ml |
1 tsp. | 5 ml |
½ Tbsp. | 7.5 ml |
1 Tbsp. | 15 ml |
⅛ c. | 30 ml | 1 fl. oz.
¼ c. | 60 ml | 2 fl. oz.
⅓ C. | 80 ml | 2.5 fl. oz.
½ c. | 125 ml | 4 fl. oz.
1 c. | 250 ml | 8 fl. oz.
2 c. (1 pt.) | 500 ml | 16 fl. oz.
1 qt. | 1 l | 32 fl. oz.
LENGTH
---
US | METRIC
⅛ in. | 3 mm
½ in. | 6 mm
½ in. | 1.25 cm
1 in. | 2.5 cm
1 ft. | 30 cm
WEIGHT
---
AVOIRDUPOIS | METRIC
¼ oz. | 7 g
½ oz. | 15 g
1 oz. | 30 g
2 oz. | 60 g
3 oz. | 90 g
4 oz. | 115 g
5 oz. | 150 g
6 oz. | 175 g
7 oz. | 200 g
8 oz. (½ lb.) | 225 g
9 oz. | 250 g
10 oz. | 300 g
11 oz. | 325 g
12 oz. | 350 g
13 oz. | 375 g
14 oz. | 400 g
15 oz. | 425 g
16 oz. (1 lb.) | 450 g
1½ lb. | 750 g
2 lb. | 900 g
2¼ lb. | 1 kg
3 lb. | 1.4 kg
4 lb. | 1.8 kg
TEMPERATURE
---
OVEN MARK | FAHRENHEIT | CELSIUS | GAS
Very cool | 250–275 | 130–140 | ½–1
Cool | 300 | 150 | 2
Warm | 325 | 165 | 3
Moderate | 350 | 175 | 4
Moderately hot | 375 | 190 | 5
| 400 | 200 | 6
Hot | 425 | 220 | 7
| 450 | 230 | 8
Very Hot | 475 | 245 | 9
Photograph by Julie Sotomura
# About the Author
MICHELLE BABB is a registered dietitian with a private practice in West Seattle, where she specializes in mind-body nutrition, weight management, and inflammatory digestive disorders.
Michelle developed a passion for cooking when she was a student at Bastyr, and now teaches nutrition-focused cooking classes at Puget Consumers Co-op. Her recipes often feature underappreciated ingredients, like beets, brussels sprouts, and Jerusalem artichokes. She takes great pleasure in converting dubious meat and potato lovers into vegetable enthusiasts.
When she's not in the kitchen, Michelle enjoys running, kayaking, sailing, and traveling. She also loves to write and is co-author of The Imperfect Perfectionist: Seasonal Secrets for a Happy and Balanced Life. Learn more about Michelle at EatPlayBe.com.
| {
"redpajama_set_name": "RedPajamaBook"
} | 14 |
Adesso o mai è il secondo album di Daniele Stefani.
Tracce
CD, Download digitale
; | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,990 |
Q: C# WPF проверка CheckBox по нумерации Доброе время суток!
Хочу создать окно авторизации, при условии что CheckBox должны быть выделены по очередности...
Если чек боксы выделяются так 1,2,3,4 то авторизация пройдена, в противном случае Ошибка! Например если пользователь будет выделять CheckBox, не по нумерации а 2,3,1,4. в таком порядке, то .
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,601 |
'use strict';
jest.useFakeTimers();
describe('timerGame', () => {
beforeEach(() => {
jest.spyOn(globalThis, 'setTimeout');
});
it('waits 1 second before ending the game', () => {
const timerGame = require('../timerGame');
timerGame();
expect(setTimeout).toHaveBeenCalledTimes(1);
expect(setTimeout).toHaveBeenCalledWith(expect.any(Function), 1000);
});
it('calls the callback after 1 second via runAllTimers', () => {
const timerGame = require('../timerGame');
const callback = jest.fn();
timerGame(callback);
// At this point in time, the callback should not have been called yet
expect(callback).not.toHaveBeenCalled();
// Fast-forward until all timers have been executed
jest.runAllTimers();
// Now our callback should have been called!
expect(callback).toHaveBeenCalled();
expect(callback).toHaveBeenCalledTimes(1);
});
it('calls the callback after 1 second via advanceTimersByTime', () => {
const timerGame = require('../timerGame');
const callback = jest.fn();
timerGame(callback);
// At this point in time, the callback should not have been called yet
expect(callback).not.toHaveBeenCalled();
// Fast-forward until all timers have been executed
jest.advanceTimersByTime(1000);
// Now our callback should have been called!
expect(callback).toHaveBeenCalled();
expect(callback).toHaveBeenCalledTimes(1);
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,587 |
Justia Patents Solid Particles Dispersed In Solid Solution Or MatrixUS Patent for Reactive compositions including metal Patent (Patent # 8,361,258)
Reactive compositions including metal
Oct 20, 2011 - Alliant Techsystems Inc.
A precursor composition of a reactive material that comprises a metal material and an energetic material, such as at least one oxidizer or at least one class 1.1 explosive. The metal material defines a continuous phase at a processing temperature of the precursor composition and the energetic material is dispersed therein. The metal material may be a fusible metal alloy having a melting point ranging from approximately 46° C. to approximately 250° C. The fusible metal alloy may include at least one metal selected from the group consisting of bismuth, lead, tin, cadmium, indium, mercury, antimony, copper, gold, silver, and zinc. The reactive composition may have a density of greater than approximately 2 g/cm3. The reactive composition may also include a polymer/plasticizer system.
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This application is a divisional of U.S. patent application Ser. No. 11/620,205, filed Jan. 5, 2007, now U.S. Pat. No. 8,075,715, issued Dec. 13, 2011, which is a continuation of U.S. patent application Ser. No. 10/801,946, filed Mar. 15, 2004, now abandoned. The disclosure of each of the previously referenced U.S. patent applications is hereby incorporated herein in its entirety by reference.
The present application is also related to U.S. Provisional Patent Application No. 60/368,284, filed Mar. 28, 2002, entitled "Low Temperature, Extrudable, High Density Reactive Materials," now abandoned; U.S. Pat. No. 6,962,634, issued Nov. 8, 2005, entitled "Low Temperature, Extrudable, High Density Reactive Materials," U.S. patent application Ser. No. 12/507,605, filed Jul. 22, 2009, entitled "Low Temperature, Extrudable, High Density Reactive Materials,"; U.S. Provisional Patent Application No. 60/184,316, filed Feb. 23, 2000, entitled "High Strength Reactive Materials," now abandoned; U.S. Pat. No. 6,593,410, issued Jul. 15, 2003, entitled "High Strength Reactive Materials," U.S. Pat. No. 7,307,117, issued Dec. 11, 2007, entitled "High Strength Reactive Materials And Methods Of Making," U.S. Provisional Application No. 60/553,430, filed Mar. 15, 2004, entitled "Reactive Material Enhanced Projectiles and Related Methods," now abandoned; U.S. Pat. No. 7,603,951, issued Oct. 20, 2009, entitled "Reactive Material Enhanced Projectiles and Related Methods," U.S. patent application Ser. No. 10/801,948, filed Mar. 15, 2004, entitled "Reactive Material Enhanced Munition Compositions and Projectiles Containing Same," now abandoned; U.S. patent application Ser. No. 12/127,627, filed May 27, 2008, entitled "Reactive Material Enhanced Munition Compositions and Projectiles Containing Same,"; U.S. Provisional Application No. 60/723,465, filed Oct. 4, 2005, entitled "Reactive Material Enhanced Projectiles And Related Methods," now abandoned; U.S. patent application Ser. No. 11/538,763, filed Oct. 4, 2006, entitled "Reactive Material Enhanced Projectiles And Related Methods," now U.S. Pat. No. 8,122,833, issued Feb. 28, 2012; U.S. Pat. No. 7,614,348, issued Nov. 10, 2009, entitled "Weapons And Weapon Components Incorporating Reactive Materials," U.S. patent application Ser. No. 11/697,005, filed Apr. 5, 2007, entitled "Consumable Reactive Material Fragments, Ordnance Incorporating Structures For Producing The Same, And Methods Of Creating The Same," pending; and U.S. Pat. No. 7,977,420, issued Jul. 12, 2011, entitled "Reactive Material Compositions, Shot Shells Including Reactive Materials, and a Method of Producing Same." The disclosure of each of the previously referenced U.S. patent applications and U.S. patents is hereby incorporated herein in its entirety by reference.
This invention relates generally to an insensitive, highly energetic composition. More specifically, the invention relates to a composition that includes a metal material and an energetic material.
Many explosive, pyrotechnic, and incendiary compositions are known in the art. To form these compositions, a fuel is typically dispersed in an organic, energetic material, such as in trinitrotoluene ("TNT"). TNT is commonly used as the energetic material in explosive compositions because it is stable and insensitive. Some common examples of military explosives that include TNT are tritonal, cyclotol, Composition B, DBX, and octol. Tritonal includes 20% aluminum and 80% TNT. Cyclotol includes 65%-75% cyclo-1,3,5-trimethylene-2,4,6-trinitramine ("RDX"; also known as hexogen or cyclonite) and 25-35% TNT. Composition B includes 60-64% RDX and 36-40% TNT. DBX includes 21% RDX, 21% ammonium nitrate, 18% aluminum, and 40% TNT. Octol includes 70-75% cyclotetramethylene tetranitramine ("HMX"; also known as octogen) and 25-30% TNT. These TNT-containing explosive compositions are produced into a usable form by casting or pressing processes. Casting is more versatile and convenient for loading the explosive, pyrotechnic, or incendiary composition than pressing and, therefore, is a more desirable process.
In casting, the energetic material is heated to a temperature above its melting point to produce a liquid phase, which is also referred to as a melt phase or a casting material. The energetic material is melted by placing it in a vessel, such as a kettle, and heating to a temperature above its melting point. The fuel, which is typically a solid material, is then dispersed in the organic melt phase. In such a mixture, the energetic material forms a continuous phase and the fuel is a dispersed phase. The mixture is poured into a container, such as a mold or a charge case, and allowed to solidify by cooling to produce the explosive, pyrotechnic, or incendiary composition. This technique is known as a "melt-pour" process because the energetic material is melted, the fuel is added, and the resulting mixture is poured into the desired mold. Many explosive, pyrotechnic, or incendiary compositions that contain TNT as an energetic material are produced by melt-pour processes because TNT has a relatively low melting point compared to the other components in conventional compositions. TNT has a melting point of approximately 81° C. and remains a liquid at temperatures ranging from approximately 81° C. to 105° C. In contrast, many other chemical components of the explosive, pyrotechnic, or incendiary compositions, such as RDX and HMX, have melting points greater than 200° C. One example of an explosive composition produced by a melt-pour process is tritonal, which contains aluminum and TNT. The aluminum is present as a powder and is dispersed in the TNT.
Explosive, pyrotechnic, and incendiary compositions also typically have a density of 1.5 g/cm3-1.7 gm/cm3. However, explosive, pyrotechnic, or incendiary compositions with higher densities have improved performance attributes and, therefore, are desired. While the performance attributes cannot be expressed by a single parameter, military explosives typically require a higher performance concentration per unit volume, a faster reaction rate, an increased detonation velocity, and a larger impact effect of detonation than industrial explosives. However, the performance attributes of military explosives also depend on a desired application for the explosive composition. For instance, if the explosive, pyrotechnic, or incendiary composition is used in mines, bombs, mine projectiles, or rocket warhead charges, the composition should have a high gas impact, a large gas volume, and a high heat of explosion. If the explosive, pyrotechnic, or incendiary composition is used in grenades, the composition should have a high speed splinter formation, a high loading density, and a high detonation velocity. In shaped charges, the explosive, pyrotechnic, or incendiary composition should have a high density, a high detonation velocity, a high strength, and high brisance. Brisance is the destructive fragmentation effect of a charge on its immediate vicinity and is used to measure the effectiveness of the composition. Brisance depends on the detonation velocity, heat of explosion, gas yield, and compactness or density of the composition.
Numerous explosive compositions are known in the art. As described in U.S. Pat. No. 5,339,624, WO 93/21135, and EP 0487472, all to Calsson et al., an explosive composition having a mechanical alloy is disclosed. The mechanical alloy is formed from solid dispersions of metallic materials, with at least one of the metallic materials being a ductile metal. The metallic materials react exothermically with one another to form a fusible alloy that provides additional energy to the explosion. The metallic materials include titanium, boron, zirconium, nickel, manganese and aluminum.
It would be desirable to produce a composition that is highly insensitive and highly energetic for use in military and industrial explosives. Optionally, the desired composition would be suitable for production in existing melt-pour facilities so that new equipment and facilities do not have to be developed.
The present invention comprises a reactive composition that includes a metal material and an energetic material, such as at least one oxidizer, at least one class 1.1 explosive, or mixtures thereof. The metal material defines a continuous phase and has the energetic material dispersed therein. The metal material may have a density greater than approximately 7 g/cm3 and may be a fusible metal alloy having a melting point ranging from approximately 46° C. to approximately 250° C. The fusible metal alloy may include at least one metal selected from the group consisting of bismuth, lead, tin, cadmium, indium, mercury, antimony, copper, gold, silver, and zinc. The energetic material may be selected from the group consisting of ammonium perchlorate, potassium perchlorate, sodium nitrate, potassium nitrate, ammonium nitrate, lithium nitrate, rubidium nitrate, cesium nitrate, lithium perchlorate, sodium perchlorate, rubidium perchlorate, cesium perchlorate, magnesium perchlorate, calcium perchlorate, strontium perchlorate, barium perchlorate, barium peroxide, strontium peroxide, copper oxide, trinitrotoluene, cyclo-1,3,5-trimethylene-2,4,6-trinitramine, cyclotetramethylene tetranitramine, hexanitrohexaazaisowurtzitane, 4,10-dinitro-2,6,8,12-tetraoxa-4,10-diazatetracyclo-[5.5.0.05,9.03,11]-dodecane, 1,3,3-trinitroazetidine, ammonium dinitramide, 2,4,6-trinitro-1,3,5-benzenetriamine, dinitrotoluene, sulfur, and mixtures thereof. The reactive composition may have a density greater than approximately 2 g/cm3.
The reactive composition may further include a polymer/plasticizer system. The polymer/plasticizer system may include at least one polymer selected from the group consisting of polyglycidyl nitrate, nitratomethylmethyloxetane, polyglycidyl azide, diethyleneglycol triethyleneglycol nitraminodiacetic acid terpolymer, poly(bis(azidomethyl)oxetane), poly(azidomethylmethyl-oxetane), poly(nitraminomethyl methyloxetane), poly(bis(difluoroaminomethyl)oxetane), poly(difluoroaminomethylmethyloxetane), copolymers thereof, cellulose acetate butyrate, nitrocellulose, nylon, polyester, fluoropolymers, energetic oxetanes, waxes, and mixtures thereof. The polymer/plasticizer system may also include at least one plasticizer selected from the group consisting of bis(2,2-dinitropropyl)acetal/bis(2,2-dinitropropyl)formal, dioctyl sebacate, dimethylphthalate, dioctyladipate, glycidyl azide polymer, diethyleneglycol dinitrate, butanetrioltrinitrate, butyl-2-nitratoethyl-nitramine, trimethylolethanetrinitrate, triethylene glycoldinitrate, nitroglycerine, isodecylperlargonate, dioctylphthalate, dioctylmaleate, dibutylphthalate, di-n-propyl adipate, diethylphthalate, dipropylphthalate, citroflex, diethyl suberate, diethyl sebacate, diethyl pimelate, and mixtures thereof.
The present invention also comprises a method of producing a reactive composition. The method includes providing a metal material in a liquid state and adding an energetic material to the metal material. The metal material may be a fusible metal alloy having a melting point below a processing temperature of the reactive composition. For instance, the metal material may be a fusible metal alloy having a melting point ranging from approximately 46° C. to approximately 250° C. The fusible metal alloy may include at least one metal selected from the group consisting of bismuth, lead, tin, cadmium, indium, mercury, antimony, copper, gold, silver, and zinc. The energetic material may be selected from the group consisting of ammonium perchlorate, potassium perchlorate, sodium nitrate, potassium nitrate, ammonium nitrate, lithium nitrate, rubidium nitrate, cesium nitrate, lithium perchlorate, sodium perchlorate, rubidium perchlorate, cesium perchlorate, magnesium perchlorate, calcium perchlorate, strontium perchlorate, barium perchlorate, barium peroxide, strontium peroxide, copper oxide, trinitrotoluene, cyclo-1,3,5-trimethylene-2,4,6-trinitramine, cyclotetramethylene tetranitramine, hexanitrohexaazaisowurtzitane, 4,10-dinitro-2,6,8,12-tetraoxa-4,10-diazatetracyclo-[5.5.0.05,9.03,11]-dodecane, 1,3,3-trinitroazetidine, ammonium dinitramide, 2,4,6-trinitro-1,3,5-benzenetriamine, dinitrotoluene, sulfur, and mixtures thereof. The reactive composition may have a density greater than approximately 2 g/cm3.
The method may further include adding a polymer/plasticizer system to the reactive composition. The polymer/plasticizer system may include at least one polymer selected from the group consisting of polyglycidyl nitrate, nitratomethylmethyloxetane, polyglycidyl azide, diethyleneglycol triethyleneglycol nitraminodiacetic acid terpolymer, poly(bis(azidomethyl)-oxetane), poly(azidomethylmethyl-oxetane), poly(nitraminomethyl methyloxetane), poly(bis(difluoroaminomethyl)oxetane), poly(difluoroaminomethylmethyloxetane), copolymers thereof, cellulose acetate butyrate, nitrocellulose, nylon, polyester, fluoropolymers, energetic oxetanes, waxes, and mixtures thereof. The polymer/plasticizer system may also include at least one plasticizer selected from the group consisting of bis(2,2-dinitropropyl)acetal/bis(2,2-dinitropropyl)formal, dioctyl sebacate, dimethylphthalate, dioctyladipate, glycidyl azide polymer, diethyleneglycol dinitrate, butanetrioltrinitrate, butyl-2-nitratoethyl-nitramine, trimethylolethanetrinitrate, triethylene glycoldinitrate, nitroglycerine, isodecylperlargonate, dioctylphthalate, dioctylmaleate, dibutylphthalate, di-n-propyl adipate, diethylphthalate, dipropylphthalate, citroflex, diethyl suberate, diethyl sebacate, diethyl pimelate, and mixtures thereof.
The present invention also comprises a method of improving homogeneity of a reactive composition. The method includes providing a metal material in a liquid state. The metal material may be a fusible metal alloy having a melting point ranging from approximately 46° C. to approximately 250° C. The fusible metal alloy may include at least one metal selected from the group consisting of bismuth, lead, tin, cadmium, indium, mercury, antimony, copper, gold, silver, and zinc. The metal material may be present in the reactive composition from approximately 13.5% by weight to approximately 85% by weight. An energetic material is added to the metal material in the liquid state. The energetic material may be selected from the group consisting of ammonium perchlorate, potassium perchlorate, sodium nitrate, potassium nitrate, ammonium nitrate, lithium nitrate, rubidium nitrate, cesium nitrate, lithium perchlorate, sodium perchlorate, rubidium perchlorate, cesium perchlorate, magnesium perchlorate, calcium perchlorate, strontium perchlorate, barium perchlorate, barium peroxide, strontium peroxide, copper oxide, trinitrotoluene, cyclo-1,3,5-trimethylene-2,4,6-trinitramine, cyclotetramethylene tetranitramine, hexanitrohexaazaisowurtzitane, 4,10-dinitro-2,6,8,12-tetraoxa-4,10-diazatetracyclo-[5.5.0.05,9.03,11]-dodecane, 1,3,3-trinitroazetidine, ammonium dinitramide, 2,4,6-trinitro-1,3,5-benzenetriamine, dinitrotoluene, sulfur, and mixtures thereof.
A polymer/plasticizer system is added to a mixture of the energetic material and the metal material. The polymer/plasticizer system may include at least one polymer selected from the group consisting of polyglycidyl nitrate, nitratomethylmethyloxetane, polyglycidyl azide, diethyleneglycol triethyleneglycol nitraminodiacetic acid terpolymer, poly(bis(azidomethyl)-oxetane), poly(azidomethylmethyl-oxetane), poly(nitraminomethyl methyloxetane), poly(bis(difluoroaminomethyl)oxetane), poly(difluoroaminomethylmethyloxetane), copolymers thereof, cellulose acetate butyrate, nitrocellulose, nylon, polyester, fluoropolymers, energetic oxetanes, waxes, and mixtures thereof. The polymer/plasticizer system may also include at least one plasticizer selected from the group consisting of bis(2,2-dinitropropyl)acetal/bis(2,2-dinitropropyl)formal, dioctyl sebacate, dimethylphthalate, dioctyladipate, glycidyl azide polymer, diethyleneglycol dinitrate, butanetrioltrinitrate, butyl-2-nitratoethyl-nitramine, trimethylolethanetrinitrate, triethylene glycoldinitrate, nitroglycerine, isodecylperlargonate, dioctylphthalate, dioctylmaleate, dibutylphthalate, di-n-propyl adipate, diethylphthalate, dipropylphthalate, citroflex, diethyl suberate, diethyl sebacate, diethyl pimelate, and mixtures thereof.
While the specification concludes with claims particularly pointing out and distinctly claiming that which is regarded as the present invention, the advantages of this invention may be more readily ascertained from the following description of the invention when read in conjunction with the accompanying drawings in which:
FIGS. 1-3 illustrate compressive strength test results of reactive compositions according to the present invention that include the polymer/plasticizer system; and
FIGS. 4-7 show photographs of pellets of the reactive compositions before and after the compressive strength tests.
A reactive composition that includes a metal material and an energetic material is disclosed. The metal material defines a continuous phase into which the energetic material is dispersed. The reactive composition may produce at least one of light, heat, motion, noise, pressure, or smoke when initiated. The metal material provides a metallic melt phase into which the energetic material may be added and dispersed. By utilizing a metal material that is capable of providing a metallic melt phase, the reactive composition may have an improved performance compared to conventional reactive compositions. The reactive composition may be highly energetic when intentionally discharged but also insensitive to accidental discharge. As such, the reactive composition may have utility in a wide range of ordnance, such as in bullets, reactive bullets, grenades, warheads (including shape charges), mines, mortar shells, artillery shells, bombs, and demolition charges.
The metal material may be a metal or a metal alloy having a melting point lower than a temperature used to process the reactive composition. The melting point of the metal material may range from approximately 46° C. to approximately 250° C., such as from approximately 75° C. to approximately 105° C. The metal material may have a density of greater than approximately 7 g/cm3 and may be unreactive with other components of the reactive composition, such as the energetic material. If the metal material is an elemental metal, the elemental metal may include gallium ("Ga"), indium ("In"), lithium ("Li"), potassium ("K"), sodium ("Na"), or tin ("Sn"). The metal material may also be a fusible metal alloy. As used herein, the term "fusible metal alloy" refers to an eutectic or noneutectic alloy that includes transition metals, post-transition metals, or mixtures thereof, such as metals from Group III, Group IV, and/or Group V of the Periodic Table of the Elements. The metals used in the fusible metal alloy may include, but are not limited to, bismuth ("Bi"), lead ("Pb"), tin ("Sn"), cadmium ("Cd"), indium ("In"), mercury ("Hg"), antimony ("Sb"), copper ("Cu"), gold ("Au"), silver ("Ag"), and/or zinc ("Zn"). Fusible metal alloys are known in the art and are commercially available from sources including, but not limited to, Indium Corp. of America (Utica, N.Y.), Alchemy Castings (Ontario, Canada), and Johnson Mathey PLC (Wayne, Pa.). While the fusible metal alloy may include any of the previously mentioned metals, the fusible metal alloy may be free of toxic metals, such as lead and mercury, to minimize environmental concerns associated with clean-up of the reactive composition.
For the sake of example only, the fusible metal alloy may be Wood's Metal, which has 50% Bi, 25% Pb, 12.5% Sn, and 12.5% Cd and is available from Sigma-Aldrich Co. (St. Louis, Mo.). Wood's Metal has a melting point of approximately 70° C. and a density of 9.58 g/cm3. The fusible metal alloy may also be INDALLOY® 174, which has 57% Bi, 26% In, and 17% Sn. INDALLOY® 174 has a melting point of 174° F. (approximately 79° C.), a density of 8.54 g/cm3, and is commercially available from Indium Corp. of America (Utica, N.Y.). INDALLOY® 162, which has 33.7% Bi and 66.3% In, may also be used as the fusible metal alloy. INDALLOY® 162 has a melting point of 162° F. (approximately 72° C.), a density of 7.99 g/cm3, and is commercially available from Indium Corp. of America (Utica, N.Y.). Other INDALLOY® materials are available from Indium Corp. of America and may be used in the reactive composition. These INDALLOY® materials are available in a range of melting points (from approximately 60° C. to approximately 300° C.) and include a variety of different metals. As such, the fusible metal alloy may be selected depending on a desired melting point and the metals used in the fusible metal alloy.
The energetic material used in the reactive composition may be an organic or inorganic energetic material, such as at least one class 1.1 explosive, at least one oxidizer, or mixtures thereof. Any conventional energetic material may be used in the reactive composition provided that the energetic material does not decompose at the temperature used to process the reactive composition. The energetic material may be a solid material at ambient temperature and either a solid or a liquid material at the processing temperature. The energetic material may also have a density that is less than the density of the metal material. Preferably, the energetic material has a density of less than approximately 2.5 g/cm3. For instance, if the energetic material is an organic material, it may have a density less than approximately 2.0 g/cm3. If the energetic material is an inorganic material, the density may be less than approximately 2.5 g/cm3. The class 1.1 explosive may include, but is not limited to, TNT, RDX, HMX, hexanitrohexaazaisowurtzitane ("CL-20"; also known as HNIW), 4,10-dinitro-2,6,8,12-tetraoxa-4,10-diazatetracyclo-[5.5.0.05,9.03,11]-dodecane ("TEX"), ammonium dinitramide ("ADN"), 1,3,3-trinitroazetidine ("TNAZ"), 2,4,6-trinitro-1,3,5-benzenetriamine ("TATB"), dinitrotoluene ("DNT"), and mixtures thereof. The oxidizer may be sulfur or a nitrate, perchlorate, or oxide, such as an alkali or alkaline metal nitrate, an alkali or alkaline metal perchlorate, or an alkaline metal peroxide including, but not limited to, ammonium nitrate ("AN"), ammonium perchlorate ("AP"), sodium nitrate ("SN"), potassium nitrate ("KN"), lithium nitrate, rubidium nitrate, cesium nitrate, lithium perchlorate, sodium perchlorate, potassium perchlorate ("KP"), rubidium perchlorate, cesium perchlorate, magnesium perchlorate, calcium perchlorate, strontium perchlorate, barium perchlorate, barium peroxide, strontium peroxide, copper oxide, and mixtures thereof. While the examples described herein disclose that the reactive composition includes a single energetic material and a single fusible metal alloy, the reactive composition may also include more than one energetic material as well as more than one fusible metal alloy. Therefore, the reactive composition may be described as including at least one energetic material and at least one fusible metal alloy.
The relative amounts of the metal material and the energetic material present in the reactive composition may vary depending on the desired application for the reactive composition. For instance, the metal material may be present in the reactive composition from approximately 10% to approximately 90%. The energetic material may be present from approximately 10% to approximately 90%.
The reactive composition may optionally include additional components depending on a desired application for the reactive composition. The additional components may optionally be present in the reactive composition at a minimum amount sufficient to provide the desired properties. For instance, the reactive composition may optionally include a second metal material that remains solid at the processing temperature. The second metal material may enhance blast effects, such as to increase blast overpressures and thermal output. The second metal material may include, but is not limited to, aluminum, nickel, magnesium, silicon, boron, beryllium, zirconium, hafnium, zinc, tungsten, molybdenum, copper, or titanium, or mixtures thereof, such as aluminum hydride ("AlH3" or alane), magnesium hydride ("MgH2"), or borane compounds ("BH3"). In addition to BH3, the borane compounds may include stabilized compounds, such as NH3—BH3. Sulfur may also be used in the reactive composition. The second metal material may be in a powdered or granular form. The second metal material may be present in the reactive composition from approximately 0.5% to approximately 60%. Percentages of each of the components in the reactive composition are expressed herein as percentages by weight of the total reactive composition.
The reactive composition may also optionally include conventional binders or filler materials. Energetic polymers, inert polymers, or fluoropolymers may also optionally be used to optimize the rheological properties of the reactive composition or as a processing aid. The polymer may soften or melt at the processing temperature. The polymer may be present in the reactive composition from approximately 0.5% to approximately 50%, such as from approximately 0.5% to approximately 5%. The polymer may include, but is not limited to, polyglycidyl nitrate ("PGN"), nitratomethylmethyloxetane ("polyNMMO"), polyglycidyl azide ("GAP"), diethyleneglycol triethyleneglycol nitraminodiacetic acid terpolymer ("9DT-NIDA"), poly(bis(azidomethyl)oxetane) ("polyBAMO"), poly(azidomethylmethyloxetane) ("polyAMMO"), poly(nitraminomethyl methyloxetane) ("po1yNAMMO"), poly(bis(difluoroaminomethyl)oxetane) ("polyBFMO"), poly(difluoroaminomethylmethyloxetane) ("polyDFMO"), copolymers thereof, and mixtures thereof. The polymer may also include cellulosic polymers, such as cellulose acetate butyrate ("CAB") or nitrocellulose; nylons; polyesters; fluoropolymers; energetic oxetanes; waxes; and mixtures thereof.
Graphite, silica, or polytetrafluoroethylene (TEFLON®) compounds may also optionally be used in the reactive composition as a processing aid or for reaction enhancement. The reactive composition may also optionally include energetic plasticizers or inert plasticizers including, but not limited to, bis(2,2-dinitropropyl)acetal/bis(2,2-dinitropropyl)formal ("BDNPA/F"), dioctyl sebacate ("DOS"), dimethylphthalate ("DMP"), dioctyladipate ("DOA"), glycidyl azide polymer ("GAP"), diethyleneglycol dinitrate ("DEGDN"), butanetrioltrinitrate ("BTTN"), butyl-2-nitratoethyl-nitramine ("BuNENA"), trimethylolethanetrinitrate ("TMETN"), triethylene-glycoldinitrate ("TEGDN"), nitroglycerine ("NG"), isodecylperlargonate ("IDP"), dioctylphthalate ("DOP"), dioctylmaleate ("DOM"), dibutylphthalate ("DBP"), di-n-propyl adipate, diethylphthalate, dipropylphthalate, citroflex, diethyl suberate, diethyl sebacate, diethyl pimelate, and mixtures thereof. The plasticizer may be present in the reactive composition from approximately 0.5% to approximately 10%, such as from approximately 0.5% to approximately 5%. As discussed below, the reactive composition may optionally include a polymer/plasticizer system. Catalysts, such as graphite, silicon, iron(III) oxide, sulfur, or nano-aluminum, may also optionally be used in the reactive composition.
In the reactive composition, the metal material provides the continuous phase and the energetic material provides the dispersed phase, which is in contrast to conventional reactive compositions where the energetic material is the continuous phase. The resulting composition may have efficient combustion and reduced sensitivity because the energetic material is coated with the metal material, which provides an intimate contact between these components.
The reactive composition may be produced by adding the energetic material to the metal material to form a substantially homogenous mixture or a heterogeneous mixture. Any optional components, such as the second metal material or any fillers, may also be added to the substantially homogenous mixture. The metal material may be in a liquid state, which is also referred to herein as a "molten metal." The molten metal may be produced by heating the metal material to a temperature above its melting point. The energetic material may then be mixed into the metal material. If the energetic material is a liquid at the processing temperature, the energetic material may be melted with the liquid state metal material to form an emulsion. Energetic materials that are liquid at the processing temperature include, but are not limited to, DNT, TNT, and TNAZ, which have melting points of 71° C., 81° C. and 101° C., respectively. If the energetic material is a solid at the processing temperature, the energetic material may be dispersed in the metal material by mixing the two components. When a solid energetic material is used, the energetic material may be present in a coarse particle size to provide a well-mixed, reactive composition. For instance, the energetic material may have a particle size ranging from approximately 5 μm to approximately 400 μm. Solid energetic materials include, but are not limited to, AP, HMX, KN, KP, and TATB, which have melting points of 220° C., 285° C., 334° C., 610° C., and 450° C., respectively. The temperature at which the reactive composition is processed may depend on the melting point of the metal material and the energetic material. In one embodiment, the processing temperature ranges from approximately 46° C. to approximately 250° C., such as from approximately 75° C. to approximately 105° C.
After mixing, the substantially homogenous mixture may be formed into the reactive composition by conventional techniques. For instance, the reactive composition may be formed by placing the substantially homogenous mixture into a mold or container having a desired shape. If the substantially homogenous mixture has a low viscosity, it may be poured into the mold. However, if the substantially homogenous mixture has a higher viscosity, it may be physically transferred to the mold. The substantially homogenous mixture may then be solidified to form the reactive composition having the desired shape.
However, when large amounts of solid additives, such as the energetic material or the optional components, are added to the metal material, a high-density gradient may be produced, resulting in low homogeneity of the reactive composition. In other words, the metal material may separate from the other components in the reactive composition. As such, the metal material may be unable to bind the energetic material or the optional components when large amounts of the solid additives are present. To improve the homogeneity and the processing of the reactive composition when large amounts of these solid additives are used, the polymer/plasticizer system may optionally be present as a processing aid.
The polymer used in the polymer/plasticizer system may have a melt temperature or softening temperature that is similar to the melt temperature of the metal material. The polymer may provide sufficient intermolecular forces to allow the polymer to be evenly distributed in the liquid phase. As previously described, the polymer may be an inert polymer, an energetic polymer, or a fluoropolymer. The plasticizer may be an inert plasticizer or an energetic plasticizer as previously described. The polymer/plasticizer system may be present in the reactive composition from approximately 0.5% to approximately 50%, such as from approximately 0.5% to approximately 5%. In one embodiment, the polymer/plasticizer system includes CAB and BDNPA/F.
The polymer/plasticizer system may form a polymeric matrix that is distributed throughout the metal material in the liquid phase. As such, the metal material may be uniformly dispersed in the reactive composition, increasing the surface area of the metal material. The polymer/plasticizer system may also enable the metal material to suspend the solid additives in the reactive composition and improve the ability of the metal material to bind to the solid additives. When the solid additives are added to the metal material, the solid additives may be evenly coated with a thin layer of the polymer and the metal material. Therefore, the ratio of surface area of the metal material to the solid additives is increased.
By utilizing the polymer/plasticizer system, performance and processability of the reactive composition may be improved. The polymer/plasticizer system may trap other components of the reactive composition in its matrix, promoting uniform mixing. As such, the polymer/plasticizer system may provide increased flexibility in formulating the reactive composition and may enable each component of the reactive composition to be mixed into a uniform blend. The polymer/plasticizer system may significantly improve performance of the reactive composition because increased amounts of the solid additives, such as increased amounts of the oxidizer, may be used. The polymer/plasticizer system may also increase processability because the polymer/plasticizer system maintains a homogenous distribution of the components during pouring, mixing, casting, and pressing of the reactive composition.
The concern may be raised that the polymer/plasticizer system, while improving processability, may reduce or degrade overall energy and performance of the reactive composition since many of the polymers and plasticizers are less energetic than other components of the reactive composition. Surprisingly, however, the polymer/plasticizer system has been shown to improve the energy and performance of the reactive composition. It is believed, without being limiting of the scope of the invention, that the metal material may be uniformly dispersed in the polymer/plasticizer system, increasing the surface area of the metal material. As the solid additives are added to this mixture, the solid additives may be evenly coated with a thin layer of the polymer and the metal material, significantly increasing the ratio of the surface area of the metal material to the solid additives. Testing performed on reactive compositions lacking the polymer/plasticizer system indicated that the metal material may have difficulty acting as a fuel because large pieces of the metal material do not react rapidly. However, a uniform, high surface area dispersion of the metal material, such as is present when the polymer/plasticizer system is used, may be able to react more completely.
If the polymer/plasticizer system is not used in the reactive composition, the reactive composition may be granulated to form a heterogenous mixture that includes crystallized particles of the metal material and small particles of the energetic material and the optional components. The granules of the reactive composition may then be pressed into a solid mass having the desired shape. When no polymer/plasticizer system is used, the metal material may be present in the reactive composition from approximately 40% to 80%, which is in contrast to the higher amounts of the metal material that may be present when the polymer/plasticizer system is used. If the metal material is present beyond this range without using the polymer/plasticizer system, it may be difficult to produce a uniform composition that is reliable from one sample to the next sample. In addition, the reactive composition formulated without the polymer/plasticizer system may lack a continuous phase and may be prone to fracture. As such, the reactive composition without the polymer/plasticizer system is limited in the amounts of the solid additives that may be used relative to the amount of the metal material.
In contrast, when the reactive composition includes the polymer/plasticizer system, the reactive composition may include a wider range of the amount of the solid additives. For instance, the reactive composition may include from approximately 13.5% of the metal material and approximately 82% of the solid additives to approximately 85% of the metal material and approximately 9% of the solid additives. In addition, the reactive composition including the polymer/plasticizer system may be substantially homogenous and uniform, which enables the reactive composition to be poured, casted, and granulated without the metal material separating from the solid additives. The reactive composition may also be pressed at lower pressures than compositions lacking the polymer/plasticizer system. The polymer/plasticizer system may also enable the reactive composition to be mixed with less shear work, increasing the safety of processing of these reactive compositions. Using the polymer/plasticizer system may also reduce the friability of the reactive composition. As ductility and toughness of the reactive composition increase, safe handling of the reactive composition may also increase, both during and after processing.
The reactive composition utilizing the polymer/plasticizer system may be processed in extruders, injection molders, and similar processing equipment. If the metal material has a melting point from approximately 46° C. to approximately 250° C. and the energetic material is a liquid at the processing temperature, the reactive composition may be produced by a melt-pour process in an existing melt-pour facility. Therefore, new equipment and facilities may not be necessary to produce the reactive composition. If the metal material has a melting point ranging from approximately 75° C. to approximately 105° C. and the energetic material is a liquid at the processing temperature, the reactive composition may be produced in existing melt-pour facilities used to produce conventional TNT-containing explosives. While it is desirable for the reactive composition to be produced by a melt-pour technique, it is contemplated that the reactive composition may be produced by other techniques, especially if the energetic material is a solid material.
By utilizing the metal material as the continuous phase, the reactive composition may have an increased detonation rate compared to the detonation rate of a conventional reactive composition. The reactive composition may also have a higher density than that of a conventional reactive composition. In addition, the reactive composition may be more insensitive to accidental discharge than conventional compositions, as measured by sensitivity tests known in the art. For instance, the reactive composition may be insensitive to friction, electrostatic, impact, and thermal incompatibility. The reactive composition may also have a high initiation threshold.
The reactive composition of the present invention may be used in ordnance, such as bullets, reactive bullets, grenades, warheads (including shape charges), mines, mortar shells, artillery shells, bombs, and demolition charges. For instance, the reactive composition may be used as a fill material in a reactive material bullet. The reactive composition may also be used as a shape charge liner, such as in a warhead. The reactive composition may also be used to provide enhanced blast, such as by adding the second metal material, such as AlH3, to the reactive composition. The reactive composition may also be formulated for use as a propellant or a gas generant.
The following examples serve to explain embodiments of the present invention in more detail. These examples are not to be construed as being exhaustive or exclusive as to the scope of this invention.
Preparation of Reactive Compositions Including INDALLOY® 174 and TNAZ
To form a reactive composition having 77.5% INDALLOY® 174 and 22.5% TNAZ (Formulation A), 775 grams of INDALLOY® 174 and 225 grams TNAZ were melted in separate, plastic, heat-resistant beakers and stirred with wood or TEFLON® rods. During melting of the TNAZ, care was taken to avoid a buildup of subliming reactive composition on the interior of the oven. The melted TNAZ was then poured into the INDALLOY® 174 and stirred thoroughly. The INDALLOY® 174/TNAZ mixture was heated at 100° C. for 5 minutes while stirring. The INDALLOY® 174/TNAZ mixture was removed from the oven and stirred until the viscosity had increased sufficiently to suspend the TNAZ. The INDALLOY® 174/TNAZ mixture was then cast into an item, such as a mold, that had been previously heated to 100° C. The item was overcast and pressed down on the top until set.
Reactive compositions having 63% INDALLOY® 174 and 37% TNAZ (Formulation B) and 50% INDALLOY® 174 and 50% TNAZ (Formulation C) were prepared as described above by varying the relative amounts of INDALLOY® 174 and TNAZ.
Preparation of Reactive Compositions Including Wood's Metal and TNAZ
A reactive composition having 63% Wood's Metal and 37% TNAZ (Formulation E) was prepared as described in Example 1, except that Wood's Metal was used instead of the INDALLOY® 174.
Preparation of Reactive Compositions Including INDALLOY® 174 and TNT
A reactive composition having 70% INDALLOY® 174 and 30% TNT (Formulation G) was prepared as described in Example 1, except that TNT was used instead of TNAZ.
Preparation of Reactive Compositions Including INDALLOY® 174 and DNT
To form a reactive composition having 75% INDALLOY® 174 and 25% DNT (Formulation F), 750 grams of INDALLOY® 174 and 250 grams DNT were melted in separate, plastic, heat-resistant beakers and stirred with wood or TEFLON® rods. The melted DNT was then poured into the INDALLOY® 174 and stirred thoroughly. The INDALLOY® 174/DNT mixture was heated at 100° C. for 5 minutes while stirring. The INDALLOY® 174/DNT mixture was removed from the oven and stirred until the viscosity had increased sufficiently to suspend the DNT. The INDALLOY® 174/DNT mixture was then cast into an item that had been previously heated to 100° C. The item was overcast and pressed down on the top until set.
Preparation of Reactive Compositions Including INDALLOY® 174 and AP
To form a reactive composition having 75% INDALLOY® 174 and 25% AP (Formulation J), 750 grams of INDALLOY® 174 and 250 grams AP were melted in a plastic, heat-resistant beaker while stirring with wood or TEFLON® rods. The AP was incorporated into the INDALLOY® 174 to produce a paste-like material. The INDALLOY® 174/AP paste was removed from the oven. The INDALLOY® 174/AP paste was added in increments to an item that had been previously heated to 100° C. and tamped gently between additions. The item was overcast and pressed down on the top until set.
Preparation of Reactive Compositions Including INDALLOY® 174 and KN
Reactive compositions including 77.5% INDALLOY® 174 and 22.5% KN (Formulation K) and 75% INDALLOY® 174 and 25% KN (Formulation L) were prepared as described in Example 5, except that KN was used instead of AP.
Preparation of Reactive Compositions Including INDALLOY® 174 and TATB
A reactive composition including 91% INDALLOY® 174 and 9% TATB (Formulation H) was prepared as described in Example 5, except that TATB was used instead of AP.
Preparation of Reactive Compositions Including INDALLOY® 174 and HMX
A reactive composition including 63% INDALLOY® 174 and 37% HMX (Formulation I) was prepared as described in Example 5, except that HMX was used instead of AP.
Preparation of Reactive Compositions Including INDALLOY® 174, TNAZ, and AlH3
A reactive composition having 50.5% INDALLOY® 174, 29.5% TNAZ, and 20% AlH3 (Formulation D) was prepared as described in Example 1, with the addition of AlH3 to the INDALLOY® 174/TNAZ mixture.
Preparation of Reactive Compositions Including Wood's Metal, TNAZ, and AlH3
A reactive composition having 50.5% Wood's Metal, 29.5% TNAZ, and 20% AlH3 (Formulation M) is prepared as described in Example 1, with the addition of AlH3 to the Wood's Metal/TNAZ mixture.
Calculated Detonation Performance of the Reactive Compositions
CHEETAH 3.0 thermochemical code, developed by L. E. Fried, W. M. Howard, and P. C. Souers, was used to calculate detonation performance parameters for the reactive compositions described in Examples 1-10. CHEETAH 3.0 models detonation performance parameters of ideal explosives and is available from Lawrence Livermore National Laboratory (Livermore, Calif.). The detonation performance parameters of the reactive compositions were compared to those of the conventional explosive compositions, such as isopropyl nitrate ("IPN")/Mg (Formulation N); IPN/RDX/Al, (Formulation O); DNANS/methylnitroaniline/RDX/AP/Al, (Formulation P); and RM4/nitromethane (Formulation Q).
Calculated Detonation Performance Comparison at 99% Theoretical Maximum Density ("TMD")
Detonation Detonation Detonation Heat of Density 99% Pressure Velocity Temperature Combustion H2 Total Energy Formulation TMD (g/cc) (kbar) (km/s) (K) (cal/g × 103) (mol/kg × 10−40) (kJ/cc)
A 4.63 307 3.55 3448 0.61 6.34 77.5% INDALLOY ® 174 22.5% TNAZ B 3.59 359 4.60 4087 0.89 8.22 63% INDALLOY ® 174 37% TNAZ C 2.99 381 5.54 4391 1.14 9.29 50% INDALLOY ® 174 50% TNAZ D 2.79 198 5.11 5039 2.60 16.09 50.5% INDALLOY ® 174 29.5% TNAZ 20% AlH3 E 3.67 364 4.82 4111 0.92 8.33 63% Wood's Metal 37% TNAZ F 3.92 99.8 3.31 2202 0c 0c 75% INDALLOY ® 174 25% DNT G 3.76 241 3.93 3229 1.16 5.51 70% INDALLOY ® 174 30% TNT H —a —a —a —a —a —a 91% INDALLOY ® 174 9% TATB I 3.69 375 4.62 3580 0.89 7.93 63% INDALLOY ® 174 37% HMX J 4.59 329 3.60 2536 0.22 4.07 75% INDALLOY ® 174 25% AP K 5.00b,c 30.4 2.33 541 0c 0c 77.5% INDALLOY ® 174 22.5% KN L 4.80 22.7 2.22 376 0c 0c 75% INDALLOY ® 174 25% KN M 2.86 190 5.14 4898 2.71 0.3 16.46 50.5% Wood's Metal 29.5% TNAZ 20% AlH3 N 1.24 72 4.78 4905 5.27 0.4 11.84 IPN Mg O 1.53 192 7.05 4928 3.70 0.4 10.69 IPN Al RDX P 1.84 232 7.48 5043 3.58 0.2 12.90 DNANS MNA RDX AP Al Q 1.59 187 5.73 4847 3.03 0.2 9.15 50% RM4 50% Nitromethane aCHEETAH does not calculate densities above 5 g/cc. bData was generated at a density of 98.8% TMD. cCHEETAH did not calculate these parameters.
The CHEETAH program was unable to adequately calculate the heat of combustion and total energy for Formulation F, which may have been a result of the low detonation temperature. However, the CHEETAH program was able to calculate these parameters for Formulation G, which had a significantly greater detonation temperature. Formulation H had too great a density to be calculated. Formulations K and L, which included the inorganic oxidizer KN, had a relatively large negative heat of formation that caused it to be nearly inert and difficult to obtain useful detonation parameters when combined with the fusible metal alloy.
As shown in Table 1, many of the reactive compositions (Formulations A, B, F, G, I, and J) had higher calculated detonation pressures and lower calculated detonation velocities than those of Formulation N, indicating that these reactive compositions had improved, calculated, performance properties. Reactive compositions A-M also had significantly higher densities than that of Formulation N.
The reactive compositions that included AlH3 as the second metal material also had increased, calculated, detonation parameters. For instance, the addition of AlH3, as in Formulations D and M, drastically boosted the detonation temperature, heat of combustion, and total energy of the reactive compositions. A comparison of the reactive compositions having INDALLOY® 174 or Wood's Metal as the metal material and TNAZ or HMX as the energetic material showed that as the relative amount of energetic material increased, the density of the explosive composition decreased and each of the other parameters increased.
Compatibility of the Reactive Compositions
Compatibility of the metal material, the energetic material, and the second metal material was also determined. Differential Scanning Calorimetry ("DSC") compatibility data for INDALLOY® 174 with various energetic materials and AlH3 is shown in Table 2.
DSC Comparison of INDALLOY ® 174 and Energetic Materials
Components Alloy: Additive DSC (exotherm onset, ° C.) INDALLOY ® 174 1:0 — Alane (AlH3) 0:1 188 Alane (AlH3) 2:1 192 Alane (AlH3) 3:1 188 Alane (AlH3) 4:1 191 CL-20 1:1 242 CL-20 3:1 243 TEX 2:1 301 TEX 3:1 296 TNAZ 3:1 257 TNAZ 4:1 256
Sensitivity of the Reactive Compositions
Hazard properties were also determined for the reactive compositions that contained INDALLOY® 174. Laboratory scale hazard properties (impact, friction, ESD, and thermal incompatibility) were measured for the compositions that contained INDALLOY® 174, as shown in Table 3. These properties were measured by conventional techniques known in the art.
The detonation performance of these reactive compositions was measured by a Dent and Rate test. A test sample of each of the reactive compositions was held in a steel pipe (3.7 cm diameter×14 cm length) that had five holes drilled in the side for velocity switches from which the detonation velocity was calculated by regression analysis. The test sample was detonated using a booster that was 160 grams pentolite (50 pentaerythritol tetranitrate ("PETN"):50 TNT) and the depth of the dent made in a witness plate was measured. The dent depth was correlated to the detonation pressure, with a deeper dent corresponding to a higher pressure.
Laboratory Scale Hazards Property and Dent and Rate Comparison
INDALLOY ® 174 A B C D E F G H I J K L
Oxidizer Particle Fine 5-100 200 20 400 Size Density (g/cc, 8.54 3.42 2.88 3.81 3.78 4.66 5.68 measured) ABL Impact 80 1.8 1.1 800 80 13 1.8 1.8 21 80 (cm)a BOE Impact Pass Fail >8 Pass Pass Pass Pass Pass (4″)b ABL Friction 800 800 <25 @ 163 800 800 800 25 @ 25 800 800 (psi @ 8 ft/sec)c 2 3 TC ESD (J)d >8 >8 0.92 5.23 1.23 7.3 1.5 >8 >8 >8 SBAT (exotherm None 163 117 219 197 167 206 182 174 171 onset, ° C.)e DSC (exotherm — 259 334 440 onset, ° C.) VTS (ml/g)f 0.19 0.23 0.25 0.19 0.20 0.22 TGA under N2 1.8 @ 25.9 @ 35.4 @ 36.6 @ 11.5 @ 10.7 @ (% weight loss @ 188 212 248 400 754 649 x° C. Dent depth (mm) 0.0 1.4 9.9 0.0 0.0 0.0 Detonation 2.3 6.9 8.4 2.0 2.2 0.8 Velocity (km/s) aThreshold Initiation Level (TIL) for 20 no-fire drops per drop height bPass is six often no-fire impacts cTIL for 20 no-fires d50% ignition point eSimulated Bulk Autoignition Temperature measures the ability of a sample to absorb heat where an exotherm <107° C. indicates a sensitive material fVacuum Thermal Stability at 75° C. for 48 hours
As shown in Table 3, neat INDALLOY® 174 was inert and gave hazard results at the least sensitive limit of each test. The TNAZ and AP reactive compositions (Formulations A-E, J, and M) were sensitive to impact but were otherwise insensitive. Formulation E was resistant to application of a hot wire but burned with a continuous hot flame once ignited. The resulting reactive composition was resistant to application of a hot wire but burned with a continuous hot flame when ignited. The DNT and KN reactive compositions (Formulations F, K, and L) were nearly as insensitive as the neat INDALLOY® 174. The Vacuum Thermal Stability ("VTS") showed no volatile loss from any reactive composition. The thermogravimetric Analysis ("TGA") of neat INDALLOY® 174 indicated some weight loss at 188° C., which was well above the normal processing temperatures of 100-110° C. The TGA of Formulation A showed significant weight loss at 212° C. that represented all of the TNAZ in the explosive composition. However, at 100° C., the TNAZ loss was only approximately 1%, which was acceptable for short processing times. In each of the other cases, TGA weight loss occurred at a temperature that was well above the processing temperature. In addition to the Formulations shown in Table 3, an insensitive reactive composition having Wood's Metal and TEX was also produced. A formulation having 63% Wood's Metal and 37% TNAZ had a TC impact of 26.1 in, an ABL friction of 800 psi @ 8 ft/s, a TC ESD of >8 J, and an SBAT (onset) at 163° C.
As indicated in Table 3, the measured dent depth of 9.9 mm for Formulation E was significantly less than the dent depth anticipated from the calculated detonation pressure of 364 kbar, which is similar to the dent depth observed with Composition B or Composition C. However, the observed detonation velocity of 8.4 km/s was 85% greater than calculated and was similar to the detonation velocity observed for very high-energy pressed explosives, such as LX-14, which has 95.5% HMX. Similar results were observed for Formulation A. The reactive compositions that contained DNT, AP, and KN (Formulations F and J-L) gave similar results to the neat INDALLOY® 174.
Safety Results for Reactive Compositions Including the Polymer/Plasticizer System
Formulations having the components listed in Table 4 were produced and safety testing was performed on these formulations. Impact properties of the formulations were measured using an impact test developed by Thiokol Corporation ("TC"). Friction properties of the formulations were measured using a friction test developed by Allegheny Ballistics Laboratory ("ABL"). Electrostatic discharge ("ESD") of the formulations was measured using an ESD test developed by TC. Onset of ignition exotherms and sensitivity to elevated temperatures of the formulations were measured using a Simulated Bulk Autoignition Test ("SBAT"). These tests are known in the art and, therefore, details of these tests are not included herein.
Safety Properties of Reactive Compositions that Include the Polymer/Plasticizer System.
TC ABL TC SBAT Impact Friction ESD Onset Formulation (in.) (lbs) (J) (° F.)
90% INDALLOY ® 174 >46 800 @ >8 340 10% KP 8 fps 80% INDALLOY ® 174 33.55 660 @ >8 349 20% KP 8 fps 60% INDALLOY ® 174 41.2 100 @ 40% KP 6 fps 85.5% INDALLOY ® 174 43.86 50 @ >8 309 9.5% KP 4 fps 1% CAB 4% BDNPA/F 76% INDALLOY ® 174 14.33 50 @ >8 317 19% KP 3 fps 1% CAB 4% BDNPA/F 68% INDALLOY ® 174 13.91 <25 @ 7.5 308 14.5% KP 2 fps 14.5% RDX 0.4% CAB 2.6% BDNPA/F 57% INDALLOY ® 174 18.64 25 @ >8 376 38% KP 4 fps 1% CAB 4% BDNPA/F 25% INDALLOY ® 174 18.64 25 @ >8 336 28% KP 4 fps 28% RDX 10% Mg 1.5% CAB 8% BDNPA/F 20% INDALLOY ® 174 19.90 25 @ >8 310 70% CL-20 6 fps 1% CAB 9% BDNPA/F 20% INDALLOY ® 174 16.82 25 @ 7.25 345 55% CL-20 2 fps 15% Mg 1% CAB 9% BDNPA/F 18% INDALLOY ® 174 21.55 800 @ >8 287 76% RDX 8 fps 6% CBN and BDNPA/F 17% INDALLOY ® 174 18.80 800 @ >8 287 78% KP 8 fps 5% CBN and BDNPA/F 14% INDALLOY ® 174 18.67 800 @ >8 371 81% KP 8 fps 5% CBN and BDNPA/F 13.5% INDALLOY ® 174 18.45 800 @ 7.5 350 82% RDX 8 fps 4.5% CBN and BDNPA/F
The results depicted in Table 4 show that the reactive compositions including the polymer/plasticizer system have good safety properties.
Reactive Compositions Including the Polymer/Plasticizer System
A quantitative analysis of the effect of the polymer/plasticizer system was determined by testing two similar formulations of the reactive composition for compressive strength in a ½-inch diameter cylindrical pellet configuration. The first formulation included 60% INDALLOY® 174 and 40% KP and is referred to herein as the reactive material enhanced bullet-1 ("RMEB-1") formulation. The second formulation included 56.85% INDALLOY® 174, 37.9% KP, and 5.25% of the polymer/plasticizer system and is referred to as the "RMEB-1 w/binder" formulation. The polymer/plasticizer system included 1.0 wt % CAB and 4.25 wt % BDNPA/F. Both of the tested formulations had the same ratio of the INDALLOY® 174 to the oxidizer.
Each of the formulations was formed into a ½-inch diameter cylindrical pellet and compressive strength tests were performed on each of the pellets as known in the art. As shown in FIGS. 1 and 2, the RMEB-1 formulation was able to withstand a higher load. However, the RMEB-1 w/binder formulation exhibited more elastic deformation even though only a small amount of the polymer/plasticizer system was used. The RMEB-1 w/binder formulation also exhibited the ability to flow under a load and to resist deformation.
In order to determine the effect of the polymer/plasticizer system, the toughness of each form was calculated by integrating each curve. As shown in FIG. 3, the RMEB-1 w/binder formulation was almost twice as tough as the RMEB-1 formulation. As such, the RMEB-1 w/binder formulation is less likely to fracture. Fractured materials are less stable and more prone to premature initiation from external stimuli than nonfractured materials. In contrast, the RMEB-1 formulation was less tough, more brittle and more prone to fracture. Photographs of the pellets before and after the compressive strength tests are shown in FIGS. 4-7.
While the invention may be susceptible to various modifications and alternative forms, specific embodiments have been shown by way of example in the drawings and have been described in detail herein. However, it should be understood that the invention is not intended to be limited to the particular forms disclosed. Rather, the invention is to cover all modifications, equivalents, and alternatives falling within the spirit and scope of the invention as defined by the following appended claims.
1. A precursor composition of a reactive material, comprising:
a metal material comprising at least one class 1.1 explosive selected from the group consisting of cyclo-1,3,5-trimethylene-2,4,6-trinitramine, cyclotetramethylene tetranitramine, hexanitrohexaazaisowurtzitane, 4,10-dinitro-2,6,8,12-tetraoxa-4,10-diazatetracyclo-[5.5.0.05,9.03,11]-dodecane, 1,3,3-trinitroazetidine, ammonium dinitramide, trinitrotoluene, dinitrotoluene, and mixtures thereof therein, the metal material comprising bismuth, indium, and tin and defining a continuous phase at a processing temperature of a precursor composition of a reactive material.
2. The precursor composition of claim 1, wherein the metal material comprises a fusible metal alloy having a melting point ranging from approximately 46° C. to approximately 250° C.
3. The precursor composition of claim 1, wherein the metal material further comprises at least one metal selected from the group consisting of lead, cadmium, mercury, antimony, copper, gold, silver, and zinc.
5. The precursor composition of claim 1, wherein the metal material has a density of greater than approximately 7 g/cm3.
6. The precursor composition of claim 1, wherein the metal material consists essentially of bismuth, indium, and tin.
7. The precursor composition of claim 1, wherein the metal material comprises a fusible metal alloy having 57% bismuth, 26% indium, and 17% tin.
8. The precursor composition of claim 1, further comprising a second metal material selected from the group consisting of aluminum, nickel, magnesium, silicon, boron, beryllium, zirconium, hafnium, zinc, tungsten, molybdenum, copper, titanium, sulfur, aluminum hydride, magnesium hydride, a borane compound, and mixtures thereof.
a metal material comprising at least one class 1.1 explosive dispersed therein, the metal material defining a continuous phase at a processing temperature of a precursor composition of a reactive material and comprising bismuth, indium, and tin, and the at least one class 1.1 explosive selected from the group consisting of cyclo-1,3,5-trimethylene-2,4,6-trinitramine, cyclotetramethylene tetranitramine, hexanitrohexaazaisowurtzitane, 4,10-dinitro-2,6,8,12-tetraoxa-4,10-diazatetracyclo-[5.5.0.05,9.03,11]-dodecane, 1,3,3-trinitroazetidine, ammonium dinitramide, 2,4,6-trinitro-1,3,5-benzenetriamine, trinitrotoluene, dinitrotoluene, and mixtures thereof.
10. The precursor composition of claim 9, further comprising a polymer/plasticizer system, wherein the polymer/plasticizer system comprises:
at least one polymer selected from the group consisting of polyglycidyl nitrate, nitratomethylmethyloxetane, polyglycidyl azide, diethyleneglycol triethyleneglycol nitraminodiacetic acid terpolymer, poly(bis(azidomethyl)oxetane), poly(azidomethylmethyloxetane), poly(nitraminomethyl methyloxetane), poly(bis(difluoroaminomethyl)oxetane), poly(difluoroaminomethylmethyloxetane), copolymers thereof, cellulose acetate butyrate, nitrocellulose, nylon, polyester, fluoropolymers, energetic oxetanes, waxes, and mixtures thereof; and
at least one plasticizer selected from the group consisting of bis(2,2-dinitropropyl)acetal/bis (2,2-dinitropropyl)formal, dioctyl sebacate, dimethylphthalate, dioctyladipate, glycidyl azide polymer, diethyleneglycol dinitrate, butanetrioltrinitrate, butyl-2-nitratoethyl-nitramine, trimethylolethanetrinitrate, triethylene glycoldinitrate, nitroglycerine, isodecylperlargonate, dioctylphthalate, dioctylmaleate, dibutylphthalate, di-n-propyl adipate, diethylphthalate, dipropylphthalate, citroflex, diethyl suberate, diethyl sebacate, diethyl pimelate, and mixtures thereof.
11. The precursor composition of claim 9, further comprising at least one oxidizer selected from the group consisting of ammonium perchlorate, potassium perchlorate, sodium nitrate, potassium nitrate, ammonium nitrate, lithium nitrate, rubidium nitrate, cesium nitrate, lithium perchlorate, sodium perchlorate, rubidium perchlorate, cesium perchlorate, magnesium perchlorate, calcium perchlorate, strontium perchlorate, barium perchlorate, barium peroxide, strontium peroxide, copper oxide, sulfur, and mixtures thereof.
12. The precursor composition of claim 9, wherein the metal material comprises from approximately 40% by weight to 80% by weight of the precursor composition.
13. The precursor composition of claim 9, wherein the metal material comprises from approximately 13.5% by weight to approximately 85% by weight of the precursor composition.
14. The precursor composition of claim 9, wherein the precursor composition comprises a heterogeneous, granulated mixture of the metal material and the at least one class 1.1 explosive.
15. The precursor composition of claim 9, wherein the metal material consists of bismuth, indium, and tin.
16. A precursor composition of a reactive material, comprising:
a metallic melt phase comprising at least one class 1.1 explosive therein, the metallic melt phase comprising bismuth, indium, and tin.
at least one class 1.1 explosive in a molten metal, the at least one class 1.1 explosive selected from the group consisting of cyclo-1,3,5-trimethylene-2,4,6-trinitramine, cyclotetramethylene tetranitramine, hexanitrohexaazaisowurtzitane, 4,10-dinitro-2,6,8,12-tetraoxa-4,10-diaza-tetracyclo-[5.5.0.05,9.03,11]-dodecane, 1,3,3-trinitroazetidine, ammonium dinitramide, 2,4,6-trinitro-1,3,5-benzenetriamine, trinitrotoluene, dinitrotoluene, and mixtures thereof, and the molten metal comprising bismuth, indium, and tin.
Filed: Oct 20, 2011
Assignee: Alliant Techsystems Inc. (Arlington, VA)
Inventors: Benjamin N. Ashcroft (Perry, UT), Daniel B. Nielson (Tremonton, UT), Daniel W. Doll (Marriott Slaterville, UT)
Primary Examiner: James McDonough
Current U.S. Class: Solid Particles Dispersed In Solid Solution Or Matrix (149/17); Structure Or Arrangement Of Component Or Product (149/2); Containing Nitrated Organic Compound (149/88); Nitrated Acyclic, Alicyclic Or Heterocyclic Amine (149/92); Containing Free Metal Or Free Carbon (149/108.2); Miscellaneous Compositions (149/109.4)
International Classification: C06B 45/00 (20060101); C06B 45/04 (20060101); C06B 25/00 (20060101); C06B 25/34 (20060101); D03D 23/00 (20060101); D03D 43/00 (20060101); | {
"redpajama_set_name": "RedPajamaCommonCrawl"
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{"url":"https:\/\/golem.ph.utexas.edu\/category\/2012\/12\/universe_polymorphism_and_typi.html","text":"## December 9, 2012\n\n### Universe Polymorphism and Typical Ambiguity\n\n#### Posted by Mike Shulman\n\nSorry I\u2019ve been kind of quiet recently. There\u2019s been a lot of good stuff happening at IAS this year, but (aside from my being busy helping to make it happen), not so much of it has been easily bloggable. But here\u2019s something that I\u2019ve just learned: do you know the difference between universe polymorphism and typical ambiguity (or even what either of them means)?\n\nIn category theory we use \u201cuniverses\u201d of one sort or another a fair amount. Any time we talk about \u201csmall\u201d versus \u201clarge\u201d categories, we are using a universe. It might be a Grothendieck universe, or it might be the proper-class universe of sets, but whatever it is, we are employing some device to separate out some of the \u201cthings\u201d under consideration which belong to the \u201cuniverse\u201d, in such a way that we can treat the universe itself as a \u201cthing\u201d.\n\nOne of the problems with universes is that sometimes you need to change your universe. For instance, suppose you prove something about small categories. Later on, you find that you need to know that the corresponding theorem for large categories is also true. What do you do?\n\nIn practice, one may make some remark about this being obvious, or decline to remark on it at all. But when being absolutely precise, there are several options. One convenient one is to consider the theorem about small categories to have been (implicitly) universally quantified over the universe which defines the notion of \u201csmall\u201d. Thus, if we choose a different universe, with respect to which the categories that we\u2019re currently calling \u201clarge\u201d are actually \u201csmall\u201d, then our original theorem can be applied to them.\n\nIn type theory, this is called universe polymorphism: a notion (theorem, definition, proof, etc.) is universe polymorphic if it is universally quantified over one or more universes.\n\nThe quantification may be implicit or explicit. A few books and papers in category theory actually make it explicit: rather than \u201csmall category\u201d they always write \u201c$U$-small category\u201d (or \u201c$U$-category\u201d) for some universe $U$, and so on. I find this to be extremely tedious and to be eschewed at all costs. But is that ever a problem?\n\nWell, when writing a paper where there\u2019s only one universe in play, there\u2019s no problem with just saying \u201csmall category\u201d to mean \u201csmall category with respect to the universe\u201d. In a sense, the entire paper is universally quantified over a single universe. If there are multiple universes in use, then one can introduce phrases like \u201csmall category\u201d, \u201cmoderate category\u201d, \u201clarge category\u201d, \u201cvery large category\u201d, etc.\n\nHowever, if within the scope of one text, you want to prove something about small categories and then apply it to both small and large ones, then the universe polymorphism has to happen at the level of individual theorems. This is no problem if we make the quantification explicit: we prove one theorem of the form \u201cfor any $U$, blah blah $U$-small categories\u2026\u201d and later on we say \u201cby theorem such-and-such with $U=V$ [resp. $U=V'$] \u2026\u201d. Can we avoid that? Can the phrase \u201csmall category\u201d mean one thing in one place and another thing in another one? That sounds\u2026 ambiguous.\n\nIt is ambiguous, of course, but sometimes ambiguity is okay, as long as there is a consistent way to resolve it. The sort of ambiguity we\u2019re interested in is called typical ambiguity, and dates back to Bertrand Russell\u2019s original theory of types.\n\nBasically, we want to let ourselves use a single variable \u201c$U$\u201d referring to a universe, but to allow that variable to mean different things when we use it in different places. Of course, this ambiguity will carry over to anything defined in terms of $U$. For instance, if we define \u201csmall category\u201d to mean \u201ccategory whose objects and morphisms belong to $U$\u201d, then the phrase \u201csmall category\u201d will likewise be allowed to mean different things in different places.\n\nBut actually, in the world of typical ambiguity, it would be silly to define \u201csmall category\u201d at all: since every category belongs to some universe, and our universe variable $U$ could refer to any universe, we might as well just define a category to be one whose objects and morphisms belong to $U$. (Similarly, we ought to simply define \u201cset\u201d \u2014 or, in type theory, \u201ctype\u201d \u2014 to mean \u201celement of $U$\u201d. This is, I guess, the origin of the name \u201ctypical ambiguity\u201d: the meaning of the word \u201ctype\u201d is ambiguous.) Now we may prove a theorem about categories in one place, and then later on apply that theorem both to a particular category $A$, and to the category of categories of which $A$ is an object.\n\nWhy is this okay? Obviously, a completely unfettered use of language like this would allow us to reproduce the classical paradoxes. What we need to know is that there is some assignment of a particular universe to each occurrence of the variable $U$ which makes all the size issues come out okay. For instance, whenever we write the phrase \u201ccategory of categories\u201d, the universe assigned to the $U$ occurring in its first word \u201ccategory\u201d must contain the universes assigned to the $U$s occurring in all particular categories that are ever said to be objects of this \u201ccategory of categories\u201d. Of course, we might say \u201cthe category of categories\u201d in more than one place, in which case the universes that get assigned to them might also be different.\n\nThis sounds like it might be tedious, but fortunately, when we are doing formalized mathematics with a computer, our proof assistant can often check automatically for us that such an assignment of universe levels exists. For instance, Coq does this. In fact, Coq does not even support explicit universe polymorphism: the only thing you can write to refer to a universe is \u201c$Type$\u201d, which is typically ambiguous: Coq then automatically tries to assign consistent universe levels to all occurrences of \u201c$Type$\u201d.\n\n(Actually, the current version of Coq, though typically ambiguous in this way, is not fully universe polymorphic: when we define \u201ccategory\u201d the occurrence of \u201c$Type$\u201d therein might refer to any universe, but it must refer to the same universe everywhere that we use the word \u201ccategory\u201d. This is obviously problematic. Fortunately, there are better algorithms available, due to Bob Harper and others, which can handle \u201ctrue\u201d universe polymorphism with typical ambiguity. Mathieu Sozeau is almost done implementing one of these in a new version of Coq. The proof assistant Agda, on the other hand, is universe polymorphic, but not typically ambiguous: you have to write universal quantifications over universes explicitly.)\n\nI would like to argue for more use of typical ambiguity in non-formalized mathematics as well. It\u2019s true that one has to verify that there is a consistent assignment of universe levels, which to do completely precisely might be tedious. However, in ordinary mathematics we do plenty of other things that would also be tedious if made completely precise, and which one might hope could be automated when we go to a formalized world. We often mark them with \u201cit is easy to show that\u201d or \u201cit is obvious that\u201d or \u201cit is left to the reader to show that\u201d, or just assert them without justification.\n\nTypical ambiguity with universe polymorphism takes a little getting used to, but once you\u2019ve internalized it, it is very convenient. For instance, here\u2019s an example I was writing about the other day in the informal type theory project: we can simply define a \u201ccardinal number\u201d to be an isomorphism class of sets. Since there is a set of sets, there is no problem forming this quotient. (In univalent type theory, this would be the 0-truncation of the type of h-sets.) There is then a set of cardinal numbers, which inherits addition and multiplication operations making it into a semiring. Likewise, we can define an \u201cordinal number\u201d to be an isomorphism class of well-ordered sets. There is a set of ordinal numbers, which is itself an ordinal number. And there is a function from ordinals to cardinals which forgets the orderings, which by the well-ordering principle is surjective and has a section, assigning to each cardinal number the least element of the fiber over it.\n\nEven in a set-theoretic foundation, I find this use of language to be much more congenial than the usual circumlocutions regarding proper classes, axioms of replacement, representatives of least rank, etc. etc.\n\nWhat about Cantor\u2019s paradox that there is no set of sets, or Burali-Forti\u2019s paradox that the set\/class of ordinals is not an ordinal? Those proofs can be written in the language of typical ambiguity, but they fail the universe consistency check. However, as long as you aren\u2019t deliberately perverse, you won\u2019t usually have to worry about universe inconsistencies. Moreover, in non-formalized mathematics, we are always free to drop into explicit universe polymorphism if it ever seems necessary for clarity. But most of the time, it isn\u2019t.\n\nPosted at December 9, 2012 8:11 PM UTC\n\nTrackBack URL for this Entry:\u00a0\u00a0 http:\/\/golem.ph.utexas.edu\/cgi-bin\/MT-3.0\/dxy-tb.fcgi\/2579\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nSince you\u2019ve mentioned typical ambiguity, I cannot resist mentioning that NF has typical ambiguity built right into its construction \u2013 and that is how it avoids Cantor\u2019s paradox, Burali-Forti\u2019s paradox, etc. Unfortunately, as you are doubtless aware, NF has its own problems\u2026\n\nPosted by: Zhen Lin on December 9, 2012 10:49 PM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nIn what sense does NF have typical ambiguity? I thought in NF there was literally one set of all sets.\n\nMaybe you\u2019re thinking of the stratification requirement on formulas in the comprehension scheme? That is similar, but not I think the same\u2026.\n\nPosted by: Mike Shulman on December 10, 2012 12:55 AM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nIt goes back to Russellian type theory: the typing here refers not to set-theoretic universes but how many levels of sets of sets of \u2026 of sets we can have. The stratification (or rather, stratifiability) requirement in NF is essentially what we get by saying \u201ceverything that can be constructed in a stratified way exists, but we don\u2019t bother distinguishing between different types once we\u2019ve constructed what we want\u201d. As far as I can tell, the difference is only in how finely we stratify the universe.\n\nPosted by: Zhen Lin on December 10, 2012 8:44 AM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nIt sounds like what you\u2019re saying is that Russellian type theory (and NF) mean something different by \u201ctypical ambiguity\u201d than is meant in modern type theory. That doesn\u2019t surprise me in principle (although it does surprise me a little in this case, given what I thought I knew), since modern type theory bears little resemblance to the Russellian original. Anyway, if that\u2019s right, then it sounds like the Russellian\/NF version also won\u2019t be solving the same problems (although of course NF has its own approach to \u201csolving\u201d the problem of universes).\n\nPosted by: Mike Shulman on December 10, 2012 3:04 PM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nHmmm. I think it is the same \u2018typical ambiguity\u2019: in both cases we have some stratification of the universe, and in both cases we are basically working locally and ignoring the precise level of the hierarchy we are at and focusing more on the relative levels of the various constructions under consideration. At least to me, there seems to be no conceptual difference between typical ambiguity in NF and typical ambiguity in say, the system $\\mathrm{ZFC}\/U_{\\lt \\omega}$ in [Feferman, 2004].\n\nPosted by: Zhen Lin on December 10, 2012 6:47 PM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nIn the systems I was writing about, there are many different universes, and when we write $Type:Type$ its meaning is ambiguous: it could mean $Type_n : Type_{n+1}$ for any $n$.\n\nSimilarly, in Feferman\u2019s system there are, actually, a countable infinity of universes, and a given statement can be interpreted relative to any of them. His reflectivity property of the universes means that its truth in one universe is equivalent to its truth in any other, but there still are all the different universes.\n\nBy contrast, my understanding of NF is that there is one set of all sets, $V = \\{ x | \\top \\}$, and that when we write $V\\in V$ this is just a single true statement, not ambiguous at all.\n\nPosted by: Mike Shulman on December 11, 2012 2:09 AM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nHmmm. Perhaps I was not clear. Essentially, NF is the _result_ of extracting the typically ambiguous fragment of Russellian type theory TST. (If I understand correctly, Specker [1962] showed that NF is consistent if and only if TST with typical ambiguity is consistent.) For example, although $X : Type_n$ implies $\\mathcal{P} X : Type_{n+1}$ in TST, once we erase the types we get the NF theorem \u201c$X \\in V$ implies $\\mathcal{P} X \\in V$\u201d. Similarly, $V \\in V$ in NF corresponds to the uncontroversial $Type_n : Type_{n+1}$ in TST. The fact that the category of NF sets fails to be cartesian closed corresponds to the fact that the graph of the evaluation map cannot be consistently typed. (In TST, the entries of an ordered pair must have the same type.)\n\nAnyway, maybe what I really want to say is that typical ambiguity is a subtle beast. But perhaps we can get away with it in the case where our universes $Type_n$ look like a model of ZFC\u2026\n\nPosted by: Zhen Lin on December 11, 2012 9:10 AM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nOkay, I think I get what you\u2019re saying. NF is one thing that might result if you start from a typically ambiguous system and say \u201cnow let\u2019s suppose instead that we actually have only one universe, but let\u2019s restrict the things that we can do with it in some way roughly motivated by the typically ambiguous statements that passed the universe consistency check.\u201d\n\nI also didn\u2019t realize that Russell\u2019s type theory was as bizarre as NF is! Typical ambiguity in modern type theory is completely unproblematic and known to be consistent (with no need for any ZFC or reflection principles). Bob Harper\u2019s algorithm just tells you how to assign universe levels, and if it can\u2019t be done, then you fail.\n\nPosted by: Mike Shulman on December 11, 2012 4:43 PM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nHey Mike, you forgot to mention Bob Harper here.\nHe is behind Mathieu\u2019s work on this, and much earlier work.\n\nSteve\n\nPosted by: Steve Awodey on December 10, 2012 2:26 AM | Permalink | Reply to this\n\n### Re: Universe Polymorphism and Typical Ambiguity\n\nSorry! I was aiming for mere exposition and hoping I could get away without giving lots of references. But I made the mistake of mentioning at least one person\u2019s name, so to be fair I should try to mention everyone else as well. As Steve says, the formal theory of universe polymorphism and typical ambiguity in type theory which Mathieu is implementing is essentially from this paper.\n\nOn the set-theoretic side, one should perhaps also mention Solomon Feferman, who has been working on modifications of ZFC that support a related sort of typical ambiguity using reflection principles. I blogged about one of those theories back here (with some discussion in the comments that is interesting to look back on from three years down the road). There are a bunch of other interesting-looking things in his list of papers that I still intend to read\u2026\n\nPosted by: Mike Shulman on December 10, 2012 3:49 AM | Permalink | Reply to this\n\nPost a New Comment","date":"2016-07-26 00:39:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 34, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7452943325042725, \"perplexity\": 614.807912404398}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-30\/segments\/1469257824499.16\/warc\/CC-MAIN-20160723071024-00181-ip-10-185-27-174.ec2.internal.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.jobilize.com\/online\/course\/distributed-parameters-transmission-lines-by-openstax","text":"# Distributed parameters\n\n Page 1 \/ 1\nThis document examines how signals behave as they travel along transmission lines. It also introduces distributed parameters.\n\nHaving learned something about how we generate signals with bipolar and field effect transistors, wenow turn our attention to the problem of getting those signals from one place to the next. Ever since Samuel Morse (and thefounder of my alma mater , Ezra Cornell) demonstrated the first working telegraph, engineers andscientists have been working on the problem of describing and predicting how electrical signals behave as they travel downspecific structures called transmission lines .\n\nAny electrical structure which carries a signal from one point to another can be considered a transmissionline. Be it a long-haul coaxial cable used in the Internet, a twisted pair in a building as part of a local-area network, acable connecting a PC to a printer, a bus layout on a motherboard, or a metallization layer on a integrated circuit,the fundamental behavior of all of these structures are described by the same basic equations. As computer switchingspeeds run into the 100s of MHz, into the GHz range, considerations of transmission line behavior are ever morecritical, and become a more dominant force in the performance limitations of any system.\n\nFor our initial purposes, we will introduce a \"generic\" transmission line , which will incorporate most (but not all) features of real transmissionlines. We will then make some rather broad simplifications, which, while rendering our results less applicable to real-lifesituations, nevertheless greatly simplify the solutions, and lead us to insights that we can indeed applyto a broad range of situations.\n\nThe generic line consists of two conductors. We will suppose a potential difference $V(x)$ exists between the two conductors, and that a current $I(x)$ flows down one conductor, and returns via the other. For the time being, we will let the transmission line be\"semi-infinite\", which means we have access to the line at some point $x$ , but the line then extends out in the $x$ direction to infinity. (Such lines are a bit difficult to handle in the lab!)\n\nIn order to be able to describe how $V(x)$ and $I(x)$ behave on this line, we have to make some kind of model of the electrical characteristics of the line itself. We can not just make up any model we want however;we have to base the model on physical realities.\n\nLet's start out by just considering one of the conductors and the physical effects of current flowing though thatconductor. We know from freshman physics that a current flowing in a wire gives rise to a magnetic field, $H$ ( ). Multiply $H$ by and you get $B$ , the magnetic flux density, and then integrate $B$ over a plane parallel to the wires and you get , the magnetic flux \"linking\" the circuit. This is shown in for at least part of the surface. The definition of $L$ , the inductance of a circuit element, is just\n\n$L\\equiv \\frac{}{I}$\nwhere is the flux linking the circuit element, and $I$ is the current flowing through it. Our only problem in finding is that the longer a section of wire we take, the more we have for the same $I$ . Thus, we will introduce the concept of a distributed parameter.\ndistributed parameter\nA distributed parameter is a parameter which is spread throughout astructure and is not confined to a lumped element such as a coil of wire.\nLikewise, if we have two conductors separated by some distance, and if there is a potential difference $V$ between the conductors, thenthere must be some charge $(Q)$ on the two conductors which gives rise to that potential difference. We can imagine a linear charge distribution on thetransmission line, (C\/m), where we have Coulombs\/m on one conductor, and $-$ Coulombs\/m on the other conductor. For a line of length ${x}_{0}$ , we would have $Q=({x}_{0})$ on each section of wire. Whenever you have two charged conductors with a voltage difference between them, you candescribe the ratio of the charge to the voltage as a capacitance. The two conductors would have a capacitance\n$C=\\frac{Q}{V}=\\frac{{x}_{0}}{V}$\nand a distributed capacitance $C$ (F\/m) which is just $\\frac{}{V}$ . A length of line ${x}_{0}$ long would have a capacitance $C=C{x}_{0}$ Farads associated with it . Thus, we see that the transmission line has both a distributed inductance $L$ and a distributed capacitance $C$ which are tied up with each other. There is really no way in which we can separate one from the other. In other words, we cannot have only the capacitance, or only the inductance, there will always be some of each associated with each section of linenow matter how small or how big we make it.\n\nWe are now ready to build our model. What we want to do is to come up with some arrangement of inductors andcapacitors which will represent electrically, the properties of the distributed capacitance and inductance we discussedabove. As a length of line gets longer, its capacitance increases, so we had better put the distributed capacitances inparallel with one another, since that is the way capacitors add up. Also, as the line gets longer, its total inductanceincreases, so we had better put the distributed inductances in series with one another, for that is the way inductances addup. is a representation of the distributed inductance and capacitance of the generic transmission line.\n\nWe break the line up into sections $(x)$ long, each one with an inductance $L(x)$ and a capacitance $C(x)$ . If we halve $(x)$ , we would halve the inductance and capacitance of each section, but we'd have twice as many of them per unitlength. Duh! The point is no matter how fine we make $C(x)$ , we still have Ls and Cs arranged like we see in , with the two kinds of components intermixed.\n\nWe could make a more realistic model and realize that all real wires have seriesresistance associated with them and that whatever we use to keep the two conductors separated will have some leakage conductanceassociated it. To account for this we would introduce a series resistance $R$ (ohms\/unit length) and a series conductance $G$ (ohms\/unit length). One section of our line model then looks like .\n\nAlthough this is a more realistic model, it leads to much more complicated math. We will start out anyway,ignoring the series resistance $R$ and the shunt conductance $G$ . This \"approximation\" turns out to be pretty good as long as eitherthe line is not too long, or the frequencies of the signals we are sending down the line do not get too high. Without theseries resistance or parallel conductance we have what is called an ideal lossless transmission line .\n\n#### Questions & Answers\n\nApplication of nanotechnology in medicine\nwhat is variations in raman spectra for nanomaterials\nI only see partial conversation and what's the question here!\nwhat about nanotechnology for water purification\nplease someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.\nDamian\nyes that's correct\nProfessor\nI think\nProfessor\nwhat is the stm\nis there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?\nRafiq\nindustrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong\nDamian\nHow we are making nano material?\nwhat is a peer\nWhat is meant by 'nano scale'?\nWhat is STMs full form?\nLITNING\nscanning tunneling microscope\nSahil\nhow nano science is used for hydrophobicity\nSantosh\nDo u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq\nRafiq\nwhat is differents between GO and RGO?\nMahi\nwhat is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq\nRafiq\nif virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION\nAnam\nanalytical skills graphene is prepared to kill any type viruses .\nAnam\nwhat is Nano technology ?\nwrite examples of Nano molecule?\nBob\nThe nanotechnology is as new science, to scale nanometric\nbrayan\nnanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale\nDamian\nIs there any normative that regulates the use of silver nanoparticles?\nwhat king of growth are you checking .?\nRenato\nWhat fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?\nwhy we need to study biomolecules, molecular biology in nanotechnology?\n?\nKyle\nyes I'm doing my masters in nanotechnology, we are being studying all these domains as well..\nwhy?\nwhat school?\nKyle\nbiomolecules are e building blocks of every organics and inorganic materials.\nJoe\nanyone know any internet site where one can find nanotechnology papers?\nresearch.net\nkanaga\nsciencedirect big data base\nErnesto\nIntroduction about quantum dots in nanotechnology\nhi\nLoga\nwhat does nano mean?\nnano basically means 10^(-9). nanometer is a unit to measure length.\nBharti\nhow did you get the value of 2000N.What calculations are needed to arrive at it\nPrivacy Information Security Software Version 1.1a\nGood\nGot questions? Join the online conversation and get instant answers!","date":"2020-07-11 14:02:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 42, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6205873489379883, \"perplexity\": 902.1496412858288}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-29\/segments\/1593655933254.67\/warc\/CC-MAIN-20200711130351-20200711160351-00457.warc.gz\"}"} | null | null |
Q: Rotating a specific area of an array I have an NxN matrix and I want to pick a specific part of this matrix and rotate that area by 90 degrees clockwise. I have to specify this rotation under these conditions:
*
*(a, b) position as upper left corner,
*(a + c, b + c) position as lower right corner, and
*whose side length is c + 1.
So, when I give a=2, b=3, and c=3, the output will be like this:
Code:
import numpy as np
N = int(input())
S = int(input())
arr = np.array(range(0,N*N))
arr.shape = N,N
for i in range(S):
a,b,c = [int(x) for x in input().split()]
arr[a:a+c+1,b:b+c+1] = np.rot90(arr[a:a+c+1,b:b+c+1], axes=(1,0))
print(arr);
A: A 90-degree clockwise rotation corresponds to a vertical flip followed by a transpose. So:
def rotate_clockwise(x):
return x[::-1].T
To use it on a particular region:
arr[a:a+s, b:b+s] = rotate_clockwise(arr[a:a+s, b:b+s])
In your case, s = c + 2.
EDIT: Perhaps even better is to use np.rot90:
def rotate_clockwise(x):
return np.rot90(x, 3)
| {
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Joan Bonet de Paredes, conegut com a Juan de Paredes, (Oriola, meitat del - Toledo, 1710), va ser un mestre de capella i compositor valencià. Segons López Calo va començar a exercir de mestre de capella l'any 1680 a Berlanga i, posteriorment a Palència i Segòvia durant el 1684. Prop del 1686 va ser mestre de capella al reial monestir de la Encarnación de Madrid. Durant els últims anys de la seva vida va obtenir la plaça de la seu metropolitana i primada de Toledo. La major part de la seva obra es conserva a les catedrals de Segòvia i Toledo, tot i que algunes obres seves també es localitzen al fons musical CMar (Fons de l'església parroquial de Sant Pere i Sant Pau de Canet de Mar), així com a la Bibloteca de Catalunya i a l'Arxiu de Montserrat.
Referències
Músics oriolans
Mestres de capella valencians històrics
Compositors valencians històrics
Compositors valencians del segle XVII | {
"redpajama_set_name": "RedPajamaWikipedia"
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\section{Summary:} Biospectrogam is an open-source software for the spectral analysis of DNA and protein sequences. The software can fetch (from NCBI server), import and manage biological data. One can analyze the data using Digital Signal Processing (DSP) techniques since the software allows the user to convert the symbolic data into numerical data using $23$ popular encodings and then apply popular transformations such as Fast Fourier Transform (FFT) etc. and export it. The ability of exporting (both encoding files and transform files) as a MATLAB\textsuperscript{\textregistered} .m file gives the user an option to apply variety of techniques of DSP. User can also do window analysis (both sliding in forward and backward directions and stagnant) with different size windows and search for meaningful spectral pattern with the help of exported MATLAB\textsuperscript{\textregistered} file in a dynamic manner by choosing time delay in the plot using Biospectrogram. Random encodings and user choice encoding allows software to search for many possibilities in spectral space.
\section{Availability:}
Biospectrogam is written in Java\textsuperscript{\textregistered} and is available to download freely from http://www.guptalab.org/biospectrogram. Software has been optimized to run
on Windows, Mac OSX and Linux. User manual and you-tube (product demo) tutorial is also available on the website. We are in the process of acquiring open source license for it.
\section{Contact:} \href{mankg@computer.org}{mankg@computer.org}
\end{abstract}
\section{Introduction}
Molecular biology has shown tremendous progress in the last decade because of various genome projects producing vast amount of biological data. This has resulted in Encode project (http://encodeproject.org) that classifies all the basic DNA elements of Human genome. This also gives us new insight into numerous molecular mechanism. In order to understand the digital biological data, people use different techniques from mathematics, computer science, etc. Digital signal processing (DSP) is a fundamental concept in information and communication technology (ICT). A natural question arises ``Can DSP techniques help us to understand the digital biology?" It turns out that the DSP techniques are playing a major role in biology and have given birth to a new branch called genomic signal processing \citep{shmulevich2007genomic}. To analyse the genomic data, researchers first convert the symbolic data (example DNA or protein data) into numerical data by applying a suitable map \citep{Kwan,Arniker2012} and then by applying signal processing transforms such as Fourier etc. to study the desired biological properties \citep{citeulike:3895919}. In this work, we present a tool, Biospectrogram, which can help researchers to apply different encodings on the biological data and apply certain transformations to do the spectral analysis. User can also export the files (encoded or transformed) to popular MATLAB\textsuperscript{\textregistered} software \citep{MATLAB:2010} to do the direct analysis.
\section{Implementation and Features}
The tool Biospectrogram has $4$ major components viz. data collector, encode, transforms and export $\&$ plot. One can use the tool in DNA or Protein mode by using the switch button. The tool has two main windows viz display window for displaying the data (collected or encoded data) and work window (encoded or transform data) to show the work. Data collector module provides a direct fetching of DNA data (both fasta and genebank file formats) from National Center of Biotechnology Information (NCBI) server by taking accession number from user which can be encoded using encode button. User can also import the files from his own machine/network. One can also select a portion of the data from the window and do further processing.
One popular encoding map is the Voss representation \citep{1195219} which maps the nucleotides $A, C, G,$ and $T$ from DNA space into the four binary indicator sequences $x_A[n]$, $x_C[n]$, $x_G[n]$, and $x_T[n]$ showing the presence (e.g. $1$) or absence (e.g. $0$) of the respective nucleotides. Similar indicator maps are available for protein space.
\begin{figure*}[!tpb
\centerline{\includegraphics[scale=0.46]{biospecwitharrows.png}}
\caption{Basic architecture of Biospectrogram showing $4$ major components (data collector, encode, transforms and export $\&$ plot). Different relationships between $23$ encodings and transformations (with solid arrows possible in our tool) and others possible broken arrows using third party software MATLAB.}\label{fig:02}
\end{figure*}
Different possible encodings ($23$ available in our tool) and transformations ($6$ available in our tool) are shown in Figure ~\ref{fig:02}. While applying encoding user has to select the fetched file from the first dropdown list and encoding scheme from the second dropdown list. The fasta file of the DNA sequence is shown in the display window and the encoded output is shown in the work window. After encoding the fetched DNA sequence or protein sequence, one can apply suitable transforms (see Figure ~\ref{fig:02}) available in the tool. To apply other transforms (not available in our tool) and filters etc. one can export the encoded files to MATLAB\textsuperscript{\textregistered} .m files and do the further analysis. For exhaustive search of a pattern, a window analysis can be done with our tool. The window button allows the user to set a window size while moving the window in both directions (forward and backward) using sliding window option whereas using stagnant window option user can select a portion of the sequence for the power spectrum from all its indicator sequences. By choosing appropriate delay time in the preferences, one can plot the transformation's output of our tool by exporting the transformation files to MATLAB\textsuperscript{\textregistered} .m files and observe the signal in a smooth automatic manner with a delay of time set by user.
\nocite{1195219,SilvermanLinsker,zhangzhang,citeulike:4180094,939833,1346354,liao,Yau2003,citeulike:6778043,4365821,cristea, rosen, chakravarthy,1227391,DanCristea:2003:LSF:774474.774488,Vaidyanathan05genomicsand}
\bibliographystyle{natbib}
| {
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} | 6,811 |
Bournemouth maintaining their lead at the top of English Championship League table after grabbing a comfortable win over Fulham on Boxing Day.
Playing in front of their home crowd, Bournemouth immediately took that attacking initiative as they went one goal up after just nine minutes through Brett Pitman.
That slender lead lasted until half time break as Fulham failed to grab any equalizing goal during the first 45 minutes.
After the break, both teams continue to push forward in search for goals, but it was the home side that eventually netted that important winner.
It was Harry Arter who secured all three points for his team after scoring Bournemouth's second in stoppage time.
AFC Bournemouth vs Fulham . Score: 2-0.
AFC Bournemouth 1-0: Brett Pitman goal (9').
AFC Bournemouth 2-0: Harry Arter goal (90+1'). | {
"redpajama_set_name": "RedPajamaC4"
} | 4,193 |
Rallies ramp up ahead of India's election
Politicians are trying to convince 800 million voters that they can lead the country and fix its problems.
Political parties in India have begun their final push to win support from the 800 million people eligible to vote, ahead of the first phase of national elections.
The ruling Congress Party is using public gatherings to try to convince voters that it still has what it takes to lead the world's largest democracy and fix some of its biggest problems.
But India's longest serving politicians are being challenged by its newest.
Arvind Kejriwal, the leader of the Aam Aadmi - or common man's party - has engaged voters by campaigning on an anti-corruption platform.
The first ballots will be cast on April 7.
Al Jazeera's Nidhi Dutt reports from New Delhi. | {
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Cockthorpe – wieś w Anglii, w Norfolk, w dystrykcie North Norfolk, w civil parish Binham. W 1931 wieś liczyła 55 mieszkańców. Cockthorpe jest wspomniana w Domesday Book (1086) jako Torp.
Przypisy
Linki zewnętrzne
http://www.origins.org.uk/genuki/NFK/places/c/cockthorpe/
Wsie w hrabstwie Norfolk | {
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} | 5,130 |
\section{Introduction}\label{s:intro}
It is a common problem in Differential Geometry to produce examples of
(possibly immersed) Riemannian manifolds $(L,g)$ satisfying a given geometric
constraint, usually a nonlinear PDE, on the metric (Einstein, constant scalar curvature,
\textit{etc.}) or on the immersion (constant mean curvature, minimal,
\textit{etc.}). If $L$ (or the immersion) happens to be singular, one then faces
the problem of ``desingularizing'' it to produce a new, smooth,
Riemannian manifold satisfying the same constraint. Often, one actually hopes to
produce a family $(L_t,g_t)$ of manifolds satisfying the constraint and which
converges in some sense to $(L,g)$ as $t\rightarrow 0$.
Roughly speaking, the usual method for
solving such desingularization problems is as follows. For simplicity, we focus
on
the situation where $L$ has only isolated point singularities and the
constraint is on the metric.
\
\textbf{Step 1:} For each singular point $x\in L$, we look for an explicit
smooth ``local model'': \textit{i.e.}, a manifold $(\hat{L},\hat{g})$ which
satisfies a related, scale-invariant, constraint and which, outside of some compact region, is
topologically and metrically similar to an annulus $B(x,\epsilon_1)\setminus
B(x,\epsilon_2)$ in $L$, centered in the singularity. We can then glue $\hat{L}$
onto the manifold $L\setminus B(x,\epsilon_2)$, using the ``neck region''
$B(x,\epsilon_1)\setminus B(x,\epsilon_2)$ to interpolate between the two
metrics. The fact that the neck region is ``small'' is usually not a problem: one can simply
rescale $\hat{g}$ to $t^2\hat{g}$ so that now $(\hat{L},t^2\hat{g})$ is of
similar size. The resulting manifold, which we denote
$(\hat{L}\#L, \hat{g}\#g)$, satisfies the constraints outside of the neck region
simply by construction. If the interpolation is done carefully we also get very good control over what happens on the neck. We think of $(\hat{L}\#L, \hat{g}\#g)$ as an ``approximate solution'' to the gluing problem. Rescaling also gives a way to build families: the idea is
to glue $(\hat{L},t^2\hat{g})$ into $B(x,\epsilon_1)\setminus
B(x,t\epsilon_2)$, producing a family $(L_t,g_t)$; intuitively, as $t\rightarrow
0$ the compact region in $\hat{L}$ collapses to the singular point $x$ and $L_t$
converges to $L$.
\
\textbf{Step 2:} We now need to perturb each $(L_t, g_t)$ so
that the resulting family satisfies the constraint globally. Thanks to a
linearization process, the perturbation process often boils down to studying a
linear elliptic system on $g_t$. One of the main problems is to verify that this
system satisfies estimates which are uniform in $t$. This is the key to
obtaining the desired perturbation for all sufficiently small $t$. Roughly
speaking, there is often a delicate balance to be found as $t\rightarrow 0$:
on the one hand, if $L_t$ was built properly, as $t\rightarrow 0$ it will get
closer to solving the constraint; on the other hand, it becomes more
singular. Uniform estimates are important in proving that this balance can be
reached.
\
The geometric problem defines the differential operator to be studied. However, this operator is often fairly intrinsic, and can be defined independently of the geometric specifics. The necessary estimates may likewise be of a much more general nature. Filtering out the geometric ``super-structure'' and concentrating on the analysis of the appropriate category of abstract Riemannian manifolds will then enhance the understanding of the problem, leading to improved results and clarity. The first goal of this paper is thus to set up an abstract framework for dealing with gluing constructions and the corresponding uniform estimates. Here, ``abstract'' means: independent of any specific geometric problem. We focus on gluing constructions concerning Riemannian manifolds with isolated
conical singularities. These are perhaps the simplest singularities possible,
but in the gluing literature they often appear as an interesting and
important case. Our framework involves two steps, parallel to those outlined above.
\
\textbf{Step A:} In Section \ref{s:sums_sobolev} we define a general \textit{connect sum} construction between Riemannian manifolds, extrapolating from standard desingularization procedures.
\
\textbf{Step B:} We show how to produce uniform estimates on these connect sum manifolds, by presenting a detailed analysis of three important problems:
(i) Sobolev Embedding Theorems, (ii) invertibility of the
Laplace operator, (iii) Poincar\'{e} and
Gagliardo-Nirenberg-Sobolev type inequalities. The main results are Theorems
\ref{thm:normstequivalent},
\ref{thm:sum_injective}, \ref{thm:cpt_sum_injective}, \ref{thm:d_invertible} and Corollary \ref{cor:improved_sob}.
\
Our Step A is actually
much more general than Step 1, as described above: it is specifically
designed to deal with both compact and non-compact manifolds and it allows us to replace
the given singularity not only with smooth compact regions but also with
non-compact ``asymptotically conical ends'' or even with new singular regions.
It also allows for
different ``neck-sizes'' around each singularity. In this sense it offers a very
broad and flexible framework to work with.
The range of possible estimates covered by our framework is clearly much wider than the set of Problems (i)-(iii) listed in Step B. Indeed, the underlying, well-known, theory of elliptic operators on conifolds is extremely general. Within this paper, this choice is to be intended as fairly arbitrary: amoungst the many possible, we choose 3 estimates of general interest but differing one from the other in flavour: Problem (i) is of a mostly
local nature, Problems (ii) and (iii) are global. Correspondingly, we split the paper into 2 parts: Part 1 deals with Sobolev Embedding Theorems and the connect sum construction, Part 2 deals with elliptic estimates.
In reality, however, our choice of Problems (i)-(iii) is based on the very specific geometric problems we happen to be interested in. The second goal of this paper is thus to lay down the analytic foundations for our papers \cite{pacini:sldefs}, \cite{pacini:slgluing} concerning deformations and desingularizations of submanifolds whose
immersion map satisfies the \textit{special Lagrangian} constraint. The starting point for this work was a collection of gluing results concerning special Lagrangian submanifolds due to Arezzo-Pacard \cite{arezzopacard}, Butscher \cite{butscher}, Lee \cite{lee} and Joyce \cite{joyce:III}, \cite{joyce:IV}, and parallel results concerning \textit{coassociative} submanifolds due to Lotay \cite{lotay}. It slowly became apparent, thanks also to many conversations with some of these authors, that several parts of these papers could be simplified, improved or generalized: related work is currently still in progress. In particular, building approximate solutions and setting up the perturbation problem requires making several choices which then influence the analysis rather drastically. A third goal of the paper is thus to present a set of choices which leads to very clean, simple and general results. Although some of these choices may seem obvious to some members of the ``gluing community'', it still seems useful to emphasize this point.
One such choice concerns the parametrization of the approximate solutions: parametrizing the necks so that they depend explicitly on the parameter $t$ is one ingredient in obtaining uniform estimates. A second ingredient is the consistent use, even when dealing with compact manifolds, of weighted rather than standard Sobolev spaces. Given the lack of
references analogous to Hebey \cite{hebey}, which deals with standard Sobolev
spaces, we
devote a fair amount of attention to their definition and
properties. Our main result here is Theorem
\ref{thm:weighted_ok}, which concerns the validity of the Sobolev Embedding
Theorems under fairly general hypotheses on the ``scale'' and
``weight'' functions with which we define our spaces. The
analogous result for conifolds, Corollary \ref{cor:embedding}, can be
seen as a special case of Theorem \ref{thm:weighted_ok} and is
well-known. However, Problem (i) above requires keeping
close track of how the corresponding Sobolev constants depend on the conifolds
and on the other data used in the connect sum construction. This explains why
we need to set up the general theory so carefully.
Another key tool for our estimates is the Fredholm theory of
elliptic operators on manifolds with asymptotically conical ends. This theory
is well-known but, for the reader's convenience, we review it (together with its
asymptotically cylindrical counterpart) in Sections \ref{s:acyl_analysis} and
\ref{s:accs_analysis}. Sections \ref{s:weightcrossing} and \ref{s:accs_harmonic}
contain instead some useful consequences of the Fredholm theory.
We conclude with one last comment. Depending on the details, the connect sum construction can have two outcomes: compact or non-compact manifolds. In the context of weighted spaces, Problem (i) does not notice the difference. Problems (ii) and (iii) require instead that
the kernels of the operators in question vanish. On non-compact manifolds this
can be achieved very simply, via an a-priori choice of weights: roughly
speaking, we require that there exist non-compact ``ends'', then put weights on them which kill the kernel. This topological assumption is perfectly compatible
with the geometric applications described in \cite{pacini:slgluing}. On compact manifolds it is instead necessary to work transversally to the kernel; uniform estimates depend on allowing the subspace itself to depend on the parameter $t$. We refer to Section \ref{s:sums_laplace} for details.
\
{\bf Acknowledgments.} I would like to thank D. Joyce for many useful
suggestions and discussions concerning the material of this paper. I also thank
M. Haskins and J. Lotay for several conversations. Part of this work was carried out while I
was a Marie Curie EIF Fellow at the University of Oxford. It has also been supported by a Marie Curie ERG grant at the Scuola Normale Superiore in Pisa.
\section{Preliminaries}\label{s:prelim}
Let $(L,g)$ be an oriented $m$-dimensional Riemannian manifold. We can identify
its tangent and cotangent bundles via the maps
\begin{equation}\label{eq:sharp}
T_xL\rightarrow T_x^*L,\ \ v\mapsto v^{\#}:=g(v,\cdot), \ \ \mbox{with inverse }
T_x^*L\rightarrow T_xL,\ \ \alpha\mapsto\alpha^\flat.
\end{equation}
There are induced isomorphisms on all higher-order tensor bundles over $L$. In
particular the metric tensor $g$, as a section of $(T^*L)^2$, corresponds to a
tensor $g^\flat$, section of $(TL)^2$. This tensor defines a natural metric on
$T^*L$ with respect to which the map of Equation \ref{eq:sharp} is an isometry.
In local coordinates, if $g=g_{ij}dx^i\otimes dx^j$ then
$g^\flat=g^{ij}\partial_i\otimes \partial_j$, where $(g^{ij})$ denotes the
inverse matrix of $(g_{ij})$.
Given any $x\in L$ we denote by $i_x(g)$ the \textit{injectivity radius at $x$},
\textit{i.e.} the radius of the largest ball in $T_xL$ on which the exponential
map is a diffeomorphism. We then define the \textit{injectivity radius of $L$}
to be the number $i(g):=\mbox{inf}_{x\in L} i_x(g)$. We denote by $Ric(g)$ the
\textit{Ricci curvature tensor} of $L$: for each $x\in L$, this gives an element
$Ric_x(g)\in T_x^*L\otimes T_x^*L$.
Let $E$ be a vector bundle over $L$. We denote by $C^\infty(E)$ (respectively,
$C^\infty_c(E)$) the corresponding space of smooth sections (respectively, with
compact support). If $E$ is a metric bundle we can define the notion of a
\textit{metric connection} on $E$: namely, a connection $\nabla$ satisfying
\begin{equation*}
\nabla(\sigma,\tau)=(\nabla\sigma,\tau)+(\sigma,\nabla\tau),
\end{equation*}
where $(\cdot,\cdot)$ is the appropriate metric. We then say that $(E,\nabla)$
is a \textit{metric pair}.
Recall that coupling the Levi-Civita connection on $TL$ with a given connection
on $E$ produces induced connections on all tensor products of these bundles and
of their duals. The induced connections depend linearly on the initial
connections. Our notation will usually not distinguish between the initial
connections and the induced connections: this is apparent when we write, for
example, $\nabla^2\sigma$ (short for $\nabla\nabla\sigma$). Recall also that the
difference between two connections $\nabla$, $\hat{\nabla}$ defines a tensor
$A:=\nabla-\hat{\nabla}$. For example, if the connections are on $E$ then $A$ is
a
tensor in $T^*L\otimes E^*\otimes E$. Once again, we will not distinguish
between this $A$ and the $A$ defined by any induced connections.
Let $E$, $F$ be vector bundles over $L$. Let $P:C^\infty(E)\rightarrow
C^\infty(F)$ be a linear differential operator with smooth coefficients, of
order $n$. We can then write $P=\sum_{i=0}^n A_i\cdot\nabla^i$, where $A_i$ is a
global section of $(TL)^i\otimes E^*\otimes F$ and $\cdot$ denotes an
appropriate contraction. Notice that since $P$ is a local operator it is
completely defined by its behaviour on compactly-supported sections.
\begin{remark} \label{rem:A}
Assume $P=\sum_{i=0}^n A_i\cdot\nabla^i$. Choose a second connection
$\hat{\nabla}$ on $E$ and set $A:=\nabla-\hat{\nabla}$. Substituting
$\nabla=\nabla-\hat{\nabla}+\hat{\nabla}=A+\hat{\nabla}$ allows us to write $P$
in terms of $\hat{\nabla}$. Notice that the new coefficient tensors $\hat{A}_i$
will depend on $A$ and on its derivatives $\hat{\nabla}^k A$.
\end{remark}
Now assume $E$ and $F$ are metric bundles. Then $P$ admits a \textit{formal
adjoint} $P^*:C^\infty(F)\rightarrow C^\infty(E)$, uniquely defined by imposing
\begin{equation}
\int_L(P\sigma,\tau)_F \,\mbox{vol}_g=\int_L(\sigma,P^*\tau)_E \,\mbox{vol}_g, \
\ \forall \sigma\in C^\infty_c(E), \ \tau\in C^\infty_c(F).
\end{equation}
$P^*$ is also a linear differential operator, of the same order as $P$.
\begin{example} \label{e:nablalaplace}
The operator $\nabla:C^\infty(E)\rightarrow C^\infty(T^*L\otimes E)$ has a
formal adjoint $\nabla^*: C^\infty(T^*L\otimes E)\rightarrow C^\infty(E)$. Given
$P=\sum_{i=0}^n A_i\cdot\nabla^i$, we can write $P^*$ in terms of $\nabla^*$.
For example, choose a smooth vector field $X$ on $L$ and consider the operator
$P:=\nabla_X=X\cdot\nabla:C^\infty(E)\rightarrow C^\infty(E)$. Then
$(\nabla_X)^*\sigma=\nabla^*(X^{\#}\otimes\sigma)$.
\end{example}
The \textit{$\nabla$-Laplace} operator on $E$ is defined as
$\Delta:=\nabla^*\nabla:C^\infty(E)\rightarrow C^\infty(E)$. When $E$ is the
trivial $\mathbb{R}$-bundle over $L$ and we use the Levi-Civita connection, this
coincides with the standard positive Laplace operator acting on functions
\begin{equation}\label{eq:laplace}
\Delta_g:=-\mbox{tr}_g(\nabla^2)=-g^\flat\cdot\nabla^2:C^\infty(L)\rightarrow
C^\infty(L).
\end{equation}
Furthermore $\nabla=d$ and $\nabla^*=d^*$ so this Laplacian also coincides with
the Hodge Laplacian $d^*d$.
On differential $k$-forms the Levi-Civita $\nabla$-Laplacian and the Hodge
Laplacian coincide only up to curvature terms.
\
To conclude, let us recall a few elements of Functional Analysis. We now let $E$
denote a Banach space. Then $E^*$ denotes its dual space and
$\langle\cdot,\cdot\rangle$ denotes the duality map
$E^*\times E\rightarrow\mathbb{R}$.
Let $P:E\rightarrow F$ be a continuous linear map between Banach spaces. Recall
that the \textit{norm} of $P$ is defined as
$\|P\|:=\mbox{sup}_{|e|=1}|P(e)|=\mbox{sup}_{e\neq 0} (|P(e)|/|e|)$. This
implies that, $\forall e\neq 0$, $|P(e)|\leq\|P\|\cdot |e|$. If $P$ is injective
and surjective then it follows from the Open Mapping Theorem that its inverse
$P^{-1}$ is also continuous. In this case $\mbox{inf}_{|e|=1}|P(e)|>0$ and we
can calculate the norm of $P^{-1}$ as follows:
\begin{equation}\label{eq:normofinverse}
\|P^{-1}\|=\mbox{sup}_{f\neq 0}\frac{|P^{-1}(f)|}{|f|}=\mbox{sup}_{e\neq
0}\frac{|e|}{|P(e)|}=\mbox{sup}_{|e|=1}\frac{1}{|P(e)|}=\frac{1}{\mbox{inf}_{
|e|=1}|P(e)|}.
\end{equation}
Recall that, given any subspace $Z\leq F$, the \textit{annihilator} of $Z$ is
defined as
$$\mbox{Ann}(Z):=\{\phi\in F^*:\langle\phi,z\rangle=0, \ \forall z\in Z\}.$$
Notice that $\mbox{Ann}(\overline{Z})=\mbox{Ann}(Z)$. Let $P^*:F^*\rightarrow
E^*$ be the dual map, defined by $\langle
P^*(\phi),e\rangle:=\langle\phi,P(e)\rangle$. It is simple to check that
$\mbox{Ann(Im$(P)$)}=\mbox{Ker}(P^*)$.
Recall that the \textit{cokernel} of $P$ is defined to be the quotient space
$\mbox{Coker}(P):=F/\mbox{Im}(P)$. Assume the image $\mbox{Im}(P)$ of $P$ is a
closed subspace of $F$, so that $\mbox{Coker}(P)$ has an induced Banach space
structure. The projection $\pi:F\rightarrow \mbox{Coker}(P)$ is surjective so
its dual map $\pi^*:(\mbox{Coker}(P))^*\rightarrow F^*$ is injective. The image
of $\pi^*$ coincides with the space $\mbox{Ann(Im$(P)$)}$ so $\pi^*$ defines an
isomophism between $(\mbox{Coker}(P))^*$ and $\mbox{Ann(Im$(P)$)}$. We conclude
that there exists a natural isomorphism
$(\mbox{Coker}(P))^*\simeq\mbox{Ker}(P^*)$.
\begin{remark} \label{rem:characterizations}
It is clear that $\mbox{Ker}(P^*)$ can be characterized as follows:
$$\phi\in\mbox{Ker}(P^*)\Leftrightarrow\langle\phi,P(e)\rangle=0,\ \ \forall
e\in E.$$
On the other hand, the Hahn-Banach Theorem shows that $f\in \overline{Z}$ iff
$\langle\phi,f\rangle=0$, $\forall \phi \in \mbox{Ann}(Z)$. Applying this to
$Z:=\mbox{Im}(P)$, we find the following characterization of
$\overline{\mbox{Im}(P)}$:
$$f\in\overline{\mbox{Im}(P)}\Leftrightarrow \langle\phi,f\rangle=0,\ \ \forall
\phi\in\mbox{Ker}(P^*).$$
\end{remark}
We say that $P$ is \textit{Fredholm} if its image $\mbox{Im}(P)$ is closed in
$F$ and both $\mbox{Ker}(P)$ and $\mbox{Coker}(P)$ are finite-dimensional. We
then define the \textit{index} of $P$ to be
$$i(P):=\mbox{dim(Ker$(P)$)}-\mbox{dim(Coker$(P)$)}=\mbox{dim(Ker$(P)$)}-\mbox{
dim(Ker$(P^*)$)}.$$
\
\textbf{Important remarks}: Throughout this paper we will often encounter chains
of inequalities of the form
\begin{equation*}
|e_0|\leq C_1 |e_1|\leq C_2 |e_2|\leq\dots
\end{equation*}
The constants $C_i$ will often depend on factors that are irrelevant within the
given context. In this case we will sometimes simplify such expressions by
omitting the subscripts of the constants $C_i$, \textit{i.e.} by using a single
constant $C$.
We assume all manifolds are oriented. In Part 2 of the paper we will work under
the assumption $m\geq 3$.
\part{Sobolev Embedding Theorems}
\section{Review of the theory of standard Sobolev spaces}\label{s:std}
We now introduce and discuss Sobolev spaces on manifolds. A good reference,
which at times we follow closely, is Hebey \cite{hebey}.
Let $(E,\nabla)$ be a metric pair over $(L,g)$. The \textit{standard Sobolev
spaces}
are defined by
\begin{equation}\label{eq:std_sob}
W^p_k(E):=\mbox{Banach space completion of the space }\{\sigma\in
C^\infty(E):\|\sigma\|_{W^p_k}<\infty\},
\end{equation}
where $p\in [1,\infty)$, $k\geq 0$ and we use the norm
$\|\sigma\|_{W^p_k}:=\left(\Sigma_{j=0}^k\int_L|\nabla^j\sigma|^p\,\mbox{vol}
_g\right)^ { 1/p}$. We will sometimes use $L^p$ to denote the space $W^p_0$.
\begin{remark}
At times we will want to emphasize the metric $g$ rather than the specific
Sobolev spaces. In these cases we will use the notation $\|\cdot\|_g$.
\end{remark}
It is important to find conditions ensuring that two metrics $g$, $\hat{g}$ on
$L$ (corresponding to Levi-Civita connections $\nabla$, $\hat{\nabla}$), define
\textit{equivalent} Sobolev norms, \textit{i.e.} such that there exists $C>0$
with $(1/C)\|\cdot\|_g\leq \|\cdot\|_{\hat{g}}\leq C\|\cdot\|_g$. In
this case the corresponding two completions, \textit{i.e.} the two spaces
$W^p_k$, coincide.
\begin{definition}\label{def:equivalentmetrics}
We say that two Riemannian metrics $g$, $\hat{g}$ on a manifold $L$ are
\textit{equivalent} if they satisfy the following assumptions:
\begin{description}
\item[A1] There exists $C_0>0$ such that
\begin{equation*}
(1/C_0)g\leq \hat{g}\leq C_0 g.
\end{equation*}
\item[A2] For all $j\geq 1$ there exists $C_j>0$
such that
\begin{equation*}
|\nabla^j\hat{g}|_g\leq C_j.
\end{equation*}
\end{description}
\end{definition}
\begin{remark}\label{rem:equivalence}
It may be useful to emphasize that the conditions of Definition
\ref{def:equivalentmetrics} are symmetric in $g$ and
$\hat{g}$. Assumption A1 is obviously
symmetric. Assumption A2 is also symmetric. For
$j=1$, for example, this follows from the following calculation which uses the
fact that the connections are metric:
\begin{equation}\label{eq:Abounded}
|\nabla\hat{g}|_g=|\nabla\hat{g}-\hat{\nabla}\hat{g}|_g=|A(\hat{g})|_g\simeq
|A(g)|_{\hat{g}}=|\hat{\nabla} g|_{\hat{g}},
\end{equation}
where $\simeq$ replaces multiplicative constants.
Notice that in Equation \ref{eq:Abounded} $A$ is the difference of the induced
connections on $T^*L\otimes T^*L$. This tensor depends linearly on the tensor
defined as the difference of the connections on $TL$. It is simple to see that
these two tensors have equivalent norms so that Assumption A2 provides a
pointwise bound on the norms of either one. From here we easily obtain bounds on
the norms of the tensor defined as the difference of the induced connections on
any tensor product of $TL$ and $T^*L$. Similar statements hold for
bounds on the derivatives of $A$.
Assumptions 1 and 2 can be unified as follows. Assume that, for
all $j\geq 0$, there exists $C_j>0$ such that
$$|\nabla^j(\hat{g}-g)|_g\leq C_j.$$
As long as $C_0$ is sufficiently small, for $j=0$ this condition implies
Assumption 1. Since $\nabla^j g=0$, it is clear that for $j>0$ it is equivalent
to Assumption 2.
\end{remark}
\begin{lemma}\label{l:equivalentstdnorms}
Assume $g$, $\hat{g}$ are equivalent.
Then the Sobolev norms defined by $g$ and $\hat{g}$ are equivalent.
\end{lemma}
\begin{proof}
Consider the Sobolev spaces of functions on $L$. Recall that $\nabla u=du$.
This implies that the $W^p_1$ norms depend only pointwise on the metrics. In
this case Assumption A1 is sufficient to ensure equivalence. In general,
however, the $W^p_k$ norms use the induced connections on tensor bundles. For
example, assume $j=2$. Then
\begin{equation*}
|\nabla^2u|=|(A+\hat{\nabla})(A+\hat{\nabla})u|\leq |A^2u|+|A\cdot\hat{\nabla}
u|+|\hat{\nabla} (Au)|+|\hat{\nabla}^2u|,
\end{equation*}
where $A:=\nabla-\hat{\nabla}$ is the difference of the appropriate connections.
It is clearly sufficient to obtain pointwise bounds on $A$ and its derivative
$\hat{\nabla} A$. As mentioned in Remark \ref{rem:equivalence}, these
follow from Assumption A2. The same is true
for Sobolev spaces of sections of tensor bundles over $L$.
Now consider the Sobolev spaces of sections of $E$. Since we are not changing
the connection on $E$, Assumption A1 ensures equivalence of the $W^p_1$ norms.
The equivalence of the $W^p_k$ norms is proved as above.
\end{proof}
For $p>1$ we define $p'$ via
\begin{equation}\label{eq:p'}
\frac{1}{p}+\frac{1}{p'}=1, \ \ \mbox{\textit{i.e. }} p'=\frac{p}{p-1}.
\end{equation}
For $p\geq 1$ we define $p^*$ via
\begin{equation}\label{eq:p*}
\frac{1}{p^*}=\frac{1}{p}-\frac{1}{m},\ \ \mbox{\textit{i.e. }}
p^*=\frac{mp}{m-p}.
\end{equation}
It is simple to check that
\begin{equation}\label{eq:sobconstantsfundrelation}
\frac{1}{p^*}+\frac{1}{p'}=\frac{m-1}{m}.
\end{equation}
More generally, for $p\geq 1$ and $l=\{1,2,\dots\}$ we define $p^*_l$ via
\begin{equation}\label{eq:p*l}
\frac{1}{p^*_l}=\frac{1}{p}-\frac{l}{m},\ \ \mbox{\textit{i.e. }}
p^*_l=\frac{mp}{m-lp},
\end{equation}
so that $p^*=p^*_1$. Notice that $p^*_l$ is obtained by $l$ iterations of the
operation
\begin{equation*}
p\mapsto p^*
\end{equation*}
and that $\frac{1}{p^*_l}<\frac{1}{p^*_{l-1}}<\frac{1}{p}$, so if $p^*_l>0$
(equivalently, $lp<m$) then $p^*_l>p^*_{l-1}>p$. In other words, under
appropriate conditions $p^*_l$ increases with $l$.
The Sobolev Embedding Theorems come in two basic forms, depending on the product
$lp$.
The \textit{Sobolev Embedding Theorems, Part I} concern the existence of
continuous embeddings of the form
\begin{equation}\label{eq:std_partI}
W^p_{k+l}(E)\hookrightarrow W^{p^*_l}_k(E)\ \ (\mbox{for }lp<m),
\end{equation}
\textit{i.e.} the existence of some constant $C>0$ such that, $\forall \sigma\in
W^p_{k+l}(E)$,
\begin{equation}\label{eq:std_partIbis}
\|\sigma\|_{W^{p^*_l}_{k}(E)}\leq C\|\sigma\|_{W^p_{k+l}(E)}.
\end{equation}
We call $C$ the \textit{Sobolev constant}.
In words, bounds on the higher derivatives of $\sigma$ enhance the integrability
of $\sigma$. Otherwise said, one can sacrifice derivatives to improve
integrability; the more derivatives one sacrifices, the higher the value of
$p^*_l$.
The \textit{exceptional case} of Part I concerns the existence of continuous
embeddings of the form
\begin{equation}\label{eq:except_partI}
W^p_{k+l}(E)\hookrightarrow W^{q}_k(E)\ \ (\mbox{for }lp=m),\ \ \forall q\in
[p,\infty).
\end{equation}
The \textit{Sobolev Embedding Theorems, Part II} concern the existence of
continuous embeddings of the form
\begin{equation}\label{eq:std_partII}
W^p_{k+l}(E)\hookrightarrow C^k(E)\ \ (\mbox{for }lp>m).
\end{equation}
Roughly speaking, this means that one can sacrifice derivatives to improve
regularity.
\
The validity of these theorems for a given manifold $(L,g)$ depends on its
Riemannian properties. It is a useful fact that the properties of $(E,\nabla)$
play no extra role: more precisely, if an Embedding Theorem holds for functions
on $L$ it then holds for sections of any metric pair $(E,\nabla)$. This is a
consequence of the following result.
\begin{lemma}[Kato's inequality]\label{l:kato} Let $(E,\nabla)$ be a metric
pair. Let $\sigma$ be a smooth section of $E$. Then, away from the zero set of
$\sigma$,
\begin{equation}\label{eq:kato}
|d|\sigma||\leq |\nabla\sigma|.
\end{equation}
\end{lemma}
\begin{proof}
\begin{equation*}
2|\sigma||d|\sigma||=|d|\sigma|^2|=2(\nabla\sigma,\sigma)\leq
2|\nabla\sigma||\sigma|.
\end{equation*}
\end{proof}
The next result shows that if Part I holds in the simplest cases it then holds
in all cases. Likewise, the general case of Part II follows from combining the
simplest cases of Part II with the general case of Part I.
\begin{prop}\label{prop:complexfromsimple}
\
\begin{enumerate}
\item Assume Part I, Equation \ref{eq:std_partI}, holds for all $p<m$ with $l=1$
and $k=0$. Then Part I holds for all $p$ and $l$ satisfying $lp<m$ and for all
$k\geq 0$.
\item Assume Part I, Equation \ref{eq:std_partI}, holds in all cases and that
the exceptional case, Equation \ref{eq:except_partI}, holds for $l=1$ and $k=0$.
Then the exceptional case holds for all $p$ and $l$ satisfying $lp=m$ and for
all $k\geq 0$.
\item Assume Part I, Equation \ref{eq:std_partI}, and the exceptional case,
Equation \ref{eq:except_partI}, hold in all cases and that Part II, Equation
\ref{eq:std_partII}, holds for all $p>m$ with $l=1$ and $k=0$. Then Part II
holds for all $p$ and $l$ satisfying $lp>m$ and for all $k\geq 0$.
\end{enumerate}
\end{prop}
\begin{proof}
As discussed above, it is sufficient to prove that the result holds for
functions: as a result of Kato's inequality it will then hold for arbitrary
metric pairs $(E,\nabla)$.
(1) Assume $l=1$. Given $u\in W^p_{k+1}$, Kato's inequality shows that
$|u|,\dots\, |\nabla^k u|\in W^p_1$. Applying Part I to each of these then shows
that $W^p_{k+1}\hookrightarrow W^{p^*}_k$. The general case follows from the
composition of the embeddings
\begin{equation*}
W^p_{k+l}\hookrightarrow W^{p^*}_{k+l-1}\hookrightarrow
W^{p^*_2}_{k+l-2}\hookrightarrow\dots
\end{equation*}
(2) For $l=1$ we can prove $W^p_{k+1}\hookrightarrow W^q_k$ as in (1) above. Now
assume $lp=m$ for $l\geq 2$. Then Part I yields $W^p_l\hookrightarrow
W^{p^*_{l-1}}_1$. Since $p^*_{l-1}=m$ we can now apply the exceptional case in
its simplest form.
(3) Let us consider, for example, the case $l=2$ and $k=0$. We are then assuming
that $p>m/2$. Let us distinguish three subcases, as follows. Assume $p\in
(m/2,m)$. Then Part 1 implies that $W^p_2\hookrightarrow W^{p^*}_1$. Since
$p^*>m$ we can now use the embedding $W^{p^*}_1\hookrightarrow C^0$ to conclude.
Now assume $p=m$. Then $W^p_2\hookrightarrow W^q_1$ for any $q>m$ and we can
conclude as above. Finally, assume $p>m$. Then $W^p_2\hookrightarrow
W^p_1\hookrightarrow C^0$. The other cases are similar.
\end{proof}
\begin{corollary}
Assume the Sobolev Embedding Theorems hold for $(L,g)$. Let $\hat{g}$ be a
second Riemannian metric on $L$ such that, for some $C_0>0$, $(1/C_0)g\leq
\hat{g}\leq C_0 g$. Then the Sobolev Embedding Theorems hold also for
$(L,\hat{g})$.
\end{corollary}
\begin{proof}
According to Proposition \ref{prop:complexfromsimple} it is sufficient to
verify the Sobolev Embedding Theorems in the case $l=1$ and $k=0$. These
involve only $C^0$-information on the metric. The conclusion is thus
straight-forward.
\end{proof}
\begin{remark} \label{rem:complexfromsimple}
Under a certain density condition, Proposition \ref{prop:complexfromsimple} can
be enhanced as follows.
Assume Part I, Equation \ref{eq:std_partI}, holds for $p=1$, $l=1$ and $k=0$,
\textit{i.e.} $W^1_1\hookrightarrow L^\frac{m}{m-1}$. Assume also that, for all
$p<m$, the space $C^\infty_c(L)$ is dense in $W^p_1$. Then Part I holds for all
$p<m$ with $l=1$ and $k=0$, \textit{i.e.} $W^p_1\hookrightarrow L^{p^*}$. The
proof is as follows.
Choose $u\in C^\infty_c(L)$. One can check that, for all $s>1$, $|u|^s\in
W^p_1$, cf. \textit{e.g.} \cite{hebey}. Then, using Part I and H\"{o}lder's
inequality,
\begin{align*}
\||u|^s\|_{L^{\frac{m}{m-1}}} &\leq C \int_L (|u|^s+|\nabla
|u|^s|)\,\mbox{vol}_g\\
&\leq C \int_L (|u|^{s-1}|u|+|u|^{s-1}|\nabla u|)\,\mbox{vol}_g\\
&\leq C \,\| |u|^{s-1}\|_{L^{p'}} \left(\|u\|_{L^p}+\|\nabla u\|_{L^p}\right).
\end{align*}
Let us now choose $s$ so that $(s-1)p'=sm/(m-1)$, \textit{i.e.} $s=p^*(m-1)/m$.
Substituting, we find
\begin{equation*}
\left(\int_L |u|^{p^*}\right)^{\frac{m-1}{m}}\leq C \left(\int_L
|u|^{p^*}\right)^{\frac{1}{p'}}\|u\|_{W^1_p}.
\end{equation*}
This leads to $\|u\|_{L^{p^*}}\leq C \|u\|_{W^1_p}$, for all $u\in
C^\infty_c(L)$. By density, the same is true for all $u\in W^1_p$.
To conclude, we mention that if $(L,g)$ is complete then $C^\infty_c(L)$ is
known to be dense in $W^p_1$ for all $p\geq 1$, cf. \cite{hebey} Theorem 3.1.
\end{remark}
The most basic setting in which all parts of the Sobolev Embedding Theorems hold
is when $L$ is a smooth bounded domain in $\mathbb{R}^m$ endowed with the standard
metric $\tilde{g}$. Another important class of examples is the following.
\begin{theorem}\label{thm:std_ok}
Assume $(L,g)$ satisfies the following assumptions: there exists $R_1>0$ and
$R_2\in\mathbb{R}$ such that
\begin{equation*}
i(g)\geq R_1,\ \ Ric(g)\geq R_2\,g.
\end{equation*}
Then:
\begin{enumerate}
\item The Sobolev embeddings Part I, Equation \ref{eq:std_partI}, hold for all
$p$ and $l$ satisfying $lp<m$ and for all $k\geq 0$.
\item The exceptional case of Part I, Equation \ref{eq:except_partI}, holds for
all $p$ and $l$ satisfying $lp=m$ and for all $k\geq 0$.
\item The Sobolev embeddings Part II, Equation \ref{eq:std_partII}, hold for all
$p$ and $l$ satisfying $lp>m$ and for all $k\geq 0$.
\end{enumerate}
Furthermore, when $lp>m$ and $k\geq 0$, $W^p_{k+l}$ is a Banach algebra.
Specifically, there exists $C>0$ such that, for all $u,v\in W^p_{k+l}$, the
product $uv$ belongs to $W^p_{k+l}$ and satisfies
\begin{equation*}
\|uv\|_{W^p_{k+l}}\leq C \|u\|_{W^p_{k+l}}\|v\|_{W^p_{k+l}}.
\end{equation*}
\end{theorem}
We will prove Theorem \ref{thm:std_ok} below. Roughly speaking, the reason it
holds is the following. Given any coordinate system on $L$, the embeddings hold
on every chart endowed with the flat metric $\tilde{g}$. Now recall that, given
any $(L,g)$ and any $x\in L$, it is always possible to find coordinates
$\phi_x:B\subset\mathbb{R}^m\rightarrow L$ in which the metric $g$ is a small
perturbation of the flat metric: this implies that the embeddings hold locally
also with respect to $g$. The problem is that, in general, the size of the ball
$B$, thus the corresponding Sobolev constants, will depend on $x$. Our
assumptions on $L$, however, can be used to build a special coordinate system
whose charts admit uniform bounds. One can then show that this implies that the
embeddings hold globally. The main technical step in the proof of Theorem
\ref{thm:std_ok} is thus the following result concerning the
existence and properties of \textit{harmonic coordinate systems}.
\begin{theorem}\label{thm:harmonic_coords}
Assume $(L,g)$ satisfies the assumptions of Theorem \ref{thm:std_ok}.
Then for all small $\epsilon>0$ there exists $r>0$ such that, for each $x\in L$,
there exist coordinates $\phi_x:B_r\subset\mathbb{R}^m\rightarrow L$ satisfying
\begin{enumerate}
\item $\phi_x^{-1}$ (seen as a map into $\mathbb{R}^m$) is harmonic.
\item $\|\phi_x^*g-\tilde{g}\|_{C^0}\leq\epsilon$.
\end{enumerate}
\end{theorem}
\begin{remark} \label{rem:harmonic_coords}
Theorem \ref{thm:harmonic_coords} can be heavily improved, cf. \cite{hebey}
Theorem 1.2. Firstly, it is actually a local result, \textit{i.e.} one can get
similar results for any open subset of $L$ by imposing similar assumptions on a
slightly larger subset. Secondly, these same assumptions actually yield certain
$C^{0,\alpha}$ bounds. Thirdly, assumptions on the higher derivatives of the
Ricci tensor yield certain bounds on the higher derivatives of
$\phi_x^*g-\tilde{g}$, see Remark \ref{rem:harmonic_coordsbis} for details.
To conclude, it may be useful to emphasize that imposing a global lower bound on
the injectivity radius of $(L,g)$ implies completeness.
\end{remark}
\begin{proof}[Proof of Theorem \ref{thm:std_ok}]
As seen in Proposition \ref{prop:complexfromsimple}, it is sufficient to prove
the Sobolev Embedding Theorems in the simplest cases. Concerning Part I, let us
choose $u\in W^{1,p}(L)$. Using the coordinates of Theorem
\ref{thm:harmonic_coords}, $\phi_x^*u\in W^{1,p}(B_r)$. All Sobolev Embedding
Theorems hold on $B_r$ with its standard metric $\tilde{g}$. Thus there exists a
constant $C$ such that, with respect to $\tilde{g}$,
\begin{equation}\label{eq:sob_local}
\|\phi_x^*u\|_{L^{p^*}(B_r)}\leq C \|\phi_x^*u\|_{W^{1,p}(B_r)}.
\end{equation}
The fact that $\nabla u=du$ implies that Equation \ref{eq:sob_local} involves
only $C^0$ information on the metric. Since $\phi_x^*g$ is $C^0$-close to
$\tilde{g}$, up to a small change of the constant $C$ the same inequality holds
with respect to $\phi_x^*g$. Let $B_x(r)$ denote the ball in $(L,g)$ with center
$x$ and radius $r$. Then $B_x(r/2)\subset\phi_x(B_r)\subset B_x(2r)$ so
\begin{align*}
\int_{B_x(r/2)}|u|^{p^*} \mbox{vol}_g&\leq
\int_{\phi_x(B_r)}|u|^{p^*}\mbox{vol}_g\\
&\leq
C\left(\int_{\phi_x(B_r)}(|u|^p+|du|^p)\,\mbox{vol}_g\right)^{\frac{p^*-p+p}{p}}
\\
&\leq C\left(\int_L
(|u|^p+|du|^p)\,\mbox{vol}_g\right)^{\frac{p^*-p}{p}}\left(\int_{B_x(2r)}
(|u|^p+|du|^p)\,\mbox{vol}_g\right).
\end{align*}
Let us now integrate both sides of the above equation with respect to $x\in L$.
We can then change the order of integration according to the formula
\begin{equation*}
\int_{x\in L}\left(\int_{y\in B_x(r)}
f(y)\,\mbox{vol}_g\right)\,\mbox{vol}_g=\int_{y\in L}f(y)\left(\int_{x\in
B_y(r)} \mbox{vol}_g\right)\,\mbox{vol}_g.
\end{equation*}
Reducing $r$ if necessary, the $C^0$ estimate on $g$ yields uniform
bounds (with respect to $x$) on $\mbox{vol}_g(B_x(r/2))$ and
$\mbox{vol}_g(B_x(2r))$ because analogous bounds hold for $\tilde{g}$. This
allows us to substitute the inner integrals with appropriate constants. We
conclude that
\begin{align*}
\int_L |u|^{p^*}\,\mbox{vol}_g &\leq C\left(\int_L
(|u|^p+|du|^p)\,\mbox{vol}_g\right)^{\frac{p^*-p}{p}}\left(\int_L
(|u|^p+|du|^p)\,\mbox{vol}_g\right)\\
&= C\left(\int_L (|u|^p+|du|^p)\,\mbox{vol}_g\right)^{\frac{p^*}{p}}.
\end{align*}
We conclude by raising both sides of the above equation to the power $1/{p^*}$.
Notice that the final constant $C$ can be estimated in terms of the volume of
balls in $L$ and of the constant $C$ appearing in Equation \ref{eq:sob_local}.
The exceptional case of Part I is similar: it is sufficient to replace $p^*$
with any $q>m$. Part II is also similar, though slightly simpler. Specifically,
one finds as above that
\begin{equation*}
\|u\|_{C^0(\phi_x(B_r))} \leq C \|u\|_{W^p_1(\phi_x(B_r))}\leq C
\|u\|_{W^p_1(L)}
\end{equation*}
Since this holds for all $x\in L$, we conclude that $\|u\|_{C^0(L)}\leq C
\|u\|_{W^p_1(L)}$.
The proof that $W^p_{k+l}$ is a Banach algebra is analogous to that given
in \cite{adams}, Theorem 5.23, for domains in $\mathbb{R}^m$. The idea is to use
the Leibniz rule for $\nabla$ to write
\begin{equation*}
\nabla^j(uv)=\sum_{k=0}^j\binom{j}{k}(\nabla^ku)\otimes(\nabla^{j-k}v),
\end{equation*}
then use the Sobolev Embedding Theorems and H\"older's inequality to
conclude.
\end{proof}
\begin{example}\label{e:std_ok}
Any compact oriented Riemannian manifold $(L,g)$ satisfies the assumptions of
Theorem \ref{thm:std_ok}. Thus the Sobolev Embedding Theorems hold in full
generality for such manifolds. The same is true for the non-compact manifold
$\mathbb{R}^m$, endowed with the standard metric $\tilde{g}$.
Let $(\Sigma,g')$ be a compact oriented Riemannian manifold. Consider
$L:=\Sigma\times\mathbb{R}$ endowed with the metric $\tilde{h}:=dz^2+g'$. It is clear
that $(L,\tilde{h})$ satisfies the assumptions of Theorem \ref{thm:std_ok} so
again the Sobolev Embedding Theorems hold in full generality for these
manifolds. More generally they hold for the asymptotically cylindrical manifolds
of Section \ref{s:manifolds}. Notice however that here we are using
the Sobolev spaces defined in Equation \ref{eq:std_sob}. In Section
\ref{s:manifolds} we will verify the Sobolev Embedding Theorems for a different
class of Sobolev spaces, cf. Definition \ref{def:acyl_sectionspaces}.
\end{example}
\section{Scaled Sobolev spaces}\label{s:scaled}
In applications standard Sobolev spaces are often not satisfactory for various
reasons. Firstly, they do not have good properties with respect to rescalings of
the sort $(L,t^2g)$. Secondly, uniform geometric
bounds of the sort seen in Theorem \ref{thm:std_ok} are too strong. Thirdly, the
finiteness condition in Equation
\ref{eq:std_sob} is very rigid and restrictive.
For all the above reasons it is often useful to modify the Sobolev norms. A
simple way of addressing the first two problems is to introduce an extra
piece of data, as follows.
Let $(L,g,\rho)$ be an oriented Riemannian manifold endowed with a \textit{scale
factor} $\rho>0$ or a \textit{scale function} $\rho=\rho(x)>0$. Given any metric
pair $(E,\nabla)$, the \textit{scaled Sobolev spaces} are defined by
\begin{equation}\label{eq:inv_sob}
{W^p_{k;sc}(E)}:=\mbox{Banach space completion of the space }
\left\{\sigma\in
C^\infty(E):\|\sigma\|_{W^p_{k;sc}}<\infty\right\},
\end{equation}
where we use the norm $\|\sigma\|_{W^p_{k;sc}}:=\left(\Sigma_{j=0}^k \int_L
|\rho^j\nabla^j\sigma|_g^p\rho^{-m}\,\mbox{vol}_g\right)^{1/p}$.
Notice that at the scale $\rho\equiv 1$ these norms coincide with the standard
norms.
\begin{remark}\label{rem:scalednorms}
Let us slightly change notation, using $g_L$ (respectively, $g_E$) to denote the
metric on $L$ (respectively, on $E$).
The metric $g$ used in the above norms to measure $\nabla^j\sigma$ is obtained
by tensoring $g_L$ (applied to $\nabla^j$) with $g_E$ (applied to $\sigma$): let
us write $g=g_L\otimes g_E$. We then find
\begin{equation*}
|\rho^j\nabla^j\sigma|_{g_L\otimes g_E}\rho^{-m}\,\mbox{vol}_{g_L\otimes
g_E}=|\nabla^j\sigma|_{(\rho^{-2}g_L)\otimes
g_E}\mbox{vol}_{(\rho^{-2}g_L)\otimes g_E}.
\end{equation*}
Roughly speaking, the scaled norms thus coincide with the standard norms
obtained via the conformally equivalent metric $\rho^{-2}g_L$ on $L$. It is
important to emphasize, however, that we are conformally rescaling only part of
the metric.
This can be confusing when $E$ is a tensor bundle over $L$, endowed with the
induced metric: it would then be natural to also rescale the metric of $E$.
We are also not changing the connections $\nabla$. In general these
connections are not metric connections with respect to $(\rho^{-2}g_L)\otimes
g_E$. This has important consequences regarding the Sobolev Embedding Theorems
for scaled Sobolev spaces, as follows.
Naively, one might hope that such theorems hold under the assumptions:
\begin{equation*}
i(\rho^{-2}g)\geq R_1,\ \ Ric(\rho^{-2}g)\geq R_2 \rho^{-2}g.
\end{equation*}
Indeed, these assumptions do suffice to prove the Sobolev Embedding Theorems in
the simplest case, \textit{i.e.} $l=1$ and $k=0$. However, the general case
requires Kato's inequality, Lemma \ref{l:kato}, which in turn requires metric
connections. To prove these theorems we will thus need further assumptions on
$\rho$, cf. Theorem \ref{thm:scaled_ok}.
\end{remark}
We now define \textit{rescaling} to be an action of $\mathbb{R}^+$ on the triple
$(L,g,\rho)$, via $t\cdot(L,g,\rho):=(L,t^2g, t\rho)$. Recall that the
Levi-Civita connection $\nabla$ on $L$ does not change under rescaling. Using
this fact it is simple to check that $\|\sigma\|_{W^p_{k;sc}}$, calculated with
respect to $t\cdot(L,g,\rho)$, coincides with $\|\sigma\|_{W^p_{k;sc}}$,
calculated with respect to $(L,g,\rho)$: in this sense the scaled norm is
invariant under rescaling.
\begin{remark}\label{rem:scalednormsbis}
As in Remark \ref{rem:scalednorms}, our definition of rescaling requires some
care. To explain this let us adopt the same notation as in Remark
\ref{rem:scalednorms}. Our notion of rescaling affects only the metric on $L$,
not the metric on $E$. As before, this can be confusing when $E$ is a tensor
bundle over $L$, endowed with the induced metric.
\end{remark}
As in Section \ref{s:std}, it is important to find conditions under which
$(L,g,\rho)$ and $(L,\hat{g},\rho)$ define equivalent norms.
\begin{definition}\label{def:equivalentscaledmetrics}
Let $(L,\rho)$ be a manifold endowed with a scale function. We say that two
Riemannian metrics $g$, $\hat{g}$ are \textit{scaled-equivalent} if they
satisfy the following assumptions:
\begin{description}
\item[A1] There exists $C_0>0$ such that
\begin{equation*}
(1/C_0)g\leq \hat{g}\leq C_0 g.
\end{equation*}
\item[A2] For all $j\geq 1$ there exists $C_j>0$
such that
\begin{equation*}
|\nabla^j\hat{g}|_{\rho^{-2}g\otimes g_E}\leq C_j,
\end{equation*}
where $\nabla$ is the Levi-Civita connection defined by $g$, $E=T^*L\otimes
T^*L$ and we are using the notation introduced in Remark \ref{rem:scalednorms}.
\end{description}
\end{definition}
\begin{remark}\label{rem:scaledequivalence}
As in Remark \ref{rem:equivalence}, one can check that
\begin{equation*}
|\nabla\hat{g}|_{\rho^{-2}g\otimes g_E}\leq C_1\Rightarrow
|A(\hat{g})|_{\rho^{-2}g\otimes g_E}\leq C_1.
\end{equation*}
In turn this implies that
$|A|_{\rho^{-2}g\otimes g_E}\leq C_1$, where now $A$ denotes the difference
$\nabla-\hat{\nabla}$ of the connections on $TL$ and $E=T^*L\otimes TL$.
Again as in Remark \ref{rem:equivalence}, one can check that if for all
$j\geq 0$ there exists $C_j>0$ such that
$$|\nabla^j(\hat{g}-g)|_{\rho^{-2}g\otimes g_E}\leq C_j$$
and if $C_0$ is sufficiently small then $g$, $\hat{g}$ satisfy Assumptions
A1, A2.
\end{remark}
The following
result is a simple consequence of Remark \ref{rem:scalednorms} and Lemma
\ref{l:equivalentstdnorms}.
\begin{lemma}\label{l:equivalentscalednorms}
Assume $(L,g,\rho)$, $(L,\hat{g},\rho)$ are scaled-equivalent in the sense of
Definition \ref{def:equivalentscaledmetrics}.
Then the scaled Sobolev norms are equivalent.
\end{lemma}
We can also define the \textit{scaled spaces of $C^k$ sections}
\begin{equation}\label{eq:scaled_C^k}
C^k_{sc}(E):=\left\{\sigma\in C^k(E): \|\sigma\|_{C^k_{sc}}<\infty\right\},
\end{equation}
where we use the norm $\|\sigma\|_{C^k_{sc}}:=\sum_{j=0}^k \sup_{x\in
L}|\rho^j\nabla^j\sigma|_g$. Once again, these norms define Banach spaces.
\begin{remark} \label{rem:harmonic_coordsbis}
One can analogously define $C^{k,\alpha}_{sc}$ spaces. Notice that Equation
\ref{eq:scaled_C^k} implies that $C^0_{sc}=C^0$. It is these spaces which are
relevant to the generalization to higher derivatives of Theorem
\ref{thm:harmonic_coords}. Specifically, bounds on the higher derivatives of
$Ric(g)$ yield $C^{k,\alpha}_{sc}$ bounds on $\phi_x^*g-\tilde{g}$ with respect
to the (constant) scale factor $r$ determined by the theorem.
\end{remark}
We are now ready to study the Sobolev Embedding Theorems for scaled spaces. As
mentioned in Remark \ref{rem:scalednorms}, these theorems require further
assumptions on $\rho$.
\begin{theorem}\label{thm:scaled_ok}
Let $(L,g)$ be a Riemannian manifold and $\rho$ a positive function on
$L$. Assume there exist constants $R_1>0$, $R_2\in\mathbb{R}$, $R_3>1$ and $\zeta>0$
such that:
\begin{description}
\item[A1] $\forall x\in L,\ \ i_x(g)\geq R_1\rho(x)$.
\item[A2] $\forall x\in L,\ \ Ric_x(g)\geq R_2 \rho(x)^{-2}g_x$.
\item[A3] $\forall x\in L, \forall y\in B(x,\zeta\rho(x))$,
\begin{equation*}
(1/R_3)\rho(x)\leq\rho(y)\leq R_3\rho(x).
\end{equation*}
\end{description}
Then all parts of the Sobolev Embedding Theorems hold for scaled norms and for
any metric pair $(E,\nabla)$. Furthermore, when $lp>m$ and $k\geq 0$,
$W^p_{k+l;sc}$ is a Banach algebra.
Now let $\hat{g}$ be a second Riemannian metric on $L$ such that, for some
$C_0>0$, $(1/C_0)g\leq \hat{g}\leq C_0 g$. Then the scaled Sobolev Embedding
Theorems hold also for $(L,\hat{g},\rho)$ and for any metric pair $(E,\nabla)$.
The Sobolev constants of $\hat{g}$ depend only on the Sobolev constants of $g$
and on $C_0$.
\end{theorem}
\begin{proof} Let us prove Part 1 for functions, assuming $l=1$, $k=0$. Choose
$x\in L$. Set $B_x:=B(x,\zeta\rho(x))$. For $y\in B_x$, consider the rescaled
metric $h$ defined by $h_y:=\rho(x)^{-2}g_y$. Assumption A1 shows that
$i_y(g)\geq R_1\rho(y)$. Using Assumption A3 we find
\begin{equation*}
i_y(h)=\rho(x)^{-1}i_y(g)\geq R_1\rho(y)\rho(x)^{-1}\geq R_1/R_3.
\end{equation*}
Now recall that the Ricci curvature $Ric$ is invariant under rescaling,
\textit{i.e.} $Ric(h)=Ric(g)$. Then Assumptions A2 and A3 show that
\begin{equation*}
Ric_y(h)=Ric_y(g)\geq R_2\rho(y)^{-2}\rho(x)^2h\geq (R_2/R_3^2)h.
\end{equation*}
We have thus obtained lower bounds on the injectivity radius and Ricci curvature
of $(B_x,h)$. Notice that these bounds are independent of $x$. Recall from
Remark \ref{rem:harmonic_coords} that Theorem \ref{thm:harmonic_coords} is
essentially local. Specifically, set $B'_x:=B(x, (1/2)\zeta\rho(x))$. Then for
any $\epsilon>0$ there exists $r=r(p,R_1,R_2,R_3,\epsilon,m)$ such that, for any
$x\in L$, there exist coordinates $\phi_x:B_r\rightarrow (B'_x,h)$ satisfying
$\|\phi_x^*h-\tilde{g}\|_{C^0}\leq \epsilon$.
Exactly as in the proof of Theorem \ref{thm:std_ok}, we can now use the local
Sobolev Embedding Theorems for $B_r$ to conclude that
\begin{equation}\label{eq:scaled_okzero}
\left(\int_{B'_x}|u|^{p^*}\mbox{vol}_h\right)^{1/p^*}\leq C
\left(\int_{B'_x}(|u|^p+|du|_h^p)\,\mbox{vol}_h\right)^{1/p}.
\end{equation}
Assumption A3 allows us, up to a change of constants, to replace the (locally)
constant quantity $\rho(x)$ with the function $\rho(y)$. Remark
\ref{rem:scalednorms} shows how replacing $\rho^{-2}g$ with $g$ leads to the
scaled norms. Proceeding as in the proof of Theorem \ref{thm:std_ok}, via double
integration, we then get
\begin{equation}\label{eq:scaled_ok}
\|u\|_{L^{p^*}_{sc}}\leq C\|u\|_{W^{p}_{1;sc}},
\end{equation}
where we are now using the metric $g$.
Now consider the case $k=1$, \textit{i.e.} assume $u\in W^p_{2;sc}$. Then
$\phi_x^*|\nabla u|_h\in W^p_1(B_r)$. As before, we obtain
\begin{equation}\label{eq:scaled_okbis}
\left(\int_{B'_x}|\nabla u|_h^{p^*}\mbox{vol}_h\right)^{1/p^*}\leq C
\left(\int_{B'_x}(|\nabla u|_h^p+|d(|\nabla
u|_h)|_h^p)\,\mbox{vol}_h\right)^{1/p}.
\end{equation}
Notice that the Levi-Civita connections of $g$ and $h$ coincide. We can thus
apply Kato's inequality, finding $|d|\nabla
u|_h|_h\leq|\nabla^2u|_h=|\rho(x)^2\nabla^2 u|_g$. This leads to
\begin{equation}\label{eq:scaled_okter}
\left(\int_{B'_x}|\rho(x)\nabla
u|_g^{p^*}\rho(x)^{-m}\mbox{vol}_g\right)^{1/p^*}\leq C
\left(\int_{B'_x}(|\rho(x)\nabla u|_g^p+|\rho(x)^2\nabla^2
u|_g^p)\rho(x)^{-m}\mbox{vol}_g\right)^{1/p}.
\end{equation}
We can now proceed as before, using Assumption A3, to obtain
\begin{equation*}
\|\nabla u\|_{L^{p^*}_{sc}}\leq C\|\nabla u\|_{W^{p}_{1;sc}}.
\end{equation*}
Together with Equation \ref{eq:scaled_ok}, this implies
$W^p_{2;sc}\hookrightarrow W^{p^*}_{1;sc}$.
The other cases and parts of the Sobolev Embedding Theorems can be proved
analogously.
The claim that $W^p_{k+l;sc}$ is a Banach algebra can be proved as in
Theorem \ref{thm:std_ok}, using Remark
\ref{rem:scalednorms} to write the scaled norms in terms of standard norms. In
this case the fact that the connection $\nabla$ is not a metric connection with
respect to the rescaled metric $\rho^{-2}g$ is not a problem: the proof only
uses the Leibniz rule (together with H\"older's inequality for $L^p$ norms and
the Sobolev Embedding Theorems which we have just proved).
The proof of the Sobolev Embedding Theorems for $(L,\hat{g},\rho)$ is similar.
For example, to prove Part I with $l=1$ and $k=0$ we locally define
$\hat{h}_y:=\rho^{-2}(x)\hat{g}_y$. Our assumption on $\hat{g}$ allows us to
substitute
$h$ with $\hat{h}$ in Equation \ref{eq:scaled_okzero}. The proof then continues
as before. Now consider the case $k=1$, \textit{i.e.} assume $u\in W^p_{2;sc}$
with respect to $\hat{g}$. Let $\hat{\nabla}$ denote the Levi-Civita connection
defined by $\hat{g}$. We can then study $\phi_x^*|\hat{\nabla}u|_{\hat{h}}$ as
before, obtaining the analogue of Equation \ref{eq:scaled_okbis} in terms of
$(\hat{h},\hat{\nabla})$ instead of $(h,\nabla)$. Since the Levi-Civita
connections
of $\hat{g}$ and $\hat{h}$ coincide we also obtain the analogue of Equation
\ref{eq:scaled_okter}. The proof then continues as before.
\end{proof}
\begin{remark}
Compare the proof of Theorem \ref{thm:scaled_ok} with the ideas of Remark
\ref{rem:scalednorms}. The main issue raised in Remark \ref{rem:scalednorms}
concerned Kato's inequality for the rescaled metric $\rho^{-2}g$. In the proof
of the theorem this problem is solved by Assumption A3, which essentially allows
us to locally treat $\rho$ as a constant. Assumptions A1 and A2 are then similar
to the assumptions of Remark \ref{rem:scalednorms}.
\end{remark}
\begin{example}\label{e:radius}
We now want to present two important examples of $(L,g,\rho)$ satisfying
Assumptions A1-A3 of Theorem \ref{thm:scaled_ok}.
\begin{enumerate}
\item Let $L$ be a smooth bounded domain in $\mathbb{R}^m$, endowed with the standard
metric
$\tilde{g}$. Given any $x\in L$ we
can define $\rho(x):=d(x,\partial L)$. This function satisfies Assumption A1
with $R_1=1$ and Assumption A2 with $R_2=0$. The triangle inequality shows that,
for all $y\in B(x,(1/2)\rho(x))$, $(1/2)\rho(x)\leq\rho(y)\leq(3/2)\rho(x)$.
This implies that Assumption A3 is also satisfied.
\item Given a compact oriented Riemannian manifold $(\Sigma,g')$, let
$L:=\Sigma\times (0,\infty)$ and $\tilde{g}:=dr^2+r^2g'$. Let $\theta$ denote
the generic point on $\Sigma$. There is a natural action
\begin{equation*}
\mathbb{R}^+\times L\rightarrow L,\ \ t\cdot(\theta,r):=(\theta,tr).
\end{equation*}
Given any $t\in\mathbb{R}^+$, it is simple to check that $t^*\tilde{g}=t^2\tilde{g}$.
For any $x\in L$, notice that $i_{tx}(\tilde{g})=i_x(t^*\tilde{g})$. We conclude
that $i_{tx}(\tilde{g})=ti_x(\tilde{g})$. Analogously,
$Ric_{tx}(\tilde{g})=Ric_x(\tilde{g})$. It follows that, given any strictly
positive $f=f(\theta)$, the function $\rho(\theta,r):=rf(\theta)$
satisfies A1 and A2. It is simple to check that it also satisfies Assumption A3.
The simplest example is $f(\theta)\equiv 1$, \textit{i.e.}
$\rho(\theta,r)=r$. In Section \ref{s:manifolds} we will extend this example to the
category of ``conifolds''.
\end{enumerate}
\end{example}
\begin{remark}\label{rem:tscaled_ok}
Since the norms $\|\cdot\|_{W^p_{k;sc}}$ are scale-invariant it is clear that
if the Sobolev Embedding Theorems hold for $(L,g,\rho)$ then they also hold
for $(L,t^2g,t\rho)$ with the same Sobolev constants. This is reflected in the
fact that Assumptions A1-A3 of Theorem \ref{thm:scaled_ok} are
scale-invariant.
\end{remark}
\section{Weighted Sobolev spaces}\label{s:weighted}
In Section \ref{s:scaled} we mentioned that the finiteness condition
determined by the standard Sobolev norms is very
restrictive. This problem can be addressed by introducing a \textit{weight
function} $w=w(x)>0$ into the integrand. Coupling weights with scale functions
then produces very general and useful spaces, as follows.
Let $(L,g)$ be a Riemannian manifold endowed with two positive
functions $\rho$ and $w$. Given any metric pair $(E,\nabla)$, the
\textit{weighted Sobolev spaces} are defined by
\begin{equation}\label{eq:weighted_sob}
W^p_{k;w}(E):=\mbox{Banach space completion of the space }
\left\{\sigma\in
C^\infty(E):\|\sigma\|_{W^p_{k;w}}<\infty\right\},
\end{equation}
where we use the norm
$\|\sigma\|_{W^p_{k;w}}:=\left(\Sigma_{j=0}
^k\int_L|w\rho^j\nabla^j\sigma|_g^p\rho^ {
-m }
\,\mbox{vol}_g\right)^{1/p}$.
We can also define the \textit{weighted spaces of $C^k$ sections}
\begin{equation}\label{eq:weighted_C^k}
C^k_w(E):=\left\{\sigma\in C^k(E): \|\sigma\|_{C^k_w}<\infty\right\},
\end{equation}
where we use the norm $\|\sigma\|_{C^k_w}:=\sum_{j=0}^k \mbox{sup}_{x\in
L}|w\rho^j\nabla^j\sigma|_g$. Once again, these norms define Banach spaces.
\begin{theorem}\label{thm:weighted_ok}
Let $(L,g)$ be a Riemannian manifold endowed with positive functions
$\rho$ and $w$. Assume $\rho$ satisfies the assumptions of Theorem
\ref{thm:scaled_ok} with respect to constants $R_1$, $R_2$, $R_3$ and $\zeta$.
Assume also that there exists a positive constant $R_4$ such that, $\forall x\in
L, \forall y\in B(x,\zeta\rho(x))$,
\begin{equation*}
(1/R_4)w(x)\leq w(y)\leq R_4 w(x).
\end{equation*}
Then all parts of the Sobolev Embedding Theorems hold for the weighted norms
defined by $(\rho,w)$ and for any metric pair $(E,\nabla)$.
Now let $\hat{g}$ be a second Riemannian metric on $L$ such that, for some
$C_0>0$, $(1/C_0)g\leq \hat{g}\leq C_0 g$. Then the weighted Sobolev Embedding
Theorems hold also for $(L,\hat{g},\rho,w)$ and for any metric pair
$(E,\nabla)$. The Sobolev constants of $\hat{g}$ depend only on the Sobolev
constants of $g$
and on $C_0$.
\end{theorem}
\begin{proof}
The proof is a small modification of the proof of Theorem \ref{thm:scaled_ok}:
one needs simply to take into account the weights by multiplying Equations
\ref{eq:scaled_okzero} and \ref{eq:scaled_okter} by $w(x)$. The assumption on
$w$ allows us, up to a change of constants, to replace the (locally) constant
quantity $w(x)$ with the function $w(y)$.
\end{proof}
\begin{remark}\label{rem:tweighted_ok}
Choose any constant $\beta\in\mathbb{R}$. Define \textit{rescaling} to be an
action of $\mathbb{R}^+$ on
$(L,g,\rho,w)$, via $t\cdot(L,g,\rho,w):=(L,t^2g, t\rho,t^\beta w)$. Then
$\|\sigma\|_{W^p_{k;w}}$, calculated with
respect to $t\cdot(L,g,\rho,w)$, coincides with
$t^\beta\|\sigma\|_{W^p_{k;w}}$,
calculated with respect to $(L,g,\rho,w)$: this shows that these weighted norms
are in general not invariant under rescaling. However, if the Sobolev Embedding
Theorems hold for $(L,g,\rho,w)$ then, multiplying by the factor $t^\beta$, we
see that they hold for $(L,t^2g, t\rho,t^\beta w)$ with the same Sobolev
constant. This is reflected in the fact that the hypotheses of Theorem
\ref{thm:weighted_ok} are $t$-invariant.
\end{remark}
\section{Manifolds with ends modelled on cones and cylinders}\label{s:manifolds}
We now introduce the category of ``conifolds''. These Riemannian manifolds are a well-known example for the theory of weighted Sobolev spaces. They will also provide a useful framework for our study of desingularizations. It will also be useful to define the
analogous ``cylindrical'' category, both for its affinities to conifolds and as
a tool for studying them.
\begin{definition}\label{def:manifold_ends}
Let $L^m$ be a smooth manifold. We say $L$ is a \textit{manifold with ends} if
it satisfies the following conditions:
\begin{enumerate}
\item We are given a compact subset $K\subset L$ such that $S:=L\setminus K$ has
a finite number of connected components $S_1,\dots,S_e$, \textit{i.e.}
$S=\amalg_{i=1}^e S_i$.
\item For each $S_i$ we are given a connected ($m-1$)-dimensional compact
manifold $\Sigma_i$ without boundary.
\item There exist diffeomorphisms $\phi_i:\Sigma_i\times [1,\infty)\rightarrow
\overline{S_i}$.
\end{enumerate}
We then call the components $S_i$ the \textit{ends} of $L$ and the manifolds
$\Sigma_i$ the \textit{links} of $L$. We denote by
$\Sigma$ the union of the links of $L$.
\end{definition}
\begin{definition}\label{def:metrics_ends}
Let L be a manifold with ends. Let $g$ be a Riemannian metric on $L$. Choose an
end $S_i$ with corresponding link $\Sigma_i$.
We say that $S_i$ is a \textit{conically singular} (CS) end if the following
conditions hold:
\begin{enumerate}
\item $\Sigma_i$ is endowed with a Riemannian metric $g_i'$.
We then let $(\theta,r)$ denote the generic point on the product manifold
$C_i:=\Sigma_i\times (0,\infty)$ and ${\tilde{g}}_i:=dr^2+r^2g_i'$ denote the
corresponding \textit{conical metric} on $C_i$.
\item There exist a constant $\nu_i>0$ and a diffeomorphism
$\phi_i:\Sigma_i\times (0,\epsilon]\rightarrow \overline{S_i}$ such that, as
$r\rightarrow
0$ and for all $k\geq 0$,
$$|{\widetilde{\nabla}}^k(\phi_i^*g-{\tilde{g}}_i)|_{{\tilde{g}}_i}=O(r^{\nu_i-k}),$$
where ${\widetilde{\nabla}}$ is the Levi-Civita connection on $C_i$ defined by ${\tilde{g}}_i$.
\end{enumerate}
We say that $S_i$ is an \textit{asymptotically conical} (AC) end if the
following conditions hold:
\begin{enumerate}
\item $\Sigma_i$ is endowed with a Riemannian metric $g_i'$.
We again let $(\theta,r)$ denote the generic point on the product manifold
$C_i:=\Sigma_i\times (0,\infty)$ and ${\tilde{g}}_i:=dr^2+r^2g_i'$ denote the
corresponding \textit{conical metric} on $C_i$.
\item There exist a constant $\nu_i<0$ and a diffeomorphism
$\phi_i:\Sigma_i\times [R,\infty)\rightarrow \overline{S_i}$ such that, as
$r\rightarrow
\infty$ and for all $k\geq 0$,
$$|{\widetilde{\nabla}}^k(\phi_i^*g-{\tilde{g}}_i)|_{{\tilde{g}}_i}=O(r^{\nu_i-k}),$$
where ${\widetilde{\nabla}}$ is the Levi-Civita connection on $C_i$ defined by ${\tilde{g}}_i$.
\end{enumerate}
In either of the above situations we call $\nu_i$ the \textit{convergence rate}
of
$S_i$.
\end{definition}
\begin{remark}\label{rem:metrics_ends}
Let $(L,g)$ be a manifold with ends. Assume $S_i$ is an
AC end as in Definition \ref{def:metrics_ends}. Using the
notation of Remark \ref{rem:scalednorms} we can rewrite this condition as
follows: for all $k\geq 0$,
$$|{\widetilde{\nabla}}^k(\phi_i^*g-{\tilde{g}}_i)|_{r^{-2}{\tilde{g}}_i\otimes {\tilde{g}}_i}=O(r^{\nu_i}).$$
In particular there exist constants $C_k>0$ such that
$$|{\widetilde{\nabla}}^k(\phi_i^*g-{\tilde{g}}_i)|_{r^{-2}{\tilde{g}}_i\otimes {\tilde{g}}_i}\leq C_kR^{\nu_i}.$$
By making $R$ larger if necessary, we can assume $C_0R^{\nu_i}$ is small.
This implies that $\phi_i^*g$ and $\tilde{g}_i$
are scaled-equivalent in the sense of Definition
\ref{def:equivalentscaledmetrics}, cf. Remark \ref{rem:scaledequivalence}, and
that, for any tensor $\sigma$ on $L$ and as $r\rightarrow\infty$,
\begin{equation*}
|\sigma|_{\phi_i^*g}=|\sigma|_{{\tilde{g}}_i}\left(1+O(r^{\nu_i})\right).
\end{equation*}
In particular, given any function $f$ on $L$, using $\sigma=df$ and
multiplying by $r$ we obtain
\begin{equation*}
|df|_{r^{-2}\phi_i^*g}=|df|_{r^{-2}{\tilde{g}}_i}+O(r^{\nu_i})|df|_{r^{-2}{\tilde{g}}_i}.
\end{equation*}
Furthermore, let $A:=\nabla-{\widetilde{\nabla}}$ denote the difference of the two
connections defined by
$\phi_i^*g$ and ${\tilde{g}}_i$. Then, as in Remark \ref{rem:equivalence},
Definition \ref{def:metrics_ends} implies that $|A|_{{\tilde{g}}_i}=O(r^{\nu_i-1})$.
This leads to
\begin{align*}
|\nabla^2f|_{\phi_i^*g}&=|{\widetilde{\nabla}}^2f|_{{\tilde{g}}_i}\left(1+O(r^{\nu_i})\right)+
|df|_{{\tilde{g}}_i}O(r^{\nu_i-1}),\\
|\mbox{tr}_{\phi_i^*g}\nabla^2
f|&=|\mbox{tr}_{{\tilde{g}}_i}{\widetilde{\nabla}}^2f|\left(1+O(r^{\nu_i})\right)
+|df|_{{\tilde{g}}_i} O(r^{\nu_i-1}).
\end{align*}
Multiplying these equations by $r^2$ we can re-write them as
\begin{align*}
|\nabla^2f|_{r^{-2}\phi_i^*g}&=|{\widetilde{\nabla}}^2f|_{r^{
-2}{\tilde{g}}_i}+O(r^{\nu_i})\left(|{\widetilde{\nabla}}^2f|_{r^{
-2}{\tilde{g}}_i}+|df|_{r^{-2}{\tilde{g}}_i}\right),\\
|r^2\Delta_{\phi_i^*g}f|&=|r^2\Delta_{{\tilde{g}}_i}f|+O(r^{\nu_i})\left(|r^2\Delta_{{\tilde{g}}_i}f|+|df|_{r^{-2}{\tilde{g}}_i}\right).
\end{align*}
Analogous comments apply to higher derivatives and to CS ends.
\end{remark}
\begin{definition}
Let $(L,g)$ be a manifold with ends endowed with a Riemannian metric. We say
that $L$ is a \textit{CS} (respectively, \text{AC})
manifold if all ends are conically singular (respectively, asymptotically
conical). We say that $L$ is a \textit{CS/AC}
manifold if all ends are either conically singular or asymptotically conical. We
use the generic term \textit{conifold} to indicate any CS, AC or CS/AC manifold.
When working with a CS/AC manifold we will often index the CS (``small") ends
with numbers $\{1,\dots,s\}$ and the AC (``large") ends with numbers
$\{1,\dots,l\}$. Furthermore we will denote the union of the CS links
(respectively, of the CS ends) by $\Sigma_0$ (respectively, $S_0$) and those
corresponding to the AC links and ends by $\Sigma_\infty$, $S_\infty$.
\end{definition}
\begin{remark}\label{rem:cpt_conifolds}
It is useful to include smooth compact manifolds in the category of conifolds: they are precisely those for which the set of ends is empty.
\end{remark}
We now need to choose which function spaces to work with on conifolds. It
turns out that the most useful
classes of function spaces are precisely those of Section \ref{s:weighted}. One
needs only to choose appropriate functions $\rho$ and $w$ satisfying the
assumptions of Theorem \ref{thm:weighted_ok}, as follows.
Regarding notation, given a vector
$\boldsymbol{\beta}=(\beta_1,\dots,\beta_e)\in \mathbb{R}^e$ and $j\in\mathbb{N}$ we set
$\boldsymbol{\beta}+j:=(\beta_1+j,\dots,\beta_e+j)$. We write
$\boldsymbol{\beta}\geq\hat{\boldsymbol{\beta}}$ iff $\beta_i\geq\hat{\beta_i}$.
\begin{definition}\label{def:csac_sectionspaces}
Let $L$ be a conifold with metric $g$. We say that a smooth function
$\rho:L\rightarrow (0,\infty)$ is a \textit{radius function} if
$\phi_i^*\rho=r$, where $\phi_i$ are the
diffeomorphisms of Definition \ref{def:metrics_ends}. Given any vector
$\boldsymbol{\beta}=(\beta_1,\dots,\beta_e)\in\mathbb{R}^e$, choose a function
$\boldsymbol{\beta}:L\rightarrow \mathbb{R}$ which, on each end $S_i$, restricts to the
constant
$\beta_i$. Then $\rho$ and $w:=\rho^{-\beta}$ satisfy the assumptions of
Theorem \ref{thm:weighted_ok}, cf. Example \ref{e:radius}. We call
$(L,g,\rho,\boldsymbol{\beta})$ a \textit{weighted conifold}.
Given any metric pair $(E,\nabla)$ we define weighted spaces
$C^k_{\boldsymbol{\beta}}(E)$ and $W^p_{k,\boldsymbol{\beta}}(E)$ as in Section
\ref{s:weighted}. We can equivalently define the space
$C^k_{\boldsymbol{\beta}}(E)$ to be the space of sections $\sigma\in C^k(E)$
such that $|\nabla^j \sigma|=O(r^{\boldsymbol{\beta}-j})$ as $r\rightarrow 0$
(respectively, $r\rightarrow\infty$) along each CS (respectively, AC) end.
In the case of a CS/AC manifold we will often separate the CS and AC weights,
writing $\boldsymbol{\beta}=(\boldsymbol{\mu},\boldsymbol{\lambda})$ for some
$\boldsymbol{\mu}\in \mathbb{R}^s$ and some $\boldsymbol{\lambda}\in \mathbb{R}^l$. We then
write $C^k_{(\boldsymbol{\mu},\boldsymbol{\lambda})}(E)$ and
$W^p_{k,(\boldsymbol{\mu},\boldsymbol{\lambda})}(E)$.
\end{definition}
One can extend to these weighted spaces many results valid for standard Sobolev
spaces. H\"{o}lder's inequality is one example.
\begin{lemma}[Weighted H\"{o}lder's inequality] \label{l:weightedhoelder}
Let $(L,g)$ be a conifold.
Then, for all $p\geq 1$ and
$\boldsymbol{\beta}=\boldsymbol{\beta_1}+\boldsymbol{\beta_2}$,
\begin{equation*}
\|uv\|_{L^1_{\boldsymbol{\beta}}}\leq
\|u\|_{L^p_{\boldsymbol{\beta_1}}}\|v\|_{L^{p'}_{\boldsymbol{\beta_2}}}.
\end{equation*}
More generally, assume $\frac{1}{q}=\frac{1}{q_1}+\frac{1}{q_2}$. Then
\begin{equation*}
\|uv\|_{L^q_{\boldsymbol{\beta}}}\leq
\|u\|_{L^{q_1}_{\boldsymbol{\beta_1}}}\|v\|_{L^{q_2}_{\boldsymbol{\beta_2}}}.
\end{equation*}
\end{lemma}
\begin{proof}
\begin{align*}
\|uv\|_{L^1_{\boldsymbol{\beta}}}
&=\int_L(\rho^{-\boldsymbol{\beta_1}}u\rho^{-m/p})(\rho^{-\boldsymbol{\beta_2}}
v\rho^ { -m/p' } )\, \mbox {vol}_g\\
&\leq
\|\rho^{-\boldsymbol{\beta_1}}u\rho^{-m/p}\|_{L^p}\|\rho^{-\boldsymbol{\beta_2}}
v\rho^ { -m/p' } \|_ { L^ { p' } } \\
&=\|u\|_{L^p_{\boldsymbol{\beta_1}}}\|v\|_{L^{p'}_{\boldsymbol{\beta_2}}}.
\end{align*}
The general case is similar.
\end{proof}
\begin{corollary}\label{cor:embedding}
Let $(L,g,\boldsymbol{\beta})$ be a weighted conifold. Then
all parts of the weighted Sobolev Embedding Theorems hold for any metric pair
$(E,\nabla)$.
Furthermore, assume $lp>m$ and $k\geq 0$. Then the corresponding weighted
Sobolev spaces are closed under multiplication, in the following sense. For
any $\boldsymbol{\beta}_1$ and $\boldsymbol{\beta_2}$ there exists $C>0$ such
that, for all $u\in W^p_{k+l,\boldsymbol{\beta_1}}$ and $v\in
W^p_{k+l,\boldsymbol{\beta_2}}$,
\begin{equation*}
\|uv\|_{W^p_{k+l,\boldsymbol{\beta_1}+\boldsymbol{\beta_2}}}\leq
C\|u\|_{W^p_{k+l,\boldsymbol{\beta_1}}}\|v\|_{W^p_{k+l,\boldsymbol{\beta_2}}}.
\end{equation*}
\end{corollary}
\begin{proof}
Let $(L,g)$ be a conifold. Write $L=K\cup S$ as in Definition
\ref{def:manifold_ends} and let $C_i$ denote
the cone corresponding to the end $S_i$. Example \ref{e:radius}
showed that the assumptions for the scaled Sobolev Embedding Theorems hold for
$(C_i,\tilde{g}_i,r)$. The same is true for the weighted Sobolev Embedding
Theorems. Using the compactness of $K$ we conclude that these
assumptions, thus the theorems, hold for $L$ with respect to any metric
$\hat{g}$ such that $\phi_i^*\hat{g}={\tilde{g}}_i$ on each end. As in Remark
\ref{rem:metrics_ends} one can assume that
$\phi_i^*g$ and $\tilde{g}_i$
are scaled-equivalent so there exists $C_0>0$ such that $(1/C_0){\tilde{g}}_i\leq
\phi_i^*g\leq C_0{\tilde{g}}_i$. Again using the compactness of $K$ we may thus assume
that $(1/C_0)\hat{g}\leq g\leq C_0\hat{g}$. Theorem \ref{thm:weighted_ok} now
shows that the weighted Sobolev Embedding Theorems hold for
$(L,g)$. The fact that weighted Sobolev spaces
are closed with respect to products can
be proved as in Theorem \ref{thm:scaled_ok}, using Lemma
\ref{l:weightedhoelder}.
\end{proof}
\begin{remark}\label{rem:embedding}
Let $(L,g)$ be an AC manifold. Notice that for
$\hat{\boldsymbol{\beta}}\geq\boldsymbol{\beta}$ there exist continuous
embeddings
$W^r_{k,\boldsymbol{\beta}}\hookrightarrow W^r_{k,\hat{\boldsymbol{\beta}}}$.
The analogous statement is true for the weighted $C^k$ spaces. By composition
Corollary \ref{cor:embedding} thus leads to the following statements:
\begin{enumerate}
\item If $lp<m$ then there exists a continuous embedding
$W^p_{k+l,\boldsymbol{\beta}}(E)\hookrightarrow
W^{p^*_l}_{k,\hat{\boldsymbol{\beta}}}(E)$.
\item If $lp=m$ then, for all $q\in [p,\infty)$, there exist continuous
embeddings $W^p_{k+l,\boldsymbol{\beta}}(E)\hookrightarrow
W^q_{k,\hat{\boldsymbol{\beta}}}(E)$.
\item If $lp>m$ then there exists a continuous embedding
$W^p_{k+l,\boldsymbol{\beta}}(E)\hookrightarrow
C^k_{\hat{\boldsymbol{\beta}}}(E)$.
\end{enumerate}
Notice that if $(L,g)$ is a CS
manifold then the behaviour on the ends is studied in terms of $r\rightarrow 0$
rather than $r\rightarrow \infty$. In this case the same conclusions hold for
the opposite situation $\hat{\boldsymbol{\beta}}\leq\boldsymbol{\beta}$.
Finally, let $(L,g)$ be a CS/AC manifold with
$\boldsymbol{\beta}=(\boldsymbol{\mu},\boldsymbol{\lambda})$. Then the same
conclusions hold for all
$\hat{\boldsymbol{\beta}}=(\hat{\boldsymbol{\mu}},\hat{\boldsymbol{\lambda}})$
with
$\hat{\boldsymbol{\mu}}\leq\boldsymbol{\mu}$,
$\hat{\boldsymbol{\lambda}}\geq\boldsymbol{\lambda}$.
\end{remark}
We now want to show that all the above notions and results are
scale-independent, as long as we rescale the weight function
correctly to take into account the possibility of variable weights. We start by examining the properties of $(L,t^2g)$.
\begin{lemma}\label{l:rescaledconifold}
Let $(L,g)$ be a conifold. For each AC end $S_i$ let
$\phi_i:\Sigma_i\times [R,\infty)\rightarrow \overline{S_i}$ denote the
diffeomorphism of Definition \ref{def:metrics_ends}. In particular, for all
$k\geq 0$ there exist $C_k>0$ such that, for $r\geq R$,
$$|{\widetilde{\nabla}}^k(\phi_i^*g-{\tilde{g}}_i)|_{r^{-2}{\tilde{g}}_i\otimes{\tilde{g}}_i}\leq C_kr^{\nu_i}\leq
C_kR^{\nu_i}.$$
As seen in Remark \ref{rem:metrics_ends}, we can thus assume that
$\phi_i^*g$, ${\tilde{g}}_i$ are scaled-equivalent.
Choose any $t>0$. Define the diffeomorphism
\begin{equation*}
\phi_{t,i}:\Sigma_i\times [tR,\infty)\rightarrow \overline{S_i},\ \
\phi_{t,i}(\theta,r):=\phi_i(\theta,r/t).
\end{equation*}
Then, for $r\geq tR$ and with respect to the same $C_k$,
there are $t$-uniform estimates
$$|{\widetilde{\nabla}}^k(\phi_{t,i}^*(t^2g)-{\tilde{g}}_i)|_{r^{-2}{\tilde{g}}_i\otimes {\tilde{g}}_i}\leq
C_k(r/t)^{\nu_i}\leq
C_kR^{\nu_i}.$$
Analogously, for each CS end $S_i$ let $\phi_i$ denote the
diffeomorphism of Definition \ref{def:metrics_ends}. Define the diffeomorphism
\begin{equation*}
\phi_{t,i}:\Sigma_i\times (0,t\epsilon]\rightarrow \overline{S_i},\ \
\phi_{t,i}(\theta,r):=\phi_i(\theta,r/t).
\end{equation*}
Then there are $t$-uniform estimates as above.
In particular, with respect to these diffeomorphisms, $(L,t^2g)$ is again a
conifold. If $\rho$ is a radius function for $(L,g)$ then $t\rho$ is a radius
function for $(L,t^2g)$.
\end{lemma}
\begin{proof}
Define the map
$$\delta_t:\Sigma_i\times \mathbb{R}^+\rightarrow \Sigma_i\times\mathbb{R}^+,\ \
(\theta,r)\mapsto (\theta,tr).$$
Since $\delta_t$ is simply a rescaling it preserves the Levi-Civita connection
$\tilde{\nabla}$. Notice that
$\phi_{t,i}=\phi_i\circ\delta_{1/t}$. It is simple to check that
$\delta_{1/t}^*(t^2{\tilde{g}}_i)={\tilde{g}}_i$. Thus, for $r\geq tR$,
\begin{align*}
|{\widetilde{\nabla}}^k(\phi_{t,i}^*(t^2g)-{\tilde{g}}_i)|_{{\tilde{g}}_i\otimes{\tilde{g}}_i}
&=|{\widetilde{\nabla}}^k(\delta_{1/t}^*\phi_i^*(t^2g)-{\tilde{g}}_i)|_{{\tilde{g}}_i\otimes{\tilde{g}}_i}\\
&=\delta_{1/t}^*\left(|{\widetilde{\nabla}}^k(\phi_i^*(t^2g)-t^2{\tilde{g}}_i)|_{t^2{\tilde{g}}_i\otimes t^2{\tilde{g}}_i}\right)\\
&=\delta_{1/t}^*\left(|{\widetilde{\nabla}}^k(\phi_i^*g-{\tilde{g}}_i)|_{t^2{\tilde{g}}_i\otimes {\tilde{g}}_i}\right)\\
&\leq t^{-k}C_k(r/t)^{\nu_i-k} = C_k(r/t)^{\nu_i}r^{-k}.
\end{align*}
These inequalities can be rescaled as in Remark \ref{rem:metrics_ends} to
obtain the desired $t$-uniform estimates.
Now notice that
$$\phi_{t,i}^*(t\rho)_{|(\theta,r)}=t\rho\circ\phi_{t,i}(\theta,
r)=t\rho\circ\phi(\theta, r/t)=tr/t=r, $$
so $t\rho$ is a radius function in the sense of Definition
\ref{def:csac_sectionspaces}.
CS ends can be studied analogously.
\end{proof}
The following result is a direct consequence of Theorem \ref{thm:weighted_ok}
and Remark \ref{rem:tweighted_ok}.
\begin{corollary}\label{cor:rescaledconifold}
Let $(L,g)$ be a conifold. Then,
for all $t>0$:
\begin{enumerate}
\item Choose a constant weight
$\boldsymbol{\beta}$. Define weighted Sobolev spaces
$W^p_{k,\boldsymbol{\beta}}$ as in Section \ref{s:weighted} using the metric
$t^2g$, the scale function $t\rho$ and the weight function
$w:=(t\rho)^{-\boldsymbol{\beta}}$. Then
all forms of the weighted Sobolev
Theorems hold for
$(L,t^2g,t\rho,(t\rho)^{-\boldsymbol{\beta}})$ with $t$-independent Sobolev
constants.
\item More generally, let $\boldsymbol{\beta}$ be a function as
in Definition \ref{def:csac_sectionspaces}. Choose a constant ``reference''
weight $\boldsymbol{\beta}'$ and define weighted Sobolev spaces
$W^p_{k,\boldsymbol{\beta}}$ as in Section \ref{s:weighted} using the metric
$t^2g$, the scale function $t\rho$ and the weight function
$w_t:=(t^\frac{\boldsymbol{\beta}'-\boldsymbol{\beta}}{\boldsymbol{\beta}}t\rho)^{-\boldsymbol{\beta
} } $. Then the weighted norms
$\|\cdot\|_{W^p_{k,\boldsymbol{\beta}}}$,
calculated with respect to these choices, coincide with
$t^{-\boldsymbol{\beta}'}\|\cdot\|_{W^p_{k,\boldsymbol{\beta}}}$,
calculated with
respect to $(L,g,\rho,w:=\rho^{-\boldsymbol{\beta}})$. In particular,
all forms of the weighted Sobolev Embedding Theorems hold for
$(L,t^2g,t\rho,
w_t:=(t^\frac{\boldsymbol{\beta}'-\boldsymbol{\beta}}{\boldsymbol{\beta}}t\rho)^{-\boldsymbol{\beta
} } )$
with $t$-independent Sobolev constants.
\end{enumerate}
\end{corollary}
\begin{remark}\label{rem:corrective_term}
Compare the weights used in parts (1) and (2) above. Basically, to deal with variable weights we introduce a corrective factor of the form $t^{\boldsymbol{\beta}-\boldsymbol{\beta}'}$: since the exponent is bounded, for fixed $t$ this doesn't affect the decay/growth condition on the ends. Its effect is simply to yield uniform estimates as $t\rightarrow 0$.
\end{remark}
We conclude this section by summarizing the main definitions and properties of
a second class of manifolds with ends, modelled on cylinders. We will see that
the corresponding theory is closely related to that of conifolds.
\begin{definition}\label{def:metrics_endsbis}
Let L be a manifold with ends. Let $g$ be a Riemannian metric on $L$. Choose an
end $S_i$ with corresponding link $\Sigma_i$. We say that $S_i$ is an
\textit{asymptotically cylindrical} (A.Cyl.) end if the
following conditions hold:
\begin{enumerate}
\item $\Sigma_i$ is endowed with a Riemannian metric $g_i'$.
We then let $(\theta,z)$ denote the generic point on the product manifold
$C_i:=\Sigma_i\times (-\infty,\infty)$ and $\tilde{h}_i:=dz^2+g_i'$ denote the
corresponding \textit{cylindrical metric} on $C_i$.
\item There exist a constant $\nu_i<0$ and a diffeomorphism
$\phi_i:\Sigma_i\times [R',\infty)\rightarrow \overline{S_i}$ such that, as
$z\rightarrow
\infty$ and for all $k\geq 0$,
$$|{\widetilde{\nabla}}^k(\phi_i^*g-\tilde{h}_i)|_{\tilde{h}_i}=O(e^{\nu_i z}),$$
where ${\widetilde{\nabla}}$ is the Levi-Civita connection on $C_i$ defined by $\tilde{h}_i$.
\end{enumerate}
We say
that $L$ is a \textit{A.Cyl.}
manifold if all ends are asymptotically
cylindrical.
\end{definition}
For the purposes of
this paper the function spaces of most interest on A.Cyl. manifolds are not the
ones already
encountered, cf. Section \ref{s:std} and Example \ref{e:std_ok}. Instead, we
use the following.
\begin{definition}\label{def:acyl_sectionspaces}
Let $(L,h)$ be a A.Cyl. manifold. We say that a smooth function
$\zeta:L\rightarrow [1,\infty)$ is a \textit{radius function} if
$\phi_i^*\zeta= z$, where $\phi_i$ are the
diffeomorphisms of Definition \ref{def:metrics_ends}. Given any vector
$\boldsymbol{\beta}=(\beta_1,\dots,\beta_e)\in\mathbb{R}^e$, choose a function
$\boldsymbol{\beta}$ on $L$ which, on each end $S_i$, restricts to the constant
$\beta_i$. We call $(L,h,\zeta,\boldsymbol{\beta})$ a \textit{weighted A.Cyl.
manifold}. Given any metric pair $(E,\nabla)$ we define Banach spaces of
sections of $E$ in the following two ways.
The \textit{weighted spaces of $C^k$ sections} of $E$ are defined by
\begin{equation}\label{eq:weighted_C^kcyl}
C^k_{\boldsymbol{\beta}}(E):=\left\{\sigma\in C^k(E):
\|\sigma\|_{C^k_{\boldsymbol{\beta}}}<\infty\right\},
\end{equation}
where we use the norm $\|\sigma\|_{C^k_{\boldsymbol{\beta}}}:=\sum_{j=0}^k
\mbox{sup}_{x\in L}|e^{-\boldsymbol{\beta}(x)\zeta(x)}\nabla^j\sigma|$.
The \textit{weighted Sobolev spaces} are defined by
\begin{equation}\label{eq:weighted_cyl}
W^p_{k,\boldsymbol{\beta}}(E):=\mbox{Banach space completion of the space
}\left\{\sigma\in
C^\infty(E):\|\sigma\|_{W^p_{k,\boldsymbol{\beta}}}<\infty\right\},
\end{equation}
where $p\in [1,\infty)$, $k\geq 0$ and we use the norm
$\|\sigma\|_{W^p_{k,\boldsymbol{\beta}}}:=\left(\sum_{j=0}^k \int_L
|e^{-\boldsymbol{\beta} \zeta}\nabla^j\sigma|^p\,\mbox{vol}_h\right)^{1/p}$.
Both types of spaces are independent of the particular choices made.
\end{definition}
\begin{remark}\label{rem:acyl_equivalent}
It is simple to see that the norm $\|\sigma\|_{W^p_{k,\boldsymbol{\beta}}}$ is
equivalent to the norm defined by $\sum_{j=0}^k(\int_L
|\nabla^j(e^{-\boldsymbol{\beta} \zeta}\sigma)|^p\,\mbox{vol}_h)^{1/p}$. This
leads to the following fact.
Let $W^p_k(E)$ denote the standard Sobolev spaces for $(L,h)$ introduced in
Section \ref{s:std}. Let $e^{\boldsymbol{\beta}\zeta}\cdot W^p_k$ denote the
space of all sections of $E$ of the form
$\sigma=e^{\boldsymbol{\beta}\zeta}\tau$ for some $\tau\in W^p_k(E)$, endowed
with the norm $\|\sigma\|:=\|\tau\|$. Then
$W^p_{k,\boldsymbol{\beta}}(E)=e^{\boldsymbol{\beta}\zeta}\cdot W^p_k(E)$ as
sets and the norms are equivalent. Analogously, the spaces
$C^k_{\boldsymbol{\beta}}(E)$ are equivalent to the spaces
$e^{\boldsymbol{\beta}\zeta}\cdot C^k(E)$, where $C^k(E)$ are the standard
spaces of $C^k$ sections used in Section \ref{s:std}.
\end{remark}
As before, weighted spaces defined with respect to A.Cyl. metrics and
cylindrical metrics are equivalent. Remark \ref{rem:acyl_equivalent} allows us
to reduce
the weighted Sobolev Embedding Theorems for A.Cyl. manifolds to the standard
Sobolev Embedding Theorems, obtaining results analogous to Corollary
\ref{cor:embedding} and Remark \ref{rem:embedding}. According to \cite{hebey}
Theorem 3.1 and Proposition 3.2 the spaces $C^\infty_c$
are dense in the standard Sobolev spaces defined for manifolds whose
ends are exactly cylindrical. The same is then true for weighted Sobolev
spaces on A.Cyl. manifolds.
\begin{remark}\label{rem:spacescoincide}
It is interesting to compare Definitions \ref{def:acyl_sectionspaces} and
\ref{def:csac_sectionspaces}. Assume $(L,h)$ is an A.Cyl. manifold with respect
to certain diffeomorphisms $\phi_i=\phi_i(\theta,z)$ as in Definition
\ref{def:metrics_ends}. Since the corresponding weighted Sobolev spaces are
equivalent we may assume that $h$ is exactly cylindrical on each end,
\textit{i.e.} using the notation of Definition \ref{def:metrics_ends} it can be
written $h=dz^2+g_i'$. Consider the conformally rescaled metric
$g:=e^{2\zeta}h$. Using the change of variables $r=e^z$ it is simple to check
that $g=dr^2+r^2g_i'$. This implies that $(L,g)$ is an AC manifold with respect
to the diffeomorphisms $\phi_i(\theta,\log z)$. Viceversa, any AC metric on $L$
defines a conformally equivalent A.Cyl. metric. Notice that if $z\in
(R',\infty)$ then $r\in (R,\infty)$ with $R:=e^{R'}$ and that
$r^{-m}\mbox{vol}_g=\mbox{vol}_h$. Thus, by change of variables,
\begin{equation}
\int_R^\infty\int_{\Sigma}|r^{-\boldsymbol{\beta}}\sigma|^pr^{-m}\,\mbox{vol}
_g=\int_{R'}^\infty\int_{\Sigma}|e^{-\boldsymbol{\beta}
z}\sigma|^p\,\mbox{vol}_h.
\end{equation}
This shows that the spaces $L^p_{\boldsymbol{\beta}}(E)$ of sections of $E$
coincide for $(L,g)$ and $(L,h)$, while the corresponding norms are
equivalent (but again, as in Remark \ref{rem:scalednorms}, one may need to take
into account which metric is being used on $E$ in the two cases).
The same is true also for Sobolev spaces of higher order. Specifically, an
explicit calculation shows that the Levi-Civita connections defined by $h$ and
$g$ are equivalent, \textit{i.e.} the corresponding Christoffel symbols coincide
up to constant multiplicative factors. It thus makes no difference which metric
is used to define $\nabla$. On the other hand, the norm inside the integral does
depend on the choice of metric. For example,
\begin{equation}
\int_R^\infty\int_{\Sigma}|r^{-\boldsymbol{\beta}+j}\nabla^j
f\sigma|_g^pr^{-m}\,\mbox{vol}_g=\int_{R'}^\infty\int_{\Sigma}|e^{-\boldsymbol{
\beta} z}\nabla^j \sigma|_h^p\,\mbox{vol}_h.
\end{equation}
This proves that the spaces $W^p_{k,\boldsymbol{\beta}}(E)$ are equivalent.
Analogous results hold for CS manifolds: if $h$ is A.Cyl. then $g:=e^{-2\zeta}h$
is CS. In this case
\begin{equation}
\int_0^{\epsilon}\int_{\Sigma}|r^{-\boldsymbol{\beta}}f|^pr^{-m}\,\mbox{vol}
_g=\int_{-\log \epsilon}^\infty\int_{\Sigma}|e^{\boldsymbol{\beta}
z}f|^p\,\mbox{vol}_h,
\end{equation}
so the space $L^p_{\boldsymbol{\beta}}$ for $(L,g)$ coincides with the space
$L^p_{-\boldsymbol{\beta}}$ for $(L,h)$.
These facts show, for example, that the Sobolev Embedding Theorems for
conifolds and A.Cyl. manifolds are simply two different points of view on the
same result. They also show that $C^\infty_c$ is dense in all weighted Sobolev
spaces on conifolds because, as already seen, this is true on A.Cyl.
manifolds. Finally, they show that in Remark \ref{rem:metrics_ends} we are
really using the cylindrical metric $r^{-2}{\tilde{g}}=\tilde{h}$ to ``measure''
${\widetilde{\nabla}}^k$ (in the sense of Remark \ref{rem:scalednorms}).
\end{remark}
\section{Conifold connect sums}\label{s:sums_sobolev}
The goal of this section is to introduce a certain ``parametric connect sum''
construction between conifolds. As mentioned in the Introduction, this is the
abstract analogue of
certain desingularization procedures used in Differential Geometry, in which an
isolated conical singularity is
replaced by something smooth or perhaps by a new collection of AC or CS ends.
For
this construction we prove that the scaled and weighted Sobolev
constants
are independent of the parameter $\boldsymbol{t}$. For simplicity we start with
the
non-parametric version.
\begin{definition}\label{def:marking}
Let $(L,g)$ be a conifold, not necessarily connected. Let $S$ denote
the union of its ends. A subset $S^*$ of $S$ defines a \textit{marking} on $L$.
We can then write $S=S^*\amalg S^{**}$, where $S^{**}$ is simply the complement
of $S^*$. We say $S^*$ is a \textit{CS-marking} if all ends in $S^*$ are CS; it
is an \textit{AC-marking} if all ends in $S^*$ are AC. We will denote by $d$
the number of ends in $S^*$.
If $L$ is weighted via $\boldsymbol{\beta}$ we require that
$\beta_i=\beta_j$ if $S_i$ and $S_j$ are marked ends belonging to the same
connected component of $L$.
\end{definition}
\begin{definition}\label{def:compatible}
Let $(L,g,S^*)$ be a CS-marked conifold. Let
$\Sigma^*$, $C^*$ denote the
links and cones corresponding to $S^*$, as in Definition
\ref{def:metrics_ends}. Given any end $S_i\subseteq S^*$ let
$\phi_i:\Sigma_i\times (0,\epsilon]\rightarrow \overline{S_i}$ be the
diffeomorphism of
Definition \ref{def:metrics_ends}.
Let $(\hat{L},\hat{g},\hat{S}^*)$ be an AC-marked
conifold. Let $\hat{\Sigma}^*$, $\hat{C}^*$,
$\hat{\phi}_i:\hat{\Sigma}_i\times
[\hat{R},\infty)\rightarrow \overline{\hat{S}_i}$ denote the corresponding
links, cones and diffeomorphisms, as above.
We say that $L$ and $\hat{L}$ are \textit{compatible} if they satisfy the
following assumptions:
\begin{enumerate}
\item $C^*=\hat{C}^*$. Up to relabelling the ends, we may assume that $C_i^*=\hat{C}_i^*$.
\item $\hat{R}<\epsilon$. We can then identify
appropriate
subsets of $S^*$ and $\hat{S}^*$ via the maps $\hat{\phi}_i\circ\phi_i^{-1}$.
\item On each marked AC end,
the metrics $\hat{\phi}_i^*\hat{g}$ and ${\tilde{g}}_i$ are scaled-equivalent in the
sense of Definition \ref{def:equivalentscaledmetrics}. Analogously, on each
marked CS end,
the metrics $\phi_i^*g$ and ${\tilde{g}}_i$ are scaled-equivalent in the
sense of Definition \ref{def:equivalentscaledmetrics}.
\end{enumerate}
If
$L$ is weighted via $\boldsymbol{\beta}$ and $\hat{L}$ is weighted via
$\hat{\boldsymbol{\beta}}$ we further require that, on the marked ends, the
corresponding constants satisfy
$\boldsymbol{\beta}_{|S^*}=\hat{\boldsymbol{\beta}}_{|\hat{S}^*}$.
\end{definition}
\begin{remark}\label{rem:compatible}
The condition $\hat{R}<\epsilon$ may seem rather strong. However, let
$(L,g,S^*)$ be CS-marked, $(\hat{L},\hat{g},\hat{S}^*)$ be AC-marked
and $C^*=\hat{C}^*$. As seen in Remark \ref{rem:metrics_ends}, by making
$\hat{R}$ larger if necessary it is possible to assume that the
metrics $\hat{\phi}_i^*\hat{g}$, ${\tilde{g}}_i$ on $\Sigma_i\times [\hat{R},\infty)$
are scaled-equivalent in the
sense of Definition \ref{def:equivalentscaledmetrics}. Lemma
\ref{l:rescaledconifold} then shows that the metrics
$\hat{\phi}_{t,i}^*(t^2\hat{g})$, ${\tilde{g}}_i$ on
$\Sigma_i\times [t\hat{R},\infty)$ are also scaled-equivalent, with the same
bounds. Analogously, by making $\epsilon$ smaller if necessary, we can assume
that the metrics $\phi_i^*g$, ${\tilde{g}}_i$ on $\Sigma_i\times
(0,\epsilon]$ are scaled-equivalent. By first making
$\hat{R}$ large and $\epsilon$ small and then rescaling to satisfy the
condition $\hat{R}<\epsilon$ we thus obtain
compatible conifolds in the sense of Definition \ref{def:compatible}.
\end{remark}
\begin{definition}\label{def:connectsum}
Let $(L,g,S^*)$, $(\hat{L},\hat{g},\hat{S}^*)$ be
compatible marked conifolds. We define the \textit{connect sum}
of $L$ and $\hat{L}$ as follows. We set
\begin{equation}
\hat{L}\#L:=(\hat{L}\setminus\hat{S}^*)\cup
(\Sigma^*\times[\hat{R},\epsilon])\cup (L\setminus S^*),
\end{equation}
where the boundary of $\hat{L}\setminus\hat{S}^*$ is identified with
$\Sigma^*\times\{\hat{R}\}$ via the maps $\hat{\phi}_i$ and the boundary of
$L\setminus S^*$ is identified with $\Sigma^*\times\{\epsilon\}$ via
the maps $\phi_i$. We can endow this manifold with any metric $\hat{g}\#g$ which
restricts to $\hat{g}$ on $\hat{L}\setminus\hat{S}^*$ and to $g$ on
$L\setminus{S}^*$. Then $\hat{L}\#L$ is a conifold. Its ends are
$\hat{S}^{**}\amalg S^{**}$. We
call $\Sigma^*\times[\hat{R},\epsilon]$ the \textit{neck region} of
$\hat{L}\#L$.
Given radius functions $\rho$ on $L$ and $\hat{\rho}$ on $\hat{L}$ we can endow
$\hat{L}\#L$ with the radius function
\begin{equation*}
\hat{\rho}\#\rho:=\left\{
\begin{array}{ll}
\hat{\rho} & \mbox{ on } \hat{L}\setminus\hat{S}^*\\
r & \mbox{ on } \Sigma^*\times[\hat{R},\epsilon]\\
\rho & \mbox{ on }L\setminus S^*.
\end{array}\right.
\end{equation*}
If $L$, $\hat{L}$ are weighted via $\boldsymbol{\beta}$,
$\hat{\boldsymbol{\beta}}$ then $\hat{L}\#L$ is weighted via the function
\begin{equation*}
\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}:=\left\{
\begin{array}{ll}
\hat{\boldsymbol{\beta}} & \mbox{ on } \hat{L}\setminus\hat{S}^*\\
\boldsymbol{\beta}_{|S^*} & \mbox{ on } \Sigma^*\times[\hat{R},\epsilon]\\
\boldsymbol{\beta} & \mbox{ on }L\setminus S^*.
\end{array}\right.
\end{equation*}
\end{definition}
\begin{example}\label{e:connectsum}
Let $\overline{L}$ be a smooth $m$-dimensional submanifold of
$\mathbb{R}^n$, endowed with the induced metric. Assume that it is either compact or
that it has AC ends: \textit{e.g.}, it could be a collection of $m$-planes in
$\mathbb{R}^n$. Now assume it has transverse self-intersection
points $x_1,\dots,x_k\in \mathbb{R}^n$. For each $x_i$ choose a ball $B(x_i,\epsilon)$
in $\mathbb{R}^n$. Then $L:=\overline{L}\setminus \{x_1,\dots,x_k\}$ is a conifold with
$s$ CS ends defined by the connected components of $\left(
B(x_1,\epsilon)\cup\dots\cup B(x_k,\epsilon)\right)\cap L$. The corresponding
cones are copies of $\mathbb{R}^m$. Choose a pair $S_1$, $S_2$ of
connected components of $B(x_1,\epsilon)\cap L$ and an appropriately rescaled
$m$-dimensional hyperboloid $\hat{L}\subseteq \mathbb{R}^n$ asymptotic to the
corresponding cones $C_1$,
$C_2$. Then $L$, $\hat{L}$ are compatible and $\hat{L}\#L$ is an abstract
Riemannian manifold which we can think of as a desingularization of
$\overline{L}$. Our hypothesis in Definition \ref{def:marking} that $L$,
$\hat{L}$ are not necessarily connected allows us to extend this construction to
intersection points of distinct submanifolds and to desingularize all points
simultaneously.
\end{example}
Since $\hat{L}\#L$ is again a conifold it is clear that all versions of the
Sobolev Embedding Theorems continue to hold for it. Notice that
$\hat{S}^{**}\cup S^{**}$ might also be empty: in this case $\hat{L}\#L$ is
a smooth compact manifold. We now consider the parametric version of this
construction.
\begin{definition}\label{def:tconnectsum}
Let $(L,g,S^*)$, $(\hat{L},\hat{g},\hat{S}^*)$ be
compatible marked conifolds with $d$ marked ends. Let
$(\rho,\boldsymbol{\beta})$, respectively
$(\hat{\rho},\hat{\boldsymbol{\beta}})$, be corresponding radius functions and
weights. Choose parameters $\boldsymbol{t}=(t_1,\dots,t_d)>0$ sufficiently
small. We assume that $\boldsymbol{t}$ is compatible with the
decomposition of $\hat{L}$ into its connected components: specifically, that
$t_i=t_j$ if $\hat{S}_i$ and $\hat{S}_j$ belong to the same connected component
of $\hat{L}$. We then define the
\textit{parametric connect sum}
of $L$ and $\hat{L}$ as follows. We set
\begin{equation*}
L_{\boldsymbol{t}}:=(\hat{L}\setminus\hat{S}^*)\cup
(\cup_{\Sigma_i\subseteq\Sigma^*}\Sigma_i\times[t_i\hat{R},\epsilon])\cup
(L\setminus S^*),
\end{equation*}
where the components of the boundary of $\hat{L}\setminus\hat{S}^*$ are
identified with
the $\Sigma_i\times\{t_i\hat{R}\}$ via maps $\hat{\phi}_{t_i,i}$ defined as in
Lemma
\ref{l:rescaledconifold} and the components of the boundary of $L\setminus S^*$
are identified
with the $\Sigma_i\times\{\epsilon\}$ via the maps $\phi_i$. Choose
$\tau\in (0,1)$. If the $t_i$ are sufficiently small, we find
$t_i\hat{R}<t_i^\tau<2t_i^\tau<\epsilon$. Choose any
metric $g_{\boldsymbol{t}}$ on $L_{\boldsymbol{t}}$ such that, for each
$\Sigma_i\subseteq \Sigma^*$,
\begin{equation*}
g_{\boldsymbol{t}}:=\left\{
\begin{array}{llll}
t_i^2\hat{g} & \mbox{ on the corresponding component of }
\hat{L}\setminus\hat{S}^*\\
\hat{\phi}_{t_i,i}^*(t_i^2\hat{g}) & \mbox{ on
}\Sigma_i\times[t_i\hat{R},t_i^\tau]\\
\phi_i^*g & \mbox{ on } \Sigma_i\times[2t_i^\tau,\epsilon]\\
g & \mbox{ on } L\setminus S^*
\end{array}\right.
\end{equation*}
and such that, for all $j\geq 0$ and as
$\boldsymbol{t}\rightarrow 0$,
$$\sup_{\Sigma_i\times[t_i^\tau,2t_i^\tau]}
|{\widetilde{\nabla}}^j(g_{\boldsymbol{t}}-{\tilde{g}}_i)|_{
r^ { -2 } {\tilde{g}}_i\otimes{\tilde{g}}_i} \rightarrow 0.$$
We endow $L_{\boldsymbol{t}}$ with the radius function
\begin{equation*}
\rho_{\boldsymbol{t}}:=\left\{
\begin{array}{ll}
t_i\hat{\rho} & \mbox{ on the corresponding component of }
\hat{L}\setminus\hat{S}^*\\
r & \mbox{ on } \Sigma_i\times[t_i\hat{R},\epsilon]\\
\rho & \mbox{ on }L\setminus S^*
\end{array}\right.
\end{equation*}
and the weight
\begin{equation*}
\boldsymbol{\beta}_{\boldsymbol{t}}:=\left\{
\begin{array}{ll}
\hat{\boldsymbol{\beta}} & \mbox{ on } \hat{L}\setminus\hat{S}^*\\
\beta_i & \mbox{ on } \Sigma_i\times[t_i\hat{R},\epsilon]\\
\boldsymbol{\beta} & \mbox{ on }L\setminus S^*.
\end{array}\right.
\end{equation*}
We now need to define the weight function $w_{\boldsymbol{t}}$. As in Corollary \ref{cor:rescaledconifold}, the simplest
case is when $\hat{\boldsymbol{\beta}}$ is constant on each connected
component of $\hat{L}$. We then define
\begin{equation*}
w_{\boldsymbol{t}}:=\rho_t^{-\boldsymbol{\beta}_t}=\left\{
\begin{array}{ll}
(t_i\hat{\rho})^{-\hat{\beta}_i} & \mbox{ on the corresponding
component of }
\hat{L}\setminus\hat{S}^*\\
r^{-\beta_i} & \mbox{ on } \Sigma_i\times[t_i\hat{R},\epsilon]\\
\rho^{-\boldsymbol{\beta}} & \mbox{ on }L\setminus S^*.
\end{array}\right.
\end{equation*}
For general weights $\hat{\boldsymbol{\beta}}$ we need to modify the
weight function. As in Corollary \ref{cor:rescaledconifold}, on the $i$-th component of
$\hat{L}$ consider the constant ``reference'' weight
$\hat{\beta}_i$. We then define
\begin{equation*}
w_{\boldsymbol{t}}:=\left\{
\begin{array}{ll}
(t_i^\frac{\hat{\beta}_i-\hat{\boldsymbol{\beta}}}{\hat{\boldsymbol{\beta}}}t_i\hat{\rho})^{-\hat{
\boldsymbol {\beta}} } & \mbox{ on the
corresponding component of }\hat{L}\setminus\hat{S}^*\\
r^{-\beta_i} & \mbox{ on } \Sigma_i\times[t_i\hat{R},\epsilon]\\
\rho^{-\boldsymbol{\beta}} & \mbox{ on }L\setminus S^*.
\end{array}\right.
\end{equation*}
We may equivalently write this as
\begin{equation*}
w_{\boldsymbol{t}}:=\left\{
\begin{array}{ll}
t_i^{\hat{\boldsymbol{\beta}}-\hat{\beta}_i}\rho_{\boldsymbol{t}}^{-\boldsymbol{\beta}_{\boldsymbol{t}}}&\mbox{ on }\hat{L}\setminus \hat{S}^*\\
\rho_{\boldsymbol{t}}^{-\boldsymbol{\beta}_{\boldsymbol{t}}}&\mbox{ elsewhere}.
\end{array}\right.
\end{equation*}
Using this data we now define weighted Sobolev spaces
$W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}$ on $L_{\boldsymbol{t}}$ as in
Section \ref{s:weighted}. We call $\Sigma_i\times[t_i\hat{R},\epsilon]$ the
\textit{neck regions} of $L_{\boldsymbol{t}}$.
\end{definition}
\begin{theorem}\label{thm:normstequivalent}
Let $(L,g,S^*)$, $(\hat{L},\hat{g},\hat{S}^*)$ be
compatible weighted marked conifolds. Define $L_{\boldsymbol{t}}$, $g_{\boldsymbol{t}}$,
$\rho_{\boldsymbol{t}}$ and
$\boldsymbol{\beta}_{\boldsymbol{t}}$
as in Definition \ref{def:tconnectsum}. Then all
forms of the weighted Sobolev Embedding Theorems hold uniformly in
${\boldsymbol{t}}$,
\textit{i.e.} the corresponding Sobolev constants are independent of
${\boldsymbol{t}}$.
\end{theorem}
\begin{proof}
The proof is similar to that of Corollary \ref{cor:embedding}. Let us for the
moment pretend that the metrics $g$, $\hat{g}$ are exactly conical on all ends
of $L$, $\hat{L}$. This allows us to assume that the metrics
$g_{\boldsymbol{t}}$ are exactly
conical on
all ends and neck regions of $L_{\boldsymbol{t}}$ so the assumptions of
Theorem \ref{thm:weighted_ok} are satisfied in these regions. On
$\hat{L}\setminus\hat{S}^*$ we are using rescaled metrics, radius
functions and weights as in Corollary \ref{cor:rescaledconifold}. As seen in
Remark \ref{rem:tweighted_ok}, the assumptions of Theorem \ref{thm:weighted_ok}
are $\boldsymbol{t}$-independent so they are verified here.
These assumptions are also
verified on $L\setminus S^*$ and on the neck regions. We conclude that all forms
of the weighted Sobolev
Embedding Theorems hold for these metrics, with
$\boldsymbol{t}$-independent Sobolev constants.
Let us now go back to the metric $g_{\boldsymbol{t}}$. Recall from Lemma
\ref{l:rescaledconifold} that we can assume that, on each end of
$L_{\boldsymbol{t}}$, $g_{\boldsymbol{t}}$ is a $\boldsymbol{t}$-uniformly
small perturbation of the conical metric. The same is true also on the neck
regions. Specifically, on $\Sigma_i\times[t_i\hat{R},t_i^\tau]$ Lemma
\ref{l:rescaledconifold} shows
that
\begin{equation*}
\sup |\phi_{t,i}^*(t_i^2\hat{g})-{\tilde{g}}_i|\leq C_0\hat{R}^{\hat{\nu}_i}.
\end{equation*}
On $\Sigma_i\times[t_i^\tau,2t_i^\tau]$ our hypotheses imply
\begin{equation*}
\sup |g_{\boldsymbol{t}}-{\tilde{g}}_i|_{
r^ { -2 } {\tilde{g}}_i\otimes{\tilde{g}}_i} \leq C_0.
\end{equation*}
The analogue is true also on $\Sigma_i\times[2t_i^\tau,\epsilon]$, using the
estimates provided by Definition \ref{def:metrics_ends}.
These perturbations are all $\boldsymbol{t}$-independent so according to
Theorem \ref{thm:weighted_ok} the weighted Sobolev Embedding Theorems
hold also for $g_{\boldsymbol{t}}$, with $\boldsymbol{t}$-independent Sobolev
constants.
\end{proof}
\begin{remark} Notice that Theorem
\ref{thm:normstequivalent} actually requires only $\boldsymbol{t}$-uniform
$C^0$-bounds
over the metrics $g_{\boldsymbol{t}}$. In Definition \ref{def:tconnectsum} we
include control
over the higher derivatives and the assumption that the quantities in
question tend to zero for use in later sections. The same is also true for
various other results, \textit{e.g.} Corollary \ref{cor:embedding}.
\end{remark}
We conclude with the following result which serves to highlight certain
properties
of $g_{\boldsymbol{t}}$ as $\boldsymbol{t}\rightarrow 0$. This is
important for Section \ref{s:sums_laplace}.
\begin{lemma}\label{l:neckestimate}
Consider $g_{\boldsymbol{t}}$ as in Definition \ref{def:tconnectsum}. Choose
a neck region in $L_{\boldsymbol{t}}$ and $b\in (0,\tau)$
so that $t_i\hat{R}<t_i^\tau<2t_i^\tau<t_i^b<\epsilon$.
Then, on $\Sigma_i\times [t_i\hat{R},t_i^b]$, the metric
$g_{\boldsymbol{t}}$ converges to the rescaled metric $t_i^2\hat{\phi}_{t,i}^*\hat{g}$ in the
following sense: for all $j\geq 0$ and as
$t\rightarrow 0$,
\begin{equation*}
\sup|r^j\hat{\nabla}^j(g_{\boldsymbol{t}}-t_i^2\hat{\phi}_{t,i}^*\hat{g})|_{
t_i^2\hat{\phi}_{t,i}^*\hat{g}\otimes t_i^2\hat{\phi}_{t,i}^*\hat{g}
}\rightarrow 0,
\end{equation*}
where $\hat{\nabla}$ denotes the Levi-Civita connection defined by
$\hat{\phi}_{t_i,i}^*\hat{g}$ on $\Sigma_i\times[t_i\hat{R},t_i^b]$.
\end{lemma}
\begin{proof}
Consider the map
\begin{equation*}
\delta_{t_i}:\Sigma_i\times[\hat{R},t_i^{b-1}]\rightarrow \Sigma_i\times
[t_i\hat{R},t_i^b], \ \ (\theta,r)\mapsto(\theta,t_ir).
\end{equation*}
We can use this map to pull the estimate back to $\Sigma_i\times[\hat{R},t_i^{b-1}]$. We can then write it as follows: for all $j\geq 0$ and as
$t\rightarrow 0$,
\begin{equation}\label{eq:neckestimate}
\sup
|\hat{\nabla}^j(\delta_{t_i}^*(t_i^{-2}g_{\boldsymbol{t}})-\hat{\phi}_i^*\hat{g}
)|_ { r^ { -2 }
\hat{\phi}_i^*\hat{g}\otimes\hat{\phi}_i^*\hat{g} } \rightarrow 0,
\end{equation}
where $\hat{\nabla}$ denotes the Levi-Civita connection defined by
$\hat{\phi}_i^*\hat{g}$ on $\Sigma_i\times[\hat{R},t^{b-1}]$.
We choose to prove this form of the estimate.
On $\Sigma_i\times [\hat{R},t_i^{\tau-1}]$ it
follows from Definition
\ref{def:tconnectsum} that
$\delta_{t_i}^*(t_i^{-2}g_{\boldsymbol{t}})=\hat{\phi}_i^*\hat{g}$
so the equation is trivially true.
On $\Sigma_i\times[t_i^{\tau-1},2t_i^{\tau-1}]$,
\begin{align*}
|{\widetilde{\nabla}}^j(\delta_{t_i}^*(t_i^{-2}g_{\boldsymbol{t}})-{\tilde{g}}_i)|_{r^{-2}
{\tilde{g}}_i\otimes{\tilde{g}}_i }
&=|{\widetilde{\nabla}}^j(\delta_{t_i}^*(t_i^{-2}g_{\boldsymbol{t}})-\delta_{t_i}^*(t_i^{-2}
{\tilde{g}}_i))|_
{\delta_{t_i}^*(r/{t_i})^{-2}\delta_{t_i}^*({t_i}^{-2}{\tilde{g}}_i)\otimes\delta_{
t_i}^*(t_i^ { -2 } {\tilde{g}}_i) } \\
&=\delta_{t_i}^*
\left(|{\widetilde{\nabla}}^j(t_i^{-2}g_{\boldsymbol{t}}-t_i^{-2}{\tilde{g}}_i)|_{(r/{t_i})^{-2}t_i^{
-2 } {\tilde{g}}_i\otimes
t_i^{-2}{\tilde{g}}_i}\right)\\
&=\delta_{t_i}^*\left(|{\widetilde{\nabla}}^j(g_{\boldsymbol{t}}-{\tilde{g}}_i)|_{r^{-2}
{\tilde{g}}_i\otimes{\tilde{g}}_i }
\right)\rightarrow 0,
\end{align*}
where the last statement follows from Definition \ref{def:tconnectsum}.
Furthermore, it follows from Definition \ref{def:metrics_ends} that
\begin{equation*}
|{\widetilde{\nabla}}^j(\hat{\phi}_i^*\hat{g}-{\tilde{g}}_i)|_{r^{-2}{\tilde{g}}_i\otimes{\tilde{g}}_i}\leq
C_j{t_i}^{(\tau-1)\hat{\nu}_i}\rightarrow 0,
\end{equation*}
using $(\tau-1)\hat{\nu}_i>0$. We have thus found that both metrics of interest
converge to the same metric ${\tilde{g}}_i$. The conclusion is a simple computation.
On $\Sigma_i\times[2t_i^{\tau-1},t_i^{b-1}]$, as above and using
$g_{\boldsymbol{t}}=\phi_i^*g$,
\begin{equation*}
|{\widetilde{\nabla}}^j(\delta_{t_i}^*(t_i^{-2}g_{\boldsymbol{t}})-{\tilde{g}}_i)|_{r^{-2}
{\tilde{g}}_i\otimes{\tilde{g}}_i }
=\delta_{t_i}^*\left(|{\widetilde{\nabla}}^j(\phi_i^*g-{\tilde{g}}_i)|_{r^{-2}{\tilde{g}}_i\otimes{\tilde{g}}_i}
\right)\leq
C_jt_i^{b\nu_i}\rightarrow 0 ,
\end{equation*}
using $b\nu_i>0$. Furthermore,
\begin{equation*}
|{\widetilde{\nabla}}^j(\hat{\phi}_i^*\hat{g}-{\tilde{g}}_i)|_{r^{-2}{\tilde{g}}_i\otimes{\tilde{g}}_i}\leq
C_j(2t_i^{\tau-1})^{\hat{\nu}_i}\rightarrow 0.
\end{equation*}
Again, combining these estimates implies the claim.
\end{proof}
\part{Elliptic estimates}
\section{Fredholm results for elliptic operators on
A.Cyl. manifolds}\label{s:acyl_analysis}
We now turn to the theory of elliptic operators via weighted Sobolev spaces,
focusing on Fredholm and index results. Results of this
kind have been proved by various authors, \textit{e.g.} Lockhart-McOwen
\cite{lockhartmcowen}, Lockhart \cite{lockhart} and Melrose \cite{melrose}. We
will follow the point of view of Lockhart and McOwen to which we refer for
details, see also Joyce-Salur \cite{joycesalur}. We start with the case of
A.Cyl.
manifolds. The theory requires appropriate assumptions on the asymptotic
behaviour of the operators, which we roughly summarize as follows.
\begin{definition}\label{def:acyl_bundles}
Given a manifold $\Sigma$, consider the projection
$\pi:\Sigma\times\mathbb{R}\rightarrow \Sigma$. A vector bundle $E_{\infty}$ on
$\Sigma\times\mathbb{R}$ is \textit{translation-invariant} if it is of the form
$\pi^*E'$, for some vector bundle $E'$ over $\Sigma$. We define the notion of
translation-invariant metrics and connections analogously.
Let $P_\infty:C^\infty(E_\infty)\rightarrow C^\infty(F_\infty)$ be a
differential
operator between translation-invariant vector bundles. We say that $P_\infty$ is
\textit{translation-invariant} if it commutes with the action of $\mathbb{R}$ on
$\Sigma\times\mathbb{R}$ determined by translations; equivalently, writing
$P_\infty=\sum
A_k^\infty\cdot\nabla^k$ with respect to a translation-invariant $\nabla$, if
the
coefficient tensors $A_k^\infty$ are independent of $z$.
Let $(L,h)$ be an A.Cyl. manifold with link $\Sigma=\amalg\Sigma_i$. Let $E$,
$F$ be vector bundles
over $L$. Assume there exist translation-invariant vector
bundles $E_\infty$, $F_\infty$ over
$\Sigma\times\mathbb{R}$ such that, using
the notation of Definition \ref{def:metrics_endsbis}, $\phi_i^*(E_{|S_i})$
(respectively, $\phi_i^*(F_{|S_i})$)
coincides with the restriction to $\Sigma_i\times (R',\infty)$ of $E_\infty$
(respectively, $F_\infty$).
Let
$P_\infty=\sum A_k^\infty\cdot\nabla^k:C^\infty(E_\infty)\rightarrow
C^\infty(F_\infty)$ be a translation-invariant linear differential operator of
order $n$. Consider a linear operator $P:C^\infty(E)\rightarrow C^\infty(F)$. We
say that $P$ is
\textit{asymptotic} to $P_\infty$ if on each end there exists $\nu_i<0$ such
that, writing $P=\sum
A_k\cdot\nabla^k$ (up to identifications) and as $z\rightarrow \infty$,
$$|\nabla^j(A_k-A_k^\infty)|=O(e^{\nu_i z}),$$
where $|\cdot|$ is defined by the translation-invariant metrics. We call $\nu_i$
the \textit{convergence rates} of the operator $P$.
In what follows, to define the spaces $W^p_{k,\boldsymbol{\beta}}(E)$,
we will assume that $E$ is endowed with a metric and a metric connection which
are asymptotic to the translation-invariant data on $E_\infty$, in the
appropriate sense.
\end{definition}
Assume $P$ is a linear operator of order $n$ with bounded coefficients $A_k$. It
follows from Definition \ref{def:acyl_sectionspaces} that, for all $p>1$, $k\geq
0$ and $\boldsymbol{\beta}$, $P$ extends to a continuous map
\begin{equation}\label{eq:contextension}
P:W^p_{k+n,\boldsymbol{\beta}}(E)\rightarrow W^p_{k,\boldsymbol{\beta}}(F).
\end{equation}
\begin{remark}\label{rem:Pbeta}
It will sometimes be useful to denote by $P_{\boldsymbol{\beta}}$ the extended
operator of Equation \ref{eq:contextension}, so as to emphasize the particular
weight being used.
\end{remark}
Now assume $P$ is asymptotic to a translation-invariant operator $P_\infty$.
Then Equation \ref{eq:contextension} holds also for the operator
$e^{-\boldsymbol{\nu}\zeta}(P-P_\infty)$, where $\boldsymbol{\nu}<0$ denotes the
convergence rates of $P$ as in Definition \ref{def:acyl_bundles}. This implies
that the operator $P-P_\infty$ extends to a continuous map
\begin{equation}\label{eq:differenceoperator}
P-P_\infty:W^p_{k+n,\boldsymbol{\beta}}(E)\rightarrow
W^p_{k,\boldsymbol{\beta}+\boldsymbol{\nu}}(F).
\end{equation}
Notice that if $\boldsymbol{\beta}<\boldsymbol{\beta'}$ then
$W^p_{k+n,\boldsymbol{\beta}}(E)\subset W^p_{k+n,\boldsymbol{\beta'}}(E)$ and
that the operator $P_{\boldsymbol{\beta'}}$ extends the operator
$P_{\boldsymbol{\beta}}$. Notice also that $C_c^\infty(E)\subset
W^p_{k,\boldsymbol{\beta}}(E)$ as a dense subset. Dualizing this relation allows
us to identify the dual space $(W^p_{k,\boldsymbol{\beta}}(E))^*$ with a
subspace of the space of distributions $(C^\infty_c(E))^*$. It is customary to
denote this space $W^{p'}_{-k,-\boldsymbol{\beta}}(E)$. Endowed with the
appropriate norm, it again contains $C^\infty_c(E)$ as a dense subset. The
duality
map $W^{p'}_{-k,-\boldsymbol{\beta}}(E)\times
W^p_{k,\boldsymbol{\beta}}(E)\rightarrow\mathbb{R}$, restricted to this subset,
coincides with the map
\begin{equation}\label{eq:duality}
C^\infty_c(E)\times W^p_{k,\boldsymbol{\beta}}(E)\rightarrow\mathbb{R},\ \
<\sigma,\sigma'>:=\int_L(\sigma,\sigma')_E\,\mbox{vol}_h.
\end{equation}
This map extends by continuity to a map defined on
$W^{p'}_{l,-\boldsymbol{\beta}}(E)\times W^p_{k,\boldsymbol{\beta}}(E)$ for all
$l\geq 0$, showing that $W^{p'}_{-k,-\boldsymbol{\beta}}(E)$ also contains all
spaces $W^{p'}_{l,-\boldsymbol{\beta}}(E)$.
It can be shown that $P$ admits continuous extensions as in Equation
\ref{eq:contextension} for any $k\in\mathbb{Z}$.
\begin{lemma}\label{l:*=*}
Let $P:C^\infty(E)\rightarrow C^\infty(F)$ be a linear differential operator of
order $n$, asymptotic to a translation-invariant operator $P_\infty$. Let
$P^*:C^\infty(F)\rightarrow C^\infty(E)$ denote its formal adjoint. Consider the
continuous extension of $P^*$ to the spaces
\begin{equation}
P^*:W^{p'}_{-k,-\boldsymbol{\beta}}(F)\rightarrow
W^{p'}_{-k-n,-\boldsymbol{\beta}}(E).
\end{equation}
Under the identification of Sobolev spaces of negative order with dual spaces,
this operator coincides with the operator dual to that of Equation
\ref{eq:contextension},
\begin{equation}
P^*:(W^p_{k,\boldsymbol{\beta}}(F))^*\rightarrow
(W^p_{k+n,\boldsymbol{\beta}}(E))^*.
\end{equation}
Furthermore if $E=F$ and $P$ is self-adjoint, \textit{i.e.} $P=P^*$ on smooth
compactly-supported sections, then $P=P^*$ on any space
$W^p_{k,\boldsymbol{\beta}}$.
\end{lemma}
\begin{proof}
The formal adjoint of $P$ is asymptotic to the formal adjoint of
$P_\infty$, so the extensions exist as specified. The statement of this lemma
can be clarified by adopting the notation of Remark \ref{rem:Pbeta}: the claim
is then that $(P^*)_{-\boldsymbol{\beta}}=(P_{\boldsymbol{\beta}})^*$, where on
the left the superscript $*$ denotes the formal adjoint and on the right it
denotes the dual map.
Since both maps are continuous, it is sufficient to show that they coincide on a
dense subset: in particular that
$(P^*)_{-\boldsymbol{\beta}}(\tau)=(P_{\boldsymbol{\beta}})^*(\tau)$, for all
$\tau\in C^\infty_c(F)$. Since we are identifying
$(P^*)_{-\boldsymbol{\beta}}(\tau)$ with an element of the dual space
$(W^p_{k+n,\boldsymbol{\beta}}(E))^*$, we can again invoke continuity to claim
that it is sufficient to prove that, for all $e\in C^\infty_c(E)$,
\begin{equation}
\langle (P^*)_{-\boldsymbol{\beta}}(\tau), e\rangle=\langle
(P_{\boldsymbol{\beta}})^*(\tau),e\rangle.
\end{equation}
This claim is now a direct consequence of the definitions and of Equation
\ref{eq:duality}.
The claim concerning self-adjoint operators is a simple consequence of
continuity.
\end{proof}
\begin{remark} \label{rem:nestlingtheker}
As already remarked, $\boldsymbol{\beta'}>\boldsymbol{\beta}$ implies
$P_{\boldsymbol{\beta'}}$ extends $P_{\boldsymbol{\beta}}$. This shows that the
spaces $\mbox{Ker}(P_{\boldsymbol{\beta}})$ grow with $\boldsymbol{\beta}$. On
the other hand, as a vector space, the cokernel of $P$ in Equation
\ref{eq:contextension} is not canonically a subspace of
$W^p_{k,\boldsymbol{\beta}}(F)$ so there is no canonical way of relating
cokernels corresponding to different weights. However, consider the following
construction, for which we assume $P$, $P^*$ are Fredholm. Pick $
\tau_1\in W^p_{k,\boldsymbol{\beta}}(F)$ such that $\langle
\sigma,\tau_1\rangle\neq 0$, for some $\sigma\in \mbox{Ker}(P^*)$. According to
Remark \ref{rem:characterizations} this implies that $\tau_1$ does not belong to
$\mbox{Im}(P)$. By density we can then find $\tilde{\tau_1}$ which is smooth and
compactly-supported and does not belong to $\mbox{Im}(P)$. Now choose $\tau_2$
satisfying $\langle \sigma,\tau_2\rangle\neq 0$ for some $\sigma\in
\mbox{Ker}(P^*)$ and which is linearly independent of $\tau_1$, \textit{etc}.
After a finite number of steps we will have found a vector space spanned by
$\tilde{\tau_1},\dots,\tilde{\tau_k}$ which defines a complement to
$\mbox{Im}(P)$ and thus is isomorphic to $\mbox{Coker}(P)$. Notice that by
construction $\tilde{\tau_i}$ belong to all spaces
$W^p_{k,\boldsymbol{\beta}}(F)$. On the other hand, as $\boldsymbol{\beta}$
decreases the dual weight $-\boldsymbol{\beta}$ increases, so $\mbox{Ker}(P^*)$
increases, so the $\tilde{\tau_i}$ chosen for the weight $\boldsymbol{\beta}$
can be used also for any weight $\boldsymbol{\beta'}<\boldsymbol{\beta}$. The
conclusion is that we can construct spaces representing the cokernel which grow
as $\boldsymbol{\beta}$ decreases, \textit{i.e.} as the function spaces become
smaller.
\end{remark}
Now assume $P$ is elliptic. We are interested in conditions ensuring that the
extended map of Equation \ref{eq:contextension} is Fredholm.
\begin{definition}\label{def:exceptional} Let $\Sigma$ be a compact oriented
Riemannian manifold with connected components $\Sigma_1,\dots,\Sigma_e$. Let
$P_\infty$ be a translation-invariant operator on $\Sigma\times\mathbb{R}$.
Consider the complexified operator $P_\infty:E_\infty\otimes \mathbb{C}\rightarrow
F_\infty\otimes \mathbb{C}$. Choose a connected component $\Sigma_j\times\mathbb{R}$ and fix
$\gamma+i\delta\in\mathbb{C}$. Let us restrict our attention to the space of sections of
$E_\infty\otimes\mathbb{C}$ of the form $e^{(\gamma+i\delta)z}\sigma(\theta)$. Consider
the subspace $V^j_{\gamma+i\delta}$ determined by the solutions to the problem
$P_\infty(e^{(\gamma+i\delta)z}\sigma(\theta))=0$ on $\Sigma_j\times\mathbb{R}$. We
define the space $\mathcal{C}^j_{P_\infty}\subseteq\mathbb{C}$ to be the space of all
$\gamma+i\delta$ such that $V^j_{\gamma+i\delta}\neq 0$. We then define the
space of \textit{exceptional weights} for $P_\infty$ on $\Sigma_j\times\mathbb{R}$ to be
the corresponding set of real values,
$\mathcal{D}_{P_\infty}^j:=\mbox{Re}(\mathcal{C}^j_{P_\infty})\subseteq\mathbb{R}$.
Now fix a multi-index $\boldsymbol{\gamma}+i\boldsymbol{\delta}\in \mathbb{C}^e$. Let
$V_{\boldsymbol{\gamma}+i\boldsymbol{\delta}}:=\oplus_{j=1}^e
V^j_{\gamma_j+i\delta_j}$. We define the space of \textit{exceptional weights}
for $P_\infty$ on $\Sigma\times\mathbb{R}$, denoted $\mathcal{D}_{P_\infty}\subseteq
\mathbb{R}^e$, to be the set of multi-indices $\gamma=(\gamma_1,\dots,\gamma_e)$ such
that, for some $j$, $\gamma_j\in\mathcal{D}_{P_\infty}^j$.
\end{definition}
\begin{remark}\label{rem:exceptional}
Definition \ref{def:exceptional} introduces the exceptional weights via the
kernel of $P_\infty$ and the space of sections with exponential growth. Along
the lines of \cite{lockhartmcowen}, the exceptional weights can equivalently be
defined as follows. Separating the $\partial\theta$ derivatives from the
$\partial z$ derivatives and setting $Dz=-i\partial z$, we can write
\begin{equation}
P_\infty=\sum A_k(\theta,\partial\theta)(\partial z)^k=\sum
A_k(\theta,\partial\theta)i^k(Dz)^k,
\end{equation}
where, to simplify the notation, $\partial\theta$ denotes any combination of
derivatives in the $\theta$ variables.
For any $\lambda\in\mathbb{C}$, set $P_\lambda:=\sum
A_k(\theta,\partial\theta)i^k\lambda^k$. Notice that
\begin{equation}
P_\infty(e^{i\lambda z}\sigma(\theta))=\sum
A_k(\theta,\partial\theta)(i\lambda)^k\sigma e^{i\lambda z}=(P_\lambda (\sigma))
e^{i\lambda z}
\end{equation}
so $P_\infty(e^{i\lambda z}\sigma(\theta))=0$ iff $P_\lambda(\sigma)=0$. We view
the latter as a \textit{generalized eigenvalue problem} on $\Sigma$ and say that
$\lambda$ is an \textit{eigenvalue} iff the corresponding generalized eigenvalue
problem admits non-trivial solutions. It follows from the above calculations
that a weight $\gamma\in\mathbb{R}$ is exceptional in the sense of Definition
\ref{def:exceptional} iff $-\gamma=\mbox{Im}(\lambda)$, for some eigenvalue
$\lambda$.
\end{remark}
For elliptic operators it turns out that the exceptional weights of $P_\infty$
determine the possible Fredholm extensions of any $P$ asymptotic to $P_\infty$.
\begin{theorem}\label{thm:acyl_fredholm}
Let $(L,h)$ be an A.Cyl. manifold with link $\Sigma=\amalg \Sigma_i$.
Let
$P:C^\infty(E)\rightarrow C^\infty(F)$ be a linear elliptic operator of order
$n$, asymptotic to an elliptic operator
$P_\infty$.
Then each $\mathcal{D}^j_{P_\infty}$ is discrete in $\mathbb{R}$ so
$\mathcal{D}_{P_\infty}$ defines a discrete set of hyperplanes in $\mathbb{R}^e$.
Furthermore, for each $p>1$ and $k\geq 0$, the extended operator
$P_{\boldsymbol{\gamma}}:W^p_{k+n,\boldsymbol{\gamma}}(E)\rightarrow
W^p_{k,\boldsymbol{\gamma}}(F)$ is Fredholm iff $\boldsymbol{\gamma}\notin
\mathcal{D}_{P_\infty}$.
\end{theorem}
In a similar vein, we can compute how the index of $P$ depends on
$\boldsymbol{\gamma}$.
\begin{definition}\label{def:indexchange}
Consider the complexified operator $P_\infty:E_\infty\otimes \mathbb{C}\rightarrow
F_\infty\otimes \mathbb{C}$. Choose a connected component $\Sigma_j\times\mathbb{R}$ of
$\Sigma\times\mathbb{R}$ and fix $\gamma+i\delta\in\mathcal{C}^j_{P_\infty}$. We denote
by $\widetilde{V}^j_{\gamma+i\delta}$ the space of solutions to the problem
$P_\infty(e^{(\gamma+i\delta)z}\sigma(\theta,z))=0$ on $\Sigma_j\times\mathbb{R}$, where
$\sigma(\theta,z)$ is polynomial in $z$. We can extend this definition to all
$\gamma+i\delta$ by setting $\widetilde{V}^j_{\gamma+i\delta}=\{0\}$ if
$\gamma+i\delta\notin \mathcal{C}^j_{P_\infty}$. Notice that
$V^j_{\gamma+i\delta}\leq\widetilde{V}^j_{\gamma+i\delta}$. Given any
$\gamma\in\mathbb{R}$ we now set
$\widetilde{V}^j_\gamma:=\bigoplus_{\delta\in\mathbb{R}}\widetilde{V}^j_{\gamma+i\delta}
$, then define the \textit{multiplicity} of $\gamma$ on $\Sigma_j\times\mathbb{R}$ by
$m^j_{P_\infty}(\gamma):=\mbox{dim}(\widetilde{V}^j_\gamma)$.
Now fix a multi-index $\boldsymbol{\gamma}\in\mathbb{R}^e$. We define the
\textit{multiplicity} of $\boldsymbol{\gamma}$ on $\Sigma\times\mathbb{R}$ to be
$m_{P_\infty}(\boldsymbol{\gamma}):=\sum_{j=1}^e m^j_{P_\infty}(\gamma_j)$.
\end{definition}
\begin{theorem} \label{thm:acyl_indexchange}
In the setting of Theorem \ref{thm:acyl_fredholm}, each multiplicity
$m_{P_\infty}(\boldsymbol{\gamma})$ is finite. Furthermore, choose
$\boldsymbol{\gamma}_1,\boldsymbol{\gamma}_2\in \mathbb{R}^e\setminus
\mathcal{D}_{P_\infty}$ with $\boldsymbol{\gamma}_1\leq\boldsymbol{\gamma}_2$.
Then
$$i_{\boldsymbol{\gamma}_2}(P)-i_{\boldsymbol{\gamma}_1}(P)=\sum_{\boldsymbol{
\gamma}\in\mathcal{D}_{P_\infty},
\boldsymbol{\gamma}_1\leq\boldsymbol{\gamma}\leq\boldsymbol{\gamma}_2}
m_{P_\infty}(\boldsymbol{\gamma}).$$
\end{theorem}
\begin{remark} Assume we can compute the value of
$i_{\boldsymbol{\gamma}}(P)$ for a specific good choice of non-exceptional
$\boldsymbol{\gamma}$. Theorem \ref{thm:acyl_indexchange} then allows us to
compute $i_{\boldsymbol{\gamma}}(P)$ for all non-exceptional
$\boldsymbol{\gamma}$ in terms of data on the link.
\end{remark}
The following result is proved in \cite{lockhartmcowen} Section 7, cf. also
\cite{joycesalur}, as a consequence of the Sobolev Embedding and
change of index theorems.
\begin{prop}\label{prop:acylindep}
In the setting of Theorem \ref{thm:acyl_indexchange}, assume
$\boldsymbol{\gamma}$ and $\boldsymbol{\gamma}'$ belong to the same connected
component of $\mathbb{R}^e\setminus\mathcal{D}_{P_\infty}$. Then
$i_{\boldsymbol{\gamma}}(P)=i_{\boldsymbol{\gamma}'}(P)$ and
$\mbox{Ker}(P_{\boldsymbol{\gamma}})=\mbox{Ker}(P_{\boldsymbol{\gamma}'})$.
Furthermore, the index and kernel are independent of the choice of $p$ and $k$.
\end{prop}
\begin{example} \label{e:acyl_harmonic}
Assume $(L,h)$ is an A.Cyl. manifold with one end with link $(\Sigma,g')$. Let
$P:=\Delta_h$ denote the positive Laplace operator on functions. Then $P$ is
asymptotic to the Laplace operator $\Delta_{\tilde{h}}$ defined on the product
$(\Sigma\times\mathbb{R}, \tilde{h}:=dz^2+g')$. One can check that
$\Delta_{\tilde{h}}=-(\partial z)^2+\Delta_{g'}$ and that
$\Delta_{\tilde{h}}e^{(\gamma+i\delta)z}\sigma(\theta)=0$ iff $\delta=0$ and
$\Delta_{g'}\sigma=\gamma^2\sigma$. In other words, the harmonic functions on
the cylinder which have exponential growth are generated by the eigenvalues of
$\Delta_{g'}$. In particular, the exceptional weights for $\Delta_h$ are of the
form $\pm\sqrt{e_n}$, where $e_n$ are the eigenvalues of $\Delta_{g'}$.
\end{example}
\section{Weight-crossing}\label{s:weightcrossing}
Let $(L,h)$ be an A.Cyl. manifold. Let $P:C^\infty(E)\rightarrow C^\infty(F)$ be
a linear elliptic operator asymptotic to some $P_\infty$ as in Definition
\ref{def:acyl_bundles}. Consider the extension of $P$ to weighted Sobolev spaces
as in Equation \ref{eq:contextension}. When $\boldsymbol{\beta}$ changes value
crossing an exceptional weight the change of index formula given in Theorem
\ref{thm:acyl_indexchange} leads us to expect that the kernel and/or cokernel of
$P$ will change. Specifically, when $\boldsymbol{\beta}$ increases we expect the
kernel of $P$ to increase and the cokernel to decrease. The process by which
this occurs can be formalized using the Fredholm and index results stated in
Section \ref{s:acyl_analysis}. The notation we rely on was introduced in
Definitions \ref{def:exceptional} and \ref{def:indexchange}. To simplify the
notation, throughout this section we forgo the distinction between bundles (or
operators) and their complexifications.
Literally speaking, given any index $\gamma\in\mathbb{R}$ and end $S_j$, the sections in
each $\widetilde{V}^j_{\gamma}$ are defined on $\Sigma_j\times\mathbb{R}$. Using the
identification $\phi_j$, we can alternatively think of them as being defined on
$S_j$. However, we can also think of them as being globally defined on $L$ by
first choosing a basis of sections $\sigma_i^j$ for each
$\widetilde{V}^j_{\gamma}$, then interpolating between them so as to get smooth
extensions $\sigma_i^j$ over $L$. In particular it may be useful to choose the
extension of each $\sigma_i^j$ so that it is identically zero on the other
ends. The construction implies that each
$P_\infty(\sigma_i^j)$ has compact support. By choosing the extensions
generically over $L\setminus S$ we can assume that all $P(\sigma_i^j)$ are
linearly independent. This implies that $P$ is injective on
$\widetilde{V}_{\boldsymbol{\gamma}}$.
Now assume $\boldsymbol{\gamma}\in\mathbb{R}^e$ is exceptional. Then, for any
$\boldsymbol{\nu}<0$ with $|\boldsymbol{\nu}|<<1$,
\begin{equation}
P:W^p_{k+n,\boldsymbol{\gamma}+\boldsymbol{\nu}}(E)\rightarrow
W^p_{k,\boldsymbol{\gamma}+\boldsymbol{\nu}}(F)
\end{equation}
is Fredholm. In particular, let $\boldsymbol{\nu}<0$ be the convergence rates of
$P$ as in Definition \ref{def:acyl_bundles}. We will assume that
$|\boldsymbol{\nu}|<<1$ as above. Writing
$P(\sigma)=(P-P_\infty)(\sigma)+P_\infty (\sigma)$ and using Equation
\ref{eq:differenceoperator} then shows that
$P(\widetilde{V}_{\boldsymbol{\gamma}})\subset
W^p_{k,\boldsymbol{\gamma}+\boldsymbol{\nu}}(F)$.
Since $P$ is injective on $\widetilde{V}_{\boldsymbol{\gamma}}$ we can define a
decomposition
\begin{equation}
\widetilde{V}_{\boldsymbol{\gamma}}=\widetilde{V}_{\boldsymbol{\gamma}}'\oplus
\widetilde{V}_{\boldsymbol{\gamma}}''
\end{equation}
by defining
$P(\widetilde{V}_{\boldsymbol{\gamma}}'):=P(\widetilde{V}_{\boldsymbol{\gamma}}
)\cap\mbox{Im}(P_{\boldsymbol{\gamma}+\boldsymbol{\nu}})$ and choosing any
complement $\widetilde{V}_{\boldsymbol{\gamma}}''$. By definition,
$P(\widetilde{V}_{\boldsymbol{\gamma}}'')\cap\mbox{Im}(P_{\boldsymbol{\gamma}
+\boldsymbol{\nu}})=0$. In other words, we can think of
$P(\widetilde{V}_{\boldsymbol{\gamma}}'')$ as belonging to the cokernel of
$P_{\boldsymbol{\gamma}+\boldsymbol{\nu}}$. On the other hand,
$P(\widetilde{V}_{\boldsymbol{\gamma}}'')$ belongs to the image of
$P_{\boldsymbol{\gamma}-\boldsymbol{\nu}}$ because
$\widetilde{V}_{\boldsymbol{\gamma}}\subset
W^p_{k+n,\boldsymbol{\gamma}-\boldsymbol{\nu}}(E)$ . Roughly speaking,
$P(\widetilde{V}_{\boldsymbol{\gamma}}'')$ thus describes the portion of the
cokernel
of $P$ which ``disappears" when crossing the exceptional weight
$\boldsymbol{\gamma}$.
By construction, for any $\sigma\in \widetilde{V}_{\boldsymbol{\gamma}}'$ there
exists $u_\sigma\in W^p_{k+n,\boldsymbol{\gamma}+\boldsymbol{\nu}}(E)$ such that
$P(\sigma)=P (u_\sigma)$. Notice that $u_\sigma$ is not necessarily uniquely
defined. However it is sufficient to fix a choice of $u_{\sigma}$ for each
element of a basis of $\widetilde{V}_{\boldsymbol{\gamma}}'$ to obtain a unique
choice of $u_\sigma$ for any $\sigma\in \widetilde{V}_{\boldsymbol{\gamma}}'$.
Notice also that $\sigma-u_\sigma\in
W^p_{k+n,\boldsymbol{\gamma}-\boldsymbol{\nu}}(E)$. We have thus defined a map
\begin{equation}\label{eq:newkernel}
\widetilde{V}_{\boldsymbol{\gamma}}'\rightarrow\mbox{Ker}(P_{\boldsymbol{\gamma}
-\boldsymbol{\nu}}), \ \
\sigma\mapsto\sigma-u_\sigma\notin
W^p_{k+n,\boldsymbol{\gamma}+\boldsymbol{\nu}}(E).
\end{equation}
The image of the map of Equation \ref{eq:newkernel} thus defines a space of
``new" elements in $\mbox{Ker}(P)$, generated by crossing the exceptional weight
$\boldsymbol{\gamma}$. Notice that $u_\sigma$ is of strictly lower order of
growth compared to $\sigma$. This shows that the map of Equation
\ref{eq:newkernel} is injective and that the elements in its image admit an
asymptotic expansion of the form $e^{\boldsymbol{\gamma}\zeta}+\mbox{lower
order}$. The following result shows that every new element in $\mbox{Ker}(P)$
arises this way.
\begin{lemma}
Let us identify $\widetilde{V}_{\boldsymbol{\gamma}}'$ with its image under the
map of Equation \ref{eq:newkernel}. Then
$$\mbox{Ker}(P_{\boldsymbol{\gamma}-\boldsymbol{\nu}})=\mbox{Ker}(P_{\boldsymbol
{\gamma}+\boldsymbol{\nu}})\oplus \widetilde{V}_{\boldsymbol{\gamma}}'.$$
\end{lemma}
\proof{}By injectivity, the inequality $\supseteq$ is clear. To prove the lemma
it is thus sufficient to prove that the inverse inequality holds on the
corresponding dimensions. Choose any $\sigma\in
\widetilde{V}_{\boldsymbol{\gamma}}''$. According to Remark
\ref{rem:characterizations},
\begin{eqnarray*}
P(\sigma)\in \mbox{Im}(P_{\boldsymbol{\gamma}-\boldsymbol{\nu}})\Leftrightarrow
\langle\tau,P(\sigma)\rangle=0,\ \forall \tau\in
\mbox{Ker}(P^*_{-\boldsymbol{\gamma}+\boldsymbol{\nu}}),\\
P(\sigma)\in \mbox{Im}(P_{\boldsymbol{\gamma}+\boldsymbol{\nu}})\Leftrightarrow
\langle\tau,P(\sigma)\rangle=0,\ \forall \tau\in
\mbox{Ker}(P^*_{-\boldsymbol{\gamma}-\boldsymbol{\nu}}).
\end{eqnarray*}
From the definition of $\widetilde{V}_{\boldsymbol{\gamma}}''$ we know that
$P(\sigma)\in \mbox{Im}(P_{\boldsymbol{\gamma}-\boldsymbol{\nu}})$ and that
$P(\sigma)\notin\mbox{Im}(P_{\boldsymbol{\gamma}+\boldsymbol{\nu}})$ unless
$\sigma=0$. Notice also that
$\mbox{Ker}(P^*_{-\boldsymbol{\gamma}+\boldsymbol{\nu}})\subseteq\mbox{Ker}(P^*_
{-\boldsymbol{\gamma}-\boldsymbol{\nu}})$. We conclude that the following map is
well-defined:
\begin{equation}
\frac{\mbox{Ker}(P^*_{-\boldsymbol{\gamma}-\boldsymbol{\nu}})}{\mbox{Ker}(P^*_{
-\boldsymbol{\gamma}+\boldsymbol{\nu}})}\times
\widetilde{V}_{\boldsymbol{\gamma}}'', \ \ ([\tau],\sigma)\mapsto
\langle\tau,P(\sigma)\rangle,
\end{equation}
and that the corresponding map
\begin{equation}
\widetilde{V}_{\boldsymbol{\gamma}}''\rightarrow
\left(\frac{\mbox{Ker}(P^*_{-\boldsymbol{\gamma}-\boldsymbol{\nu}})}{\mbox{Ker}
(P^*_{-\boldsymbol{\gamma}+\boldsymbol{\nu}})}\right)^*
\end{equation}
is injective. This proves that
\begin{equation}\label{eq:dimcalc}
\mbox{dim}(\widetilde{V}_{\boldsymbol{\gamma}}'')\leq
\mbox{dim}(\mbox{Ker}(P^*_{-\boldsymbol{\gamma}-\boldsymbol{\nu}}))-\mbox{dim}
(\mbox{Ker}(P^*_{-\boldsymbol{\gamma}+\boldsymbol{\nu}})).
\end{equation}
On the other hand, the change of index formula shows that
\begin{align}\label{eq:indexcalc}
\mbox{dim}(\widetilde{V}_{\boldsymbol{\gamma}}')+\mbox{dim}(\widetilde{V}_{
\boldsymbol{\gamma}}'')&=\mbox{dim}(\mbox{Ker}(P_{\boldsymbol{\gamma}
-\boldsymbol{\nu}}))-\mbox{dim}(\mbox{Ker}(P^*_{-\boldsymbol{\gamma}+\boldsymbol
{\nu}}))\\
&\ \
-\mbox{dim}(\mbox{Ker}(P_{\boldsymbol{\gamma}+\boldsymbol{\nu}}))+\mbox{dim}
(\mbox{Ker}(P^*_{-\boldsymbol{\gamma}-\boldsymbol{\nu}})).\nonumber
\end{align}
Substracting Equation \ref{eq:dimcalc} from Equation \ref{eq:indexcalc} proves
the desired inequality.
\endproof
\section{Fredholm results for elliptic operators on
conifolds}\label{s:accs_analysis}
We now want to see how to achieve analogous results for certain elliptic
operators on conifolds. In parallel with Section \ref{s:acyl_analysis} it
is possible to develop an abstract definition and theory of
\textit{asymptotically conical} operators, analogous to that of asymptotically
translation-invariant operators on A.Cyl. manifolds. For simplicity, however, we
will limit ourselves to the special case of the Laplace
operator acting on functions. This already
contains the main ideas of the general theory.
Let $(L,g)$ be a conifold. Consider the weighted spaces introduced in
Definition \ref{def:csac_sectionspaces}. As in Section \ref{s:acyl_analysis} we
denote the dual space $(W^p_{k,\boldsymbol{\beta}})^*$ by
$W^{p'}_{-k,-\boldsymbol{\beta}-m}$. This choice of weights is compatible with
the identifications of Remark \ref{rem:spacescoincide}, and the properties of
these dual spaces are analogous to those seen in Section \ref{s:acyl_analysis}.
It follows directly from the definitions that
\begin{equation*}
\nabla:W^p_{k,\boldsymbol{\beta}}\rightarrow W^p_{k-1,\boldsymbol{\beta}-1}
\end{equation*}
is a continuous operator. Equation \ref{eq:laplace} then implies that $\Delta_g$
extends to a continuous map
\begin{equation}\label{eq:extendedlaplacian}
\Delta_{\boldsymbol{\beta}}:W^p_{k,\boldsymbol{\beta}}\rightarrow
W^p_{k-2,\boldsymbol{\beta}-2}.
\end{equation}
The following result is closely related to Lemma \ref{l:*=*} and uses the fact
that $\Delta_g$ is formally self-adjoint.
\begin{lemma} \label{l:intbyparts}
Let $(L,g)$ be a conifold. Choose $u\in W^p_{k,\boldsymbol{\beta}}$, $v\in
W^{p'}_{2-k, 2-\boldsymbol{\beta}-m}$. Then
\begin{equation}\label{eq:intbyparts}
\langle v,\Delta_g u\rangle=\langle dv,du\rangle=\langle \Delta_g
v,u\rangle.\end{equation}
\end{lemma}
\begin{proof}
Using the appropriate dualities, each expression in Equation \ref{eq:intbyparts}
defines by composition a continuous bilinear map $(u,v)\in W^p_{k,\beta}\times
W^{p'}_{2-k, 2-\beta-m}\rightarrow\mathbb{R}$. Since $\Delta_g=d^*d$ the equalities hold
on the dense subsets $C^\infty_c\times C^\infty_c$. By continuity the equalities
thus continue to hold on the full Sobolev spaces.
\end{proof}
We now want to investigate the Fredholm properties of
$\Delta_{\boldsymbol{\beta}}$. It is initially useful to distinguish between the
AC and CS case. To begin, let $(L,g)$ be an AC manifold with ends $S_j$ and
links $\Sigma_j$. The starting point for the Fredholm theory is then the
following observation.
\begin{lemma} \label{l:conformalac=acyl}
Let $(\Sigma,g')$ be a Riemannian manifold. Let the corresponding cone
$C:=\Sigma\times (0,\infty)$ have the conical metric ${\tilde{g}}:=dr^2+r^2g'$. Let
$\Delta_{{\tilde{g}}}$ denote the corresponding Laplace operator on functions. Then,
under the substitution $r=e^z$, the operator $r^2\Delta_{{\tilde{g}}}$ coincides with
the translation-invariant operator
\begin{equation}\label{eq:rescaledlaplacian}
P_\infty:=-(\partial z)^2+(2-m)\partial z+\Delta_\Sigma
\end{equation}
on the cylinder $\Sigma\times\mathbb{R}$.
\end{lemma}
\proof{}
Recall that in any local coordinate system the Laplace operator on functions is
given by the formula
\begin{equation}\label{eq:locallaplacian}
\Delta_g=-\frac{1}{\sqrt{g}}\partial_j(\sqrt{g}g^{ij}\partial_i).
\end{equation}
Let $U$ be a local chart on $\Sigma$ so that $U\times (0,\infty)$ is a local
chart on $C$. Equation \ref{eq:locallaplacian} then shows that
\begin{equation}
\Delta_{{\tilde{g}}}=-(\partial r)^2-\frac{m-1}{r}\partial r+r^{-2}\Delta_\Sigma.
\end{equation}
The substitution $r=e^z$ implies $r\partial r=\partial z$. The claim is then a
simple calculation.
\endproof
Lemma \ref{l:conformalac=acyl} allows us to study the Fredholm properties of
$\Delta_g$ by building an equivalent problem for an A.Cyl. manifold, as follows.
We use the notation of Section \ref{s:acyl_analysis}.
Multiplication by $\rho^2$ defines an isometry
$W^p_{k-2,\boldsymbol{\beta}-2}\simeq W^p_{k-2,\boldsymbol{\beta}}$. Thus
$\Delta_{\boldsymbol{\beta}}$ in Equation \ref{eq:extendedlaplacian} is Fredholm
iff the operator
\begin{equation}\label{eq:extendedrescaledlaplacian}
\rho^2\Delta_{\boldsymbol{\beta}}:W^p_{k,\boldsymbol{\beta}}\rightarrow
W^p_{k-2,\boldsymbol{\beta}}
\end{equation}
is Fredholm. Now consider the A.Cyl. manifold $(L,h)$, where $h=\rho^{-2}g$. It
follows from Equation \ref{eq:laplace} and Lemma \ref{l:conformalac=acyl} that
the operator $P:=\rho^2\Delta_g$ is asymptotic in the sense of Definition
\ref{def:acyl_bundles} to the translation-invariant operator $P_\infty$ of
Equation \ref{eq:rescaledlaplacian}. One can check that the convergence rate
$\boldsymbol{\nu}$ of $P$ coincides with the convergence rate $\boldsymbol{\nu}$
of the AC manifold, cf. Definition \ref{def:metrics_ends}.
It is simple to verify that the equation
$P_\infty(e^{(\gamma+i\delta)z}\sigma(\theta))=0$ is equivalent to the following
eigenvalue problem on the link:
\begin{equation}\label{eq:ac_harmonicbis}
\Delta_{\Sigma_j}\sigma=[(\gamma+i\delta)^2+(m-2)(\gamma+i\delta)]\sigma.
\end{equation}
Using the fact that the eigenvalues $e_n^j$ of $\Delta_{\Sigma_j}$ are real and
non-negative, it follows that $\delta=0$ and that $\gamma$ satisfies
$\gamma^2+(m-2)\gamma=e_n^j$ for some $n$, \textit{i.e.}
\begin{equation}\label{eq:exceptionalforlaplacian}
\gamma=\frac{(2-m)\pm\sqrt{(2-m)^2+4e_n^j}}{2}.
\end{equation}
This shows that, for this particular operator,
$\mathcal{C}^j_{P_\infty}=\mathcal{D}^j_{P_\infty}$. It also follows from Lemma
\ref{l:conformalac=acyl} that the equation $P_\infty(e^{\gamma
z}\sigma(\theta))=0$ is equivalent to $\Delta_{{\tilde{g}}}(r^{\gamma}\sigma)=0$. Thus
\begin{equation}\label{eq:ac_harmonicter}
V^j_{\gamma}=\{r^\gamma\sigma(\theta): \Delta_{\tilde{g}}(r^{\gamma}\sigma)=0\},
\end{equation}
\textit{i.e.} $V^j_{\gamma}$ coincides with the space of homogeneous harmonic
functions of degree $\gamma$ on the cone $\Sigma_j\times (0,\infty)$.
Varying the choice of eigenvalue $e_n^j$ gives the set of exceptional weights
for $P_\infty$ on the end $S_j$. Repeating this for each end defines the set
$\mathcal{D}_{P_\infty}\subset\mathbb{R}^e$. According to Theorem
\ref{thm:acyl_fredholm} these are the weights for which the operator $P$ is not
Fredholm with respect to the Sobolev spaces of $(L,h)$. However, recall from
Remark \ref{rem:spacescoincide} that the Sobolev spaces of $(L,g)$ and $(L,h)$
coincide. Thus $\mathcal{D}_{P_\infty}\subset\mathbb{R}^e$ are also the weights for
which the operators of Equations \ref{eq:extendedrescaledlaplacian},
\ref{eq:extendedlaplacian} are not Fredholm.
\begin{remark}
Notice that in this particular case (and in the analogous case presented in
Example \ref{e:acyl_harmonic}) the generalized eigenvalue problem introduced in
Remark \ref{rem:exceptional} has reduced to an eigenvalue problem in the usual
sense.
\end{remark}
It is also fairly straight-forward to verify that, for this operator $P_\infty$,
the spaces $\widetilde{V}^j_{\gamma+i\delta}$ and $V^j_{\gamma+i\delta}$
coincide, cf. Joyce \cite{joyce:I} Proposition 2.4 for details. This allows us
to simplify the definition of the multiplicity $m(\boldsymbol{\gamma})$.
\
The situation for CS manifolds is similar. The change of variables $r=e^{-z}$
introduces a change of sign in Equation \ref{eq:rescaledlaplacian}. This sign is
later cancelled by a change of sign in the identification of Sobolev spaces of
$(L,g)$ and $(L,h)$. The final result is thus identical to the AC case.
Combining these results leads to the following conclusion.
\begin{corollary}\label{cor:laplaceresults}
Let $(L,g)$ be a conifold with $e$ ends. For each end $S_j$ with
link $\Sigma_j$ let $e^j_n$ denote the eigenvalues of the positive Laplace
operator $\Delta_{\Sigma_j}$ and define the set of ``exceptional weights"
$\mathcal{D}^j:=\{\gamma_j\}\subseteq\mathbb{R}$ as in Equation
\ref{eq:exceptionalforlaplacian}. Given any weight $\gamma\in\mathbb{R}$ define
$V^j_\gamma$ as in Equation \ref{eq:ac_harmonicter} and let $m^j(\gamma)$ denote
its dimension. Given any weight $\boldsymbol{\gamma}\in \mathbb{R}^e$ set
$m(\boldsymbol{\gamma}):=\sum_{j=1}^e m^j(\gamma_j)$. Let
$\mathcal{D}\subseteq\mathbb{R}^e$ denote the set of weights $\boldsymbol{\gamma}$ for
which $m(\boldsymbol{\gamma})>0$. Then each multiplicity
$m(\boldsymbol{\gamma})$ is finite and the Laplace operator
\begin{equation}
\Delta_g:W^p_{k,\boldsymbol{\beta}}\rightarrow W^p_{k-2,\boldsymbol{\beta}-2}
\end{equation}
is Fredholm iff $\boldsymbol{\beta}\notin \mathcal{D}$.
The analogue of Theorem \ref{thm:acyl_indexchange} also holds. For example,
assume $L$ is a CS/AC manifold and write
$\boldsymbol{\beta}=(\boldsymbol{\mu},\boldsymbol{\lambda})$. Choose
$(\boldsymbol{\mu}_1,\boldsymbol{\lambda}_1),
(\boldsymbol{\mu}_2,\boldsymbol{\lambda}_2) \in \mathbb{R}^e\setminus \mathcal{D}$ with
$\boldsymbol{\mu}_1\geq\boldsymbol{\mu}_2$,
$\boldsymbol{\lambda}_1\leq\boldsymbol{\lambda}_2$. Then
$$i_{\boldsymbol{\mu}_2,\boldsymbol{\lambda}_2}(\Delta_g)-i_{\boldsymbol{\mu}_1,
\boldsymbol{\lambda}_1}(\Delta_g)=\sum
m(\boldsymbol{\mu},\boldsymbol{\lambda}),$$
where the sum is taken over all
$(\boldsymbol{\mu},\boldsymbol{\lambda})\in\mathcal{D}$ such that
$\boldsymbol{\mu}_1\geq\boldsymbol{\mu}\geq\boldsymbol{\mu}_2$,
$\boldsymbol{\lambda}_1\leq\boldsymbol{\lambda}\leq\boldsymbol{\lambda}_2$.
\end{corollary}
In the same way one can also prove the analogue of Proposition
\ref{prop:acylindep}.
\section{Harmonic functions on conifolds}\label{s:accs_harmonic}
We can use the results of Sections \ref{s:weightcrossing} and
\ref{s:accs_analysis} to reach a good understanding of the properties of the
Laplace operator acting on functions on conifolds. Specifically, we will
be interested in the kernel and cokernel of $\Delta_g$.
\subsubsection*{Smooth compact manifolds} Let $(L,g)$ be a smooth compact
Riemannian manifold. Let $\Delta_g$ denote the positive Laplace operator on
functions. Consider the map
\begin{equation}\label{eq:cptextendedlaplacian}
\Delta_g:W^p_k(L)\rightarrow W^p_{k-2}(L).
\end{equation}
For all $p>1$ and $k\in\mathbb{Z}$, standard elliptic regularity shows that any
$f\in\mbox{Ker}(\Delta_g)$ is smooth. The maximum principle then proves that $f$
is constant. Thus $\mbox{Ker}(\Delta_g)=\mathbb{R}$, independently of the choice of
$p,k$.
As seen in Section \ref{s:prelim}, $f\in\mbox{Im}(\Delta_g)$ iff $<u,f>=0$, for
all $u\in Ker(\Delta_g^*)$, where $\Delta_g^*$ is the operator dual to that of
Equation \ref{eq:cptextendedlaplacian}. As in Lemma \ref{l:*=*} we can identify
this with the formal adjoint operator. However, $\Delta_g$ is formally
self-adjoint, \textit{i.e.} the operators $\Delta_g$ and $\Delta_g^*$ coincide
on smooth functions. By continuity they continue to coincide when extended to
any Sobolev space. Thus $\mbox{Ker}(\Delta_g^*)=\mbox{Ker}(\Delta_g)=\mathbb{R}$. As in
Equation \ref{eq:duality} we find $<u,f>=\int_Luf\,\mbox{vol}_g$. It follows
that $\mbox{Im}(\Delta_g)=\{f\in W^p_{k-2}(L):\int_Lf\,\mbox{vol}_g=0\}$. In
particular, $\Delta_g$ has index zero.
\subsubsection*{AC manifolds} Let $(L,g)$ be a AC manifold with convergence
rate $\boldsymbol{\nu}<0$ as in Definition \ref{def:metrics_ends}. Let
$\Delta_g$ denote the positive Laplace operator on weighted Sobolev spaces of
functions, as in Equation \ref{eq:extendedlaplacian}. For simplicity, we will
restrict our attention to the case of $L$ with 2 ends.
Each end defines exceptional weights, plotted as points on the horizontal and
vertical axes of Figure \ref{fig:1}. Each exceptional weight gives rise to an
exceptional hyperplane, plotted as a vertical or horizontal line. The Laplacian
is Fredholm for weights $\boldsymbol{\beta}=(\beta_1,\beta_2)$ which are
non-exceptional, \textit{i.e.} which do not lie on these lines. The arrow
indicates the direction in which the corresponding Sobolev spaces, thus the
kernel of $\Delta_g$, become bigger.
Choose $\boldsymbol{\beta}$ non-exceptional. For all $p>1$ and $k\in\mathbb{Z}$,
standard elliptic regularity proves that any $f\in\mbox{Ker}(\Delta_g)$ is
smooth. Furthermore, since $\mbox{Ker}(\Delta_g)$ is independent of $p$ and $k$,
the Sobolev Embedding Theorems show that $f$ has growth of the order
$O(r^{\boldsymbol{\beta}})$. If $\boldsymbol{\beta}<0$ we can thus apply the
maximum principle to conclude that $f\equiv 0$. In other words, $\Delta_g$ is
injective throughout the quadrant defined by the lower shaded region. Since
$\Delta_g$ is formally self-adjoint, the same holds for $\Delta_g^*$. Recall
from Section \ref{s:accs_analysis} how weights on AC manifolds change under
duality. We conclude, following Section \ref{s:prelim}, that
$\mbox{Coker}(\Delta_g)=0$ for $\boldsymbol{\beta}>2-m$. In other words,
$\Delta_g$ is surjective throughout the quadrant defined by the upper shaded
region. In particular, the map of Equation \ref{eq:extendedlaplacian} is an
isomorphism and has index zero for $2-m<\boldsymbol{\beta}<0$, \textit{i.e.} in
the region marked by A.
When $\boldsymbol{\beta}>2-m$ the cokernel is independent of the weight. Thus,
any
change of index corresponds entirely to a change of kernel. Furthermore,
$\mbox{Ker}(\Delta_g)=\mbox{Ker}(\rho^2\Delta_g)$. We can thus use the results
of Section \ref{s:weightcrossing}
to study how the kernel changes as $\boldsymbol{\beta}$
increases. For example, assume we are interested in harmonic functions for some
(thus any) $\boldsymbol{\beta}$ in the region B. We can reach this region by
keeping $\beta_2$ fixed and repeatedly increasing $\beta_1$, starting from the
region A. Each time we cross an exceptional line $x=\gamma$, new harmonic
functions on $(L,g)$ are generated by elements $r^\gamma\sigma(\theta)\in
V^1_\gamma$. Specifically, these new harmonic functions will be asymptotic to
$r^\gamma\sigma$ on the first end and to zero on the second end. Using the ideas
of Section \ref{s:weightcrossing} we can further show that the lower-order terms
will have rate $O(r^{\gamma+\nu_1})$ on the first end and $O(r^{\nu_2})$ on the
second. Analogous results hold for harmonic functions for $\boldsymbol{\beta}$
in the region C. The construction shows that the harmonic functions in the
regions B and C are linearly independent. We can thus apply the change of index
formula to show that harmonic functions in the generic region D are generated by
linear combinations of harmonic functions in the regions B, C.
It may be good to emphasize that the above constructions depend on the specific
$(L,g)$ only in terms of the specific exceptional weights, but are otherwise
completely independent of $(L,g)$. However, these constructions fail if D is
chosen outside the region where $\Delta_g$ is surjective.
\begin{figure}
\includegraphics[width=90mm,height=90mm]{accssldefs_fig1.jpg}
\caption{Harmonic functions on AC manifolds}\label{fig:1}
\end{figure}
\subsubsection*{CS manifolds} Let $(L,g)$ be a CS manifold with convergence
rate $\boldsymbol{\nu}>0$ as in Definition \ref{def:metrics_ends}. As before,
let $\Delta_g$ denote the positive Laplace operator on weighted Sobolev spaces
of functions, as in Equation \ref{eq:extendedlaplacian}. We again restrict our
attention to the case of $L$ with 2 ends.
Figure \ref{fig:2} plots the exceptional weights and lines in this case. Once
again the arrow indicates the direction in which the corresponding Sobolev
spaces, thus the kernel of $\Delta_g$, become bigger. Choose
$\boldsymbol{\beta}$ non-exceptional. As before, any $f\in \mbox{Ker}(\Delta_g)$
is smooth with growth of order $O(r^{\boldsymbol{\beta}})$. If
$\boldsymbol{\beta}>0$ the maximum principle shows that $f=0$. Now assume
$\boldsymbol{\beta}=\frac{2-m}{2}$. In this case
$(W^2_{k-2,\boldsymbol{\beta}-2})^*=W^2_{2-k,\boldsymbol{\beta}}$. Choose $f\in
W^2_{k,\boldsymbol{\beta}}$ and assume $\Delta_gf=0$. Then, choosing $u=v=f$ in
Lemma \ref{l:intbyparts} and using regularity, we can conclude $df=0$ so $f$ is
constant. This shows that, for any weight in the region A,
$\mbox{Ker}(\Delta_g)=\mathbb{R}$.
As before we also find that, in this region, $\mbox{Im}(\Delta_g)=\{f\in
W^p_{k-2,{\beta}-2}:\int_Lf\,\mbox{vol}_g=0\}$. In particular, the index of
$\Delta_g$ is zero.
Now assume $(\beta_1,\beta_2)>(0,\frac{2-m}{2})$. Then
$W^p_{k,\boldsymbol{\beta}}\subset W^p_{k,(\frac{2-m}{2},\frac{2-m}{2})}$ so our
integration by parts argument remains valid. On the other hand the only constant
function in $W^p_{k,\boldsymbol{\beta}}$ is zero so in this case we find that
$\Delta_g$ is injective. The same holds for
$(\beta_1,\beta_2)>(\frac{2-m}{2},0)$. Thus $\Delta_g$ is injective in the upper
shaded region. By duality we deduce that $\Delta_g$ is surjective in the lower
shaded region.
Now assume $\boldsymbol{\beta}$ crosses from A to B. In this particular case the
method used above for AC manifolds fails, because it would require $\Delta_g$ to
be surjective in the region A. We can however bypass this problem as follows:
the change of index formula shows that the index increases by one and we know
that the Laplacian is surjective in B, so $\mbox{Ker}(\Delta_g)=\mathbb{R}$ in B. The
same is true for the region C. We can use Section \ref{s:weightcrossing} to
study the harmonic functions in the lower shaded region. For example, the
harmonic functions in D will be generated by functions which are of the form
$r^\gamma\sigma+O(r^{\gamma+\nu_1})$ on the first end and of the form
$O(r^{\nu_2})$ on the second end. Notice a difference with respect to AC
manifolds: harmonic functions in B and C (more generally, in D and E) are not
necessarily linearly independent. Thus we cannot write harmonic functions in F
as the direct sum of harmonic functions in D and E, as in the AC case. Once
again, harmonic functions elsewhere will be heavily dependent on the specific
$(L,g)$.
We may also be interested in the cokernel of $\Delta_g$. The change of index
formula shows that the dimension of the cokernel increases with $\beta$. For
example, the index is -1 in the regions G,H. Since $\Delta_g$ is injective here
this implies that the cokernel has dimension 1. More generally, the change of
index formula allows us to compute the dimension of the cokernel wherever
$\Delta_g$ is injective. We can also use the ideas of Remark
\ref{rem:nestlingtheker} to build complements of $\mbox{Im}(\Delta_g)$ which
grow with $\boldsymbol{\beta}$.
\begin{figure}
\includegraphics[width=90mm,height=90mm]{accssldefs_fig2bis.jpg}
\caption{Harmonic functions on CS manifolds}\label{fig:2}
\end{figure}
\subsubsection*{CS/AC manifolds} Let $(L,g)$ be a CS/AC manifold with
convergence rate $\boldsymbol{\nu}$. Following the same conventions as before,
we now turn to Figure \ref{fig:3}. Here, the horizontal axis corresponds to the
CS end with weight $\mu$ and the vertical axis corresponds to the AC end with
weight $\lambda$.
When $\lambda<0$ and $\mu>2-m$, the maximum principle and integration by parts
show that $\Delta_g$ is injective. Dually, when $\lambda>2-m$ and $\mu<0$,
$\Delta_g$ is surjective. In the region A, $\Delta_g$ is an isomorphism with
index zero. Harmonic functions in the region B are of the form
$r^\gamma\sigma+O(r^{\gamma+\nu_2})$ on the AC end and of the form
$O(r^{\nu_1})$ on the CS end. Harmonic functions in the region C are of the form
$r^\gamma\sigma+O(r^{\gamma+\nu_1})$ on the CS end and of the form
$O(r^{\nu_2})$ on the AC end. Since these functions are linearly independent,
their linear combinations give the harmonic functions in the region D.
\begin{figure}
\includegraphics[width=90mm,height=90mm]{accssldefs_fig3.jpg}
\caption{Harmonic functions on CS/AC manifolds}\label{fig:3}
\end{figure}
\begin{example}
$\mathbb{R}^m$ with its standard metric can be viewed as a CS/AC manifold, the CS end
being a neighbourhood of the origin. In this case all harmonic functions can be
written explicitly, so in this case we have exact information on their
asymptotics.
\end{example}
\section{The Laplacian on conifold connect sums}\label{s:sums_laplace}
Let $(L,g,\rho, S^*)$,
$(\hat{L},\hat{g},\hat{\rho},\hat{S}^*)$ be compatible marked
conifolds. As seen in Section \ref{s:sums_sobolev}, we can define their connect sum
$(\hat{L}\#L,\hat{g}\#g,\hat{\rho}\#\rho)$. This is a new conifold so we can
study the properties of its Laplace operator as in Section
\ref{s:accs_harmonic}.
We start with the case in which $\hat{S}^{**}\cup S^{**}\neq\emptyset$, \textit{i.e.} the set of ends is non-empty. This case actually turns out to be easier than the alternative situation, where $\hat{L}\#L$ is smooth and compact, because we can use weights to force injectivity of the Laplacian.
\subsubsection*{Non-compact conifolds} Assume the set $\hat{S}^{**}\cup S^{**}$ of ends of $\hat{L}\#L$ is non-empty. If weights $\boldsymbol{\beta}$,
$\hat{\boldsymbol{\beta}}$ are non-exceptional for $\Delta_g$,
$\Delta_{\hat{g}}$ then the weight
$\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}$ is non-exceptional for
$\Delta_{\hat{g}\#g}$ so
$$\Delta_{\hat{g}\#g}:W^p_{k,\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}}
\rightarrow W^p_{k-2,\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}-2} $$
is Fredholm. The same holds
for the parametric connect sums
$(L_{\boldsymbol{t}},g_{\boldsymbol{t}},\rho_{\boldsymbol{t}},\boldsymbol{\beta}
_{\boldsymbol{t}})$.
We want to study the invertibility of the Laplace operator. The following result is obvious.
\begin{lemma}\label{l:sum_injective}
Let $(\hat{L},\hat{g},\hat{\rho},\hat{\boldsymbol{\beta}},\hat{S}^*)$ be a weighted
AC-marked conifold. Assume $\hat{\boldsymbol{\beta}}$ satisfies the
conditions
\begin{equation*}
\left\{
\begin{array}{l}
\hat{\beta}_i<0\mbox{ for all AC ends
}\hat{S}_i\in \hat{S}\\
\hat{\beta}_i>2-m\mbox{
for all CS ends }\hat{S}_i\in \hat{S}
\end{array}
\right.
\end{equation*}
so that $\Delta_{\hat{g}}$ is injective.
Let $(L,g,\rho,\boldsymbol{\beta}, S^*)$ be a weighted CS-marked conifold. Assume
$\boldsymbol{\beta}$ satisfies the conditions
\begin{equation*}
\left\{
\begin{array}{l}
\beta_i<0\mbox{ for all AC
ends }S_i\in S\\
\beta_i>2-m\mbox{ for all CS ends
}S_i\in S.
\end{array}
\right.
\end{equation*}
This is not yet sufficient to conclude that $\Delta_g$ is
injective because the set of AC ends might be empty. To obtain injectivity we
must furthermore assume that each connected component of $L$ has at least one end, \textit{e.g.}
$S'$, satisfying the condition
\begin{equation*}
\left\{
\begin{array}{l}
\beta_1<0 \mbox{ if $S'$ is AC}\\
\beta_1>0 \mbox{ if $S'$ is CS}.
\end{array}
\right.
\end{equation*}
Now assume that $L$, $\hat{L}$ are compatible. Then, for all ends $S_i\in S^*$,
$2-m<\beta_i<0$. This implies that $S'\in S^{**}$ so
$\hat{L}\#L$ has at least one end. Furthermore,
$\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}$ satisfies the conditions
\begin{equation*}
\left\{
\begin{array}{l}
\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}_{|S_i}<0\mbox{ for all AC ends
}S_i\in \hat{S}^{**}\cup S^{**}\\
\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}_{|S_i}>2-m\mbox{
for all CS ends }S_i\in \hat{S}^{**}\cup S^{**}.
\end{array}
\right.
\end{equation*}
Together with the condition on $S'$, this implies that
$\Delta_{\hat{g}\#g}$ is injective.
If furthermore $\boldsymbol{\beta}$, $\hat{\boldsymbol{\beta}}$ are
non-exceptional for $\Delta_g$,
$\Delta_{\hat{g}}$ then
$$\Delta_{\hat{g}\#g}:W^p_{k,\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}}
\rightarrow W^p_{k-2,\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}-2} $$
is a topological isomorphism onto its image so there exists $C>0$ such that, for
all $f\in W^p_{k,\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}}$,
$$\|f\|_{W^p_{k,\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}}}\leq C\|\Delta
f\|_{W^p_{k-2,\hat{\boldsymbol{\beta}}\#\boldsymbol{\beta}-2}}.$$
\end{lemma}
For the constant $C$ in Lemma \ref{l:sum_injective} one can choose the norm of
the inverse map $(\Delta_{\hat{g}\#g})^{-1}$,
as in Equation \ref{eq:normofinverse}. The analogous result holds also for
parametric connect sums. We now want to show that, in this case, the
invertibility constant $C$ can be chosen to be $\boldsymbol{t}$-independent. In
other words,
there exists a $\boldsymbol{t}$-uniform upper bound on the norms of the inverse maps
$(\Delta_{g_{\boldsymbol{t}}})^{-1}$.
\begin{theorem}\label{thm:sum_injective}
Let $(L,g,\rho,\boldsymbol{\beta}, S^*)$,
$(\hat{L},\hat{g},\hat{\rho},\hat{\boldsymbol{\beta}},\hat{S}^*)$ be marked
compatible conifolds satisfying all the conditions of Lemma
\ref{l:sum_injective}. Define $(L_{\boldsymbol{t}},g_{\boldsymbol{t}},\rho_{\boldsymbol{t}},\boldsymbol{\beta}_{\boldsymbol{t}})$ as in Definition
\ref{def:tconnectsum}. Then there exists $C>0$ such that, for all $f\in
W^p_{k,\boldsymbol{\beta}_t}(L_{\boldsymbol{t}})$,
$$\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}}\leq
C\|\Delta_{g_{\boldsymbol{t}}}
f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}}.$$
\end{theorem}
\begin{proof}
To simplify the notation let us assume that all $t_i$ coincide: we can then
work with a unique parameter $t$. The general case is analogous.
Let $C_g$ denote an invertibility constant for $\Delta_g$ on $L$,
\textit{i.e.} for all $f\in W^p_{k,\boldsymbol{\beta}}(L)$,
$$\|f\|_{W^p_{k,\boldsymbol{\beta}}}\leq C_g\|\Delta_g
f\|_{W^p_{k-2,\boldsymbol{\beta}-2}}.$$
Let $C_{\hat{g}}$ denote an analogous constant for $\Delta_{\hat{g}}$ on
$\hat{L}$.
Choose constants $a$, $b$ satisfying $0<b<a<\tau$ and a smooth
decreasing function $\eta:\mathbb{R}\rightarrow [0,1]$ such that $\eta(r)=1$ for
$r\leq b$ and $\eta(r)=0$ for $r\geq a$. Then the function $\eta_t(r):=\eta
(\log r/\log t):(0,\infty)\rightarrow [0,1]$ has the following properties:
\begin{enumerate}
\item $\eta_t$ is smooth increasing, $\eta_t(r)=0$ for $r\leq t^a$,
$\eta_t(r)=1$ for $r\geq t^b$.
\item For all $k\geq 1$ there exists $C_k>0$ such that
$$\left|r^k\frac{\partial^k\eta_t}{(\partial r)^k}(r)\right|\leq
\frac{C_k}{|\log t|}\rightarrow 0\ \ \mbox{ as
$t\rightarrow 0$}.$$
W set $\eta_t'(r):=\frac{\partial\eta_t}{\partial r}(r)$,
$\eta_t''(r):=\frac{\partial^2\eta_t}{(\partial r)^2}(r)$.
\end{enumerate}
Using the diffeomorphisms $\hat{\phi}_{t,i}$ and $\phi_i$ we now extend
$\eta_t$ to a smooth function on $L_{\boldsymbol{t}}$ by setting $\eta_t\equiv
0$ on $(\hat{L}\setminus \hat{S}^*)\cup(\Sigma^*\times[t\hat{R},t^a])$ and
$\eta_t\equiv 1$ on $(L\setminus S^*)\cup(\Sigma^*\times[t^b,\epsilon])$.
For any $f\in W^p_{k,\boldsymbol{\beta}_t}$,
$$\|f\|_{W^p_{k,\boldsymbol{\beta}_t}}\leq
\|\eta_tf\|_{W^p_{k,\boldsymbol{\beta}_t}}
+\|(1-\eta_t)f\|_ { W^p_{k,\boldsymbol{\beta}_t}}.$$
Notice that $\eta_tf$ has support in $(\Sigma^*\times
[t^a,\epsilon])\cup(L\setminus S^*)$, where, up to
identifications via the diffeomorphisms $\phi_i$,
$(g_{\boldsymbol{t}},\rho_{\boldsymbol{t}})=(g,\rho)$,
$\boldsymbol{\beta}_{\boldsymbol{t}}=\boldsymbol{\beta}$. Thus
\begin{align*}
\|\eta_tf\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}(g_{\boldsymbol{t}})}
&=\|\eta_tf\|_{W^p_{k,\boldsymbol{\beta}}(g)}\\
&\leq C_g\|\Delta_g(\eta_tf)\|_{W^p_{k-2,\boldsymbol{\beta}-2}(g)}\\
&=C_g\|\Delta_{g_{\boldsymbol{t}}}(\eta_tf)\|_{W^p_{k-2,\boldsymbol{\beta}_{
\boldsymbol{t}}-2} (g_{\boldsymbol{t}})}\\
&\leq
C_g\left(\|\eta_t\Delta_{g_{\boldsymbol{t}}}f\|_{W^p_{k-2,\boldsymbol{\beta}_{
\boldsymbol{t}}-2 }}+
\|\eta_t'\nabla f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}}+
\|\eta_t''f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}}\right),
\end{align*}
where we drop unnecessary constants.
Applying the Leibniz rule to expressions of the form
$\nabla^j(\eta_t\Delta_{g_{\boldsymbol{t}}}f)$ we find (again up to constants)
\begin{align*}
\|\eta_t\Delta_{g_{\boldsymbol{t}}}f\|^p_{W^p_{k-2,\boldsymbol{\beta}_{
\boldsymbol{t}}-2 }}&\leq
\sum_{j=0}^{k-2}\sum_{l=0}^{j}\int|\rho^l\nabla^l\eta_t|^p_{g_{\boldsymbol{t}}}
|\rho^{2-\boldsymbol{ \beta_{
\boldsymbol{t}} } +j-l } \nabla^ { j
-l}\Delta_{g_{\boldsymbol{t}}}f|^p_{g_{\boldsymbol{t}}}\rho^{-m}\mbox{vol}_{g_{
\boldsymbol{t}} }\\
&\leq \left(1+\left(\frac{C}{|\log
t|}\right)^p\right)\|\Delta_{g_{\boldsymbol{t}}}f\|^p_{W^p_{k-2,\boldsymbol{
\beta } _ {
\boldsymbol{t}}-2}}.
\end{align*}
Analogously,
\begin{equation*}
\|\eta_t'\nabla f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}}\leq
\frac{C}{|\log
t|}
\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}},\
\
\|\eta_t''f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}}\leq
\frac{C}{|\log
t|}
\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}}.
\end{equation*}
The function $(1-\eta_t)f$ has support in
$(\hat{L}\setminus\hat{S}^*)\cup(\Sigma^*\times[t\hat{R},t^b])$. On this space
Definition \ref{def:tconnectsum} shows that
$\boldsymbol{\beta}_{\boldsymbol{t}}=\hat{\boldsymbol{\beta}}$. Furthermore,
on the $i$-th component $\Sigma_i\times [t\hat{R},t^b]$ and up to identifications via the diffeomorphisms $\hat{\phi}_{t,i}$, Lemma \ref{l:neckestimate} shows that $g_{\boldsymbol{t}}$ is scaled-equivalent to $t^2\hat{g}$ and $\rho_{\boldsymbol{t}}=t\hat{\rho}$.
Using Corollary \ref{cor:rescaledconifold} we thus find
\begin{align*}
\|(1-\eta_t)f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}(g_{\boldsymbol{t}},\rho_{\boldsymbol
{t}})}
&\simeq
\|(1-\eta_t)f\|_ { W^p_{k,\hat{\boldsymbol{\beta}}}(t^2\hat{g},t\hat{\rho})}\\
&=t^{-\beta_i}\|(1-\eta_t)f\|_ { W^p_{k,\hat{\boldsymbol{\beta}}}(\hat{g},\hat{\rho})}\\
&\leq t^{-\beta_i}C_{\hat{g}}
\|\Delta_{\hat{g}}((1-\eta_t)f)\|_ {
W^p_{k-2,\hat{\boldsymbol{\beta}}-2}(\hat{g},\hat{\rho})}\\
&=t^{2-\beta_i}C_{\hat{g}} \|\Delta_{t^2\hat{g}}((1-\eta_t)f)\|_ {
W^p_{k-2,\hat{\boldsymbol{\beta}}-2}(\hat{g},\hat{
\rho}) }\\
&=C_{\hat{g}}\|\Delta_{t^2\hat{g}}((1-\eta_t)f)\|_ {
W^p_{k-2,\hat{\boldsymbol{\beta}}-2}(t^2\hat{g},t\hat{\rho})}\\
&\simeq C_{\hat{g}}\|\Delta_{g_{\boldsymbol{t}}}((1-\eta_t)f)\|_ {
W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}(g_{\boldsymbol{t}},\rho_{\boldsymbol{t}})},
\end{align*}
where $\simeq$ replaces multiplicative constants. We now continue as above.
Combining the above results leads to an inequality of the form
$$\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}}\leq (C_g+C_{\hat{g}})
\left(\|\Delta_{g_{\boldsymbol{t}}}f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol
{t}}-2}}+\frac{C}{|\log
t|}
\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}}\right).$$
For $t$ sufficiently small we can absorb
the second term on the
right hand side into the left hand side, proving the claim.
\end{proof}
\subsubsection*{Smooth compact manifolds} Assume the set $\hat{S}^{**}\cup S^{**}$ is empty, so that $\hat{L}\#L$ is smooth and compact. In this case the Laplace operator, acting on functions, always has kernel: the space of constants $\mathbb{R}$. We can thus achieve injectivity only by restricting ourselves to a subspace transverse to constants. Furthermore, if we want the invertibility constant to be independent of $\boldsymbol{t}$, we need our notion of ``transversality'' to depend on $\boldsymbol{t}$, as follows.
\begin{theorem}\label{thm:cpt_sum_injective}
Let $(L,g,\rho, S^*)$,
$(\hat{L},\hat{g},\hat{\rho},\hat{S}^*)$ be marked
compatible conifolds such that the parametric connect sums $(L_{\boldsymbol{t}},g_{\boldsymbol{t}},\rho_{\boldsymbol{t}})$ are smooth and compact. Choose constant weights $\boldsymbol{\beta}=\hat{\boldsymbol{\beta}}\in (2-m,0)$ and define $\boldsymbol{\beta}_{\boldsymbol{t}}$ as usual.
\begin{enumerate}
\item Assume $L$ has only one connected component.
Then there exists a constant $C>0$ and, for each $\boldsymbol{t}$ sufficiently small, a subspace $E_{\boldsymbol{t}}\subset W^p_{k,\boldsymbol{t}}(L_{\boldsymbol{t}})$ such that
\begin{equation}\label{eq:decomp_Et}
W^p_{k,\boldsymbol{t}}(L_{\boldsymbol{t}})=E_{\boldsymbol{t}}\oplus\mathbb{R}
\end{equation}
and, for all $f\in E_{\boldsymbol{t}}$,
$$\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}}\leq
C\|\Delta_{g_{\boldsymbol{t}}}
f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}}.$$
Furthermore, the image of the restricted operator $\Delta_{g_{\boldsymbol{t}}|E_{\boldsymbol{t}}}$ coincides with the image of the full operator $\Delta_{g_{\boldsymbol{t}}}$.
\item Assume $L$ has $k>1$ connected components. Then there exists a constant $C>0$ and, for each $\boldsymbol{t}$ sufficiently small, a codimension $k$ subspace $E_{\boldsymbol{t}}\subset W^p_{k,\boldsymbol{t}}(L_{\boldsymbol{t}})$ transverse to constants such that, for all $f\in E_{\boldsymbol{t}}$,
$$\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}}\leq
C\|\Delta_{g_{\boldsymbol{t}}}
f\|_{W^p_{k-2,\boldsymbol{\beta}_{\boldsymbol{t}}-2}}.$$
\end{enumerate}
\end{theorem}
\begin{proof}
Assume $L$ has one connected component. Choose any closed subspace $E\subset W^p_{k,\boldsymbol{\beta}}(L)$ such that
\begin{equation*}
W^p_{k,\boldsymbol{\beta}}(L)=E\oplus\mathbb{R}.
\end{equation*}
Define $\eta_{\boldsymbol{t}}$ as in the proof of Theorem \ref{thm:sum_injective}. Extending it to zero on the CS ends of $L$, we can think of it as an element of $W^p_{k,\boldsymbol{\beta}}(L)$. One can check that $\eta_{\boldsymbol{t}}\rightarrow 1$ in the $W^p_{k,\boldsymbol{\beta}}$ norm as $\boldsymbol{t}\rightarrow 0$ so, for small $\boldsymbol{t}$, $\eta_{\boldsymbol{t}}\notin E$.
The multiplication map
\begin{equation*}
P_{\boldsymbol{t}}:W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}(L_{\boldsymbol{t}})\rightarrow W^p_{k,\boldsymbol{\beta}}(L),\ \ f\mapsto \eta_{\boldsymbol{t}}f,
\end{equation*}
is linear and uniformly continuous with respect to the parameter $\boldsymbol{t}$, so $E_{\boldsymbol{t}}:=P_{\boldsymbol{t}}^{-1}(E)$ is linear and closed. Since $\eta_{\boldsymbol{t}}$ does not belong to $E$, constants do not belong to $E_{\boldsymbol{t}}$. To confirm that $E_{\boldsymbol{t}}$ has codimension 1, choose any linear function $Q:W^p_{k,\boldsymbol{\beta}}(L)\rightarrow\mathbb{R}$ such that $E=\mbox{Ker}(Q)$. Then $E_{\boldsymbol{t}}=\mbox{Ker}(Q\circ P_{\boldsymbol{t}})$, so it is defined by one linear condition. This proves Decomposition \ref{eq:decomp_Et}.
Consider $\Delta_{g_{\boldsymbol{t}}}$ restricted to $E_{\boldsymbol{t}}$. It is clearly injective. One can check that it is uniformly injective exactly as in Theorem \ref{thm:sum_injective}.
Now assume $L$ has multiple components $L_1,\dots,L_k$. For each $L_i$, choose a closed subspace $E_i\subset W^p_{k,\boldsymbol{\beta}}(L_i)$ as above. The multiplication map
\begin{equation*}
W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}(L_{\boldsymbol{t}})\rightarrow \bigoplus W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}(L_i),\ \ f\mapsto \eta_{\boldsymbol{t}}f,
\end{equation*}
is again linear and uniformly continuous, so we can define $E_{\boldsymbol{t}}$ as the inverse of $E_1\oplus\dots,\oplus E_k$. One can again check that it has codimension $k$ and that, restricted to this space, $\Delta_{g_{\boldsymbol{t}}}$ is uniformly injective.
\end{proof}
\begin{remark}
Notice that, even though $L_{\boldsymbol{t}}$ is smooth and compact, the proof of Theorem \ref{thm:cpt_sum_injective} requires the use of radius functions and weights on the necks.
\end{remark}
\section{Further Sobolev-type inequalities on conifold connect
sums}\label{s:improved_sob}
Given a conifold $(L,g)$, we can also apply the theory of Section
\ref{s:accs_analysis} to the elliptic operator
\begin{equation}\label{eq:D_g}
D_g=d\oplus d^*_g:W^p_{k,\boldsymbol{\beta}}(\Lambda^{even})\rightarrow
W^p_{k-1,\boldsymbol{\beta}-1}(\Lambda^{odd}),
\end{equation}
defined from the bundle of all even-dimensional forms on $L$ to the bundle of
all odd-dimensional forms. As for the Laplacian, it is possible to define and
study the exceptional weights for this operator. For any non-exceptional
weight $\boldsymbol{\beta}$, the operator $D_g$ of Equation \ref{eq:D_g} is
Fredholm. It is simple to conclude that, for such weights, the restricted
operator
\begin{equation}\label{eq:d}
d:W^p_{k,\boldsymbol{\beta}}(L)\rightarrow
W^p_{k-1,\boldsymbol{\beta}-1}(\Lambda^1)
\end{equation}
has closed image. Notice that $\mbox{Ker}(d)$ can only contain constants. If
the choice of weights is such that constants do not belong to the space
$W^p_{k,\boldsymbol{\beta}}(L)$, the operator $d$ of Equation \ref{eq:d} is a
topological isomorphism onto its image and can be inverted. In particular
there exists $C>0$ such that, for any $f\in W^p_{k,\boldsymbol{\beta}}(L)$,
\begin{equation*}
\|f\|_{W^p_{k,\boldsymbol{\beta}}}\leq
C\|df\|_{W^p_{k-1,\boldsymbol{\beta}-1}}.
\end{equation*}
We now want to show that, on conifolds obtained as parametric connect sums,
such $C$ can chosen independently of $\boldsymbol{t}$. For brevity, we restrict our attention to the non-compact case.
\begin{theorem}\label{thm:d_invertible}
Let $(\hat{L},\hat{g},\hat{\rho},\hat{\boldsymbol{\beta}},\hat{S}^*)$ be a weighted AC-marked conifold. Assume that $\hat{\boldsymbol{\beta}}$ is non-exceptional for the operator
\begin{equation*}
D_{\hat{g}}:W^p_{k,\hat{\boldsymbol{\beta}}}(\Lambda^{even})\rightarrow
W^p_{k-1,\hat{\boldsymbol{\beta}}-1}(\Lambda^{odd})
\end{equation*}
defined on the manifold $\hat{L}$ and that $\hat{\beta}_i<0$ for all ends $\hat{S}_i\in \hat{S}^*$.
Let $(L,g,\rho,\boldsymbol{\beta}, S^*)$ be a weighted CS-marked conifold. Assume $\boldsymbol{\beta}$ is non-exceptional for the operator
\begin{equation*}
D_g:W^p_{k,\boldsymbol{\beta}}(\Lambda^{even})\rightarrow
W^p_{k-1,\boldsymbol{\beta}-1}(\Lambda^{odd})
\end{equation*}
defined on the manifold $L$ and that each connected component of $L$ has at least one end, \textit{e.g.}
$S'$, satisfying the condition
\begin{equation*}
\left\{
\begin{array}{l}
\beta_1<0 \mbox{ if $S'$ is AC}\\
\beta_1>0 \mbox{ if $S'$ is CS}.
\end{array}
\right.
\end{equation*}
Now assume that $L,\hat{L}$ are compatible. Then, for all ends $S_i\in S^*$,
$\beta_i=\hat{\beta_i}<0$. This implies that $S'\in S^{**}$ so each connect sum $L_{\boldsymbol{t}}$ has at least one end.
There exists $C>0$ such that, for all $f\in
W^p_{k,\boldsymbol{\beta}_t}(L_{\boldsymbol{t}})$,
\begin{equation}\label{eq:d_invertible}
\|f\|_{W^p_{k,\boldsymbol{\beta}_{\boldsymbol{t}}}}\leq
C\|df\|_{W^p_{k-1,\boldsymbol{\beta}_{\boldsymbol{t}}-1}}.
\end{equation}
\end{theorem}
\begin{proof}
As seen above, the assumptions prove that the operator $d$ is a topological
isomorphism (onto its image) between Sobolev spaces on both manifolds $L$,
$\hat{L}$. This means that there exist constants $C_g$, $C_{\hat{g}}$ satisfying
the analogue of Equation \ref{eq:d_invertible} on both manifolds separately. We
can use $C_g$, $C_{\hat{g}}$ to build $C$ satisfying Equation
\ref{eq:d_invertible} on $L_{\boldsymbol{t}}$ using the same ideas introduced
in the proof of Theorem \ref{thm:sum_injective}. There is only one difference,
as follows. In the proof of Theorem \ref{thm:sum_injective} we use the equality
\begin{equation*}
t^{-\beta_i}C_{\hat{g}}
\|\Delta_{\hat{g}}((1-\eta_t)f)\|_{W^p_{k-2,\hat{\boldsymbol{\beta}}-2}(\hat{g},\hat{\rho})}=t^{2-\beta_i}C_{\hat{g}} \|\Delta_{t^2\hat{g}}((1-\eta_t)f)\|_ {
W^p_{k-2,\hat{\boldsymbol{\beta}}-2}(\hat{g},\hat{\rho}) }.
\end{equation*}
The factor $t^{2-\beta_i}$ is then cancelled by rescaling. In
particular, the above equality uses the fact that the
Laplacian depends on the metric and rescales in a specific
way.
In the case at hand the operator $d$ does not depend on the metric. However,
notice that it takes functions into 1-forms: it is this property that allows us
to conclude. Specifically, setting $\alpha_t=d((1-\eta_t)f)$ and assuming $\hat{\boldsymbol{\beta}}$ is constant to simplify the
notation, we find:
\begin{align*}
\|\alpha_t\|^p_{W^p_{k-1,\hat{\boldsymbol{\beta}}-1}(\hat{g},
\hat{\rho})}
&=\sum_j\int_{\hat{L}}|\hat{\rho}^{1-\hat{\boldsymbol{\beta}}+j}
\nabla^j\alpha_t|^p_{\hat{g}\otimes\hat{g}}\hat{\rho}^{-m}\mbox{vol}_{\hat{g}}\\
&=t^{p\hat{\boldsymbol{\beta}}}\sum_j\int_{\hat{L}}|(t\hat{\rho})^{1-\hat{\boldsymbol{\beta}}+j}
\nabla^j\alpha_t|^p_{t^2\hat{g}\otimes t^2\hat{g}}(t\hat{\rho})^{-m}\mbox{vol}_{t^2\hat{g}}\\
&=t^{p\hat{\boldsymbol{\beta}}}\|\alpha_t\|^p_{W^p_{k-1,\hat{\boldsymbol{\beta}}-1}(t^2\hat{g} ,
t\hat{\rho})}.
\end{align*}
The proof can now continue as for Theorem \ref{thm:sum_injective}.
\end{proof}
Combining Theorems \ref{thm:normstequivalent} and \ref{thm:d_invertible} we
obtain the following improvement of the weighted Sobolev Embedding Theorems,
Part 1, for parametric connect sums.
\begin{corollary}\label{cor:improved_sob}
Let $(L,g,\rho,\boldsymbol{\beta}, S^*)$,
$(\hat{L},\hat{g},\hat{\rho},\hat{\boldsymbol{\beta}},\hat{S}^*)$ be marked
compatible conifolds as in Theorem \ref{thm:d_invertible}. Define
$L_{\boldsymbol{t}}$ as in Definition
\ref{def:tconnectsum}. Then there exists
$C>0$ such that, for all $1\leq p<m$, $\boldsymbol{t}$ and $f\in
W^p_{1,\boldsymbol{\beta}_{\boldsymbol{t}}}(L_{\boldsymbol{t
}} )$ ,
\begin{equation*}
\|f\|_{L^{p^*}_{\boldsymbol{\beta}_{\boldsymbol{t}}}}\leq C
\|df\|_{L^p_{\boldsymbol{\beta}_{\boldsymbol{t}}-1}}.
\end{equation*}
\end{corollary}
\begin{remark}
Following standard terminology in the
literature we can refer to Equation \ref{eq:d_invertible} as a ``uniform weighted Poincar\'{e} inequality'' and to Corollary \ref{cor:improved_sob} as a ``uniform
weighted
Gagliardo-Nirenberg-Sobolev inequality''. Alternatively, following \cite{hebey}
Chapter 8, the latter is a ``uniform weighted Euclidean-type Sobolev
inequality''.
\end{remark}
\bibliographystyle{amsplain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,155 |
\subsection{Experimental Setup}
We used well-known three datasets:CIFAR10, CIFAR100~\cite{krizhevsky2009learning}, and TinyImageNet~\cite{Le2015TinyIV} and models : VGG16-BN~\cite{simonyan2014very}, ResNet56~\cite{he2016deep}, and MobileNetV2~\cite{sandler2018mobilenetv2} in various experiments. We used \textit{hyperband}~\cite{li2017hyperband} as search space pruner and
TPE~\cite{bergstra2011algorithms}, BO~\cite{balandat2020botorch}, CMA-ES~\cite{loshchilov2016cma} as representative hyperparameter optimization methods. We used SGD optimizer with momentum(=0.9) and decayed learning rate at 80 epoch and 120 epoch by 10 times in most experiments. We mainly targeted three hyperparameters, which most researchers mainly adjust: learning rate, weight decay, and batch size. Additionally, we verified the scalability of our observation by adding the other two hyperparameters~(learning rate decay term, and SGD momentum). The search spaces of hyperparameters and allocated time budget are described in Table \ref{tbl:HPO_range}. We used the learning rate, weight decay, and batch size as the main target hyperparameters, and the learning rate decay term and SGD momentum were used only in additional experiments. In most experiments, we utilized open source hyperparameter optimization framework \textit{optuna}~\cite{optuna_2019}.\\
\begin{table}[h]
\centering
\caption{The target hyperparameters and Time Budget}
\label{tbl:HPO_range}
\begin{tabular}{c|ccccc|cc}
\hline
& \multicolumn{5}{c|}{Hyperparameters} & \multicolumn{2}{c}{Time Budget} \\ \hline
Category & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Learning \\ Rate\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Batch \\ Size\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Weight \\ Decay\end{tabular}} & \multicolumn{1}{c|}{\begin{tabular}[c]{@{}c@{}}Learning Rate \\ Decay\end{tabular}} & \begin{tabular}[c]{@{}c@{}}SGD\\ Momentum\end{tabular} & \multicolumn{1}{c|}{Epoch} & \begin{tabular}[c]{@{}c@{}}HPO \\ Trials\end{tabular} \\ \hline
Value & \multicolumn{1}{c|}{1e-5 $\sim$2e-1} & \multicolumn{1}{c|}{32$\sim$1024} & \multicolumn{1}{c|}{1e-5$\sim$2e-2} & \multicolumn{1}{c|}{1e-1$\sim$1} & 1e-1$\sim$9.9e-1 & \multicolumn{1}{c|}{160} & 50 \\ \hline
\end{tabular}
\vspace{1mm}
\end{table}
\indent We pruned neural networks with a high percentile~($\rho$) value 85 in most experiments to take sufficient reduction in computational amount. And we set minimum channel remaining ratio~($\sigma$) value 0.15 to guarantee normal training of each neural network. In terms of data augmentation, we used random horizontal flip~(probability=0.5) and random crop~(zero padding=4, size=32) for CIFAR Series. In the case of TinyImageNet, we used random horizontal flip~(probability=0.5), random rotation~(degree=$15^{\circ}$) and random-crop~(zero padding=6, size=128).
\begin{table}[!t]
\centering
\caption{Experimental results on whether the neural network obtained through pruning can be used as a good proxy model. `Time Reduction' means the averagely reduced percentage of the time required from the forward pass to the backward pass for the same batch size~(The higher is better).}
\label{tbl:main_results}
\begin{tabular}{c|c|c|cc|c}
\hline
\multirow{3}{*}{Dataset} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Hyperparameter\\ Optimizer\end{tabular}} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Neural\\ Network\end{tabular}} & \multicolumn{2}{c|}{Accuracy~(\%)} & \multirow{3}{*}{\begin{tabular}[c]{@{}c@{}}Time \\ Reduction\end{tabular}} \\ \cline{4-5}
& & & \multicolumn{1}{c|}{\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Typical\\ HPO\end{tabular}}} & \multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}With \\ Pruning\end{tabular}} & \\
& & & \multicolumn{1}{c|}{} & & \\ \hline
\multirow{9}{*}{CIFAR10} & \multirow{3}{*}{TPE} & ResNet56 & \multicolumn{1}{c|}{$93.43 \pm 0.23 $} & $93.62 \pm 0.32$ & 12 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$93.41 \pm 0.42$} & $93.61 \pm 0.07$ & 23 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$94.54 \pm 0.14$} & $94.87 \pm 0.36$ & 32 \% \\ \cline{2-6}
& \multirow{3}{*}{BO} & ResNet56 & \multicolumn{1}{c|}{$93.04 \pm 0.37$} & $93.18 \pm 0.58$ & 12 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$93.70 \pm 0.14$} & $93.94 \pm 0.22$ & 23 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$94.58 \pm 0.42$} & $95.27 \pm 0.10$ & 32 \% \\ \cline{2-6}
& \multirow{3}{*}{CMA-ES} & ResNet56 & \multicolumn{1}{c|}{$93.11 \pm 0.39$} & $93.50 \pm 0.20$ & 12 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$93.30 \pm 0.36$} & $93.79 \pm 0.21$ & 23 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$94.66 \pm 0.65$} & $95.40 \pm 0.10$ & 32 \% \\ \hline
\multirow{9}{*}{CIFAR100} & \multirow{3}{*}{TPE} & ResNet56 & \multicolumn{1}{c|}{$71.24 \pm 0.23$} & $72.31\pm 0.73$ & 12 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$73.51 \pm 0.34 $} & $73.36 \pm 0.47$ & 22 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$77.94 \pm 2.22$} & $77.89 \pm 1.47$ & 37 \% \\ \cline{2-6}
& \multirow{3}{*}{BO} & ResNet56 & \multicolumn{1}{c|}{$71.23 \pm 1.19$} & $71.74 \pm 1.59$ & 12 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$73.00 \pm 0.60$} & $73.43 \pm 0.34$ & 22 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$76.20 \pm 1.66$} & $78.57 \pm 0.72$ & 37 \% \\ \cline{2-6}
& \multirow{3}{*}{CMA-ES} & ResNet56 & \multicolumn{1}{c|}{$71.32 \pm 0.54$} & $71.05 \pm 0.45$ & 12 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$72.25 \pm 0.42$} & $73.47 \pm 0.30$ & 22 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$74.78 \pm 1.52$} & $75.90 \pm 1.03$ & 37 \% \\ \hline
\multirow{9}{*}{TinyImageNet} & \multirow{3}{*}{TPE} & ResNet56 & \multicolumn{1}{c|}{$52.92 \pm 0.80$} & $52.29 \pm 0.94$ & 20 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$61.49 \pm 0.55$} & $61.03 \pm 0.44$ & 16 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$57.99 \pm 0.44$} & $56.99 \pm 1.40$ & 23 \% \\ \cline{2-6}
& \multirow{3}{*}{BO} & ResNet56 & \multicolumn{1}{c|}{$51.72 \pm 0.36$} & $51.34 \pm 0.76$ & 20 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$56.48 \pm 2.25$} & $58.44 \pm 1.96$ & 16 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$57.84 \pm 0.42$} & $56.08 \pm 1.45$ & 23 \% \\ \cline{2-6}
& \multirow{3}{*}{CMA-ES} & ResNet56 & \multicolumn{1}{c|}{$52.96 \pm 0.70$} & $52.60 \pm 0.30$ & 20 \% \\ \cline{3-6}
& & VGG16-BN & \multicolumn{1}{c|}{$56.27 \pm 2.27$} & $58.78 \pm 2.01$ & 16 \% \\ \cline{3-6}
& & MobileNetV2 & \multicolumn{1}{c|}{$57.87 \pm 1.72$} & $56.32 \pm 1.82$ & 23 \% \\ \hline
\end{tabular}
\end{table}
\begin{table}[!t]
\centering
\caption{Hyperparameter transferability experiment on ResNet Family. Each column represents the accuracy of ResNet56 for CIFAR100 validation set. The left side of $\rightarrow$ refers to the neural network performing HPO, and the right side refers to the target neural network for hyperparameter transfer.}
\label{tbl:family networks}
\begin{tabular}{c|c|c|c|c}
\hline
& Baseline & \begin{tabular}[c]{@{}c@{}}ResNet8\\ $\rightarrow$ ResNet56\end{tabular} & \begin{tabular}[c]{@{}c@{}}ResNet14\\ $\rightarrow$ ResNet56\end{tabular} & \begin{tabular}[c]{@{}c@{}}ResNet32\\ $\rightarrow$ ResNet56\end{tabular} \\ \hline
Accuracy(\%) & $72.31 \pm 0.73$ & $71.66 \pm 0.38$ & $71.76 \pm 0.39$ & $71.60 \pm 0.74$ \\ \hline
\end{tabular}
\vspace{2mm}
\end{table}
\subsection{Hyperparameter Optimization with Pruning}
We conducted experiments on commonly used three datasets and models to verify our presumption. In addition, we used three hyperparameter optimization algorithms: random search~\cite{bergstra2011algorithms}, evolutionary strategy~\cite{loshchilov2016cma}, and Bayesian optimization~\cite{balandat2020botorch} to prove that our presumption is independent of the hyperparameter optimizer. And we used \textit{Hyperband}~\cite{li2017hyperband} as a search space pruner for the overall experiments. All experiments corresponding to each row of the Table \ref{tbl:main_results} were performed using 5 random seeds. And we measured the accuracy for the validation set of each dataset to quantify the performance of hyperparameters found through HPO.\\
\indent Table \ref{tbl:main_results} shows the performance of hyperparameter sets obtained through typical HPO framework~(Fig.\ref{fig:HPO Procedures}.(a)) and the proposed HPO framework which uses a pruned neural network as a proxy model~(Fig.\ref{fig:HPO Procedures}.(b)). We can see there are marginal gaps when comparing the case where the target neural network is directly used for HPO and the case where the proxy model is used for HPO. In other words, the performance of the hyperparameter sets found through each procedure is almostly equal. If the hyperparameter optimizer is different, even if the dataset and network are the same, there are performance differences. This phenomenon may be attributed to the performance difference of the hyperparameter optimizer. In terms of time reduction, the rate of reduction was lower than expected, due to the property of the neural network~($N_{\mathcal{P}}$) obtained through the pruning method used in experiments. Since the single-shot pruning method uses an initial weight, if we use weight initializations that are commonly used~\cite{he2015delving,glorot2010understanding}, relatively more channels of the low-level layer than high-level layer are inevitably left. This is because the weights will be initialized more sparsely as it goes to a higher-level layer, and the pruning score for each weight is calculated according to Eq.\ref{eq2}. As a result, since the channels of the low level layer are remained at a relatively high rate, the time reduction rate was lower than the actual number of parameters reduction rate.\\
\indent Taken together, the experimental results are consistent with the trend of the Table \ref{tbl:motivation} that became our motivation. And this trend is independent of dataset, hyperparameter optimizer, and model. Therefore, we concluded that it is a good choice to use the neural network obtained through pruning as a proxy model to reduce the time required for hyperparameter optimization.
\subsection{Hyperparameter Transferability Analysis}
\indent In this section, we analyze the reason of why a pruned neural network can be used as a proxy model. And we discuss whether our proposal is valid even when conducting HPO for more hyperparameters. \\
\begin{wrapfigure}{r}{0.5\textwidth}{h}
\begin{center}
\vspace{-1.7cm}
\includegraphics[trim=0.0cm 0.0cm 0.0cm 0.0cm,width=0.5\textwidth]{Figures/train_loss_curves_Fig2.png}
\end{center}
\vspace{-0.8cm}
\caption{Training loss curves when $N_{\mathcal{P}}$ and $N_{\mathcal{B}}$ are trained on CIFAR 10. The shades of each curve mean the standard deviation of training loss.}
\vspace{-0.5cm}
\label{Fig:trend}
\end{wrapfigure}
\textbf{Training Trends.} Simply thinking, it can be expected that the neural network~($N_{\mathcal{P}}$) obtained by pruning will have a similar training tendency to the corresponding base neural network~($N_{\mathcal{B}}$). We checked whether the training tendency was actually similar through a simple experiment. If the training trend is similar, the logit values can be similar and consequently the loss curve will be similar. The loss curves in Fig.\ref{Fig:trend} are obtained using 5 controlled random seeds, and ResNet56 was used as the neural network and CIFAR10 was used as the dataset. In epochs 80 and 120, the learning rate was attenuated by 10 times, and SGD was used as the optimizer, 0.9 for momentum, and 5e-4 for weight decay.\\
\indent As shown in Fig.\ref{Fig:trend}, we can see that the average training loss of $N_{\mathcal{P}}$ is larger over the entire epochs than $N_{\mathcal{B}}$. This means that $N_{\mathcal{P}}$ and $N_{\mathcal{B}}$ have different training trends, that is, logit values from each model are different. However, hyperparameter transferability between $N_{\mathcal{P}}$ and $N_{\mathcal{B}}$ for ResNet56 is high according to Table \ref{tbl:main_results}. Through these two facts, we conclude that it is difficult to measure hyperparameter transferability through loss curve or logit similarity. Therefore, we took a different approach to measure the hyperparameter transferability. \\ \\
\indent \textbf{Structural Similarity.} When the performances of family networks are evaluated in network architecture literatures~\cite{he2016deep,zagoruyko2016wide,huang2017densely}, we noted that hyperparameters do not vary significantly according to the depth or channel width of the neural network. We performed an experiment to verify that the depth of each neural network would not have a significant effect on HPO if each neural network belongs to a family. We performed the following an experiment using ResNet Family~\cite{he2016deep}. First, we acquired hyperparameter sets by performing HPO for ResNet8, ResNet14, and ResNet32 according to Fig.\ref{fig:HPO Procedures}.(a). Next, the performance was measured when the hyperparameter sets obtained earlier were used for ResNet56 training. For comparison, we considered the baseline performance of ResNet56 as the validation accuracy when using the hyperparameter set acquired through Fig.\ref{fig:HPO Procedures} (b), and measured the performance when transferring the hyperparameters of ResNet8, ResNet14, and ResNet32. Each column of Table \ref{tbl:family networks} shows the performance when the hyperparameter set is transferred from each ResNet Family to ResNet56.\\
\begin{table}[!t]
\centering
\caption{Hyperparameter transfer experiments between distinct neural networks. We transferred the hyperparameters obtained from source neural network to target neural network. We measured the accuracies of the validation set. The number in parentheses means the difference between the performance obtained through the proposed HPO framework and the performance when source hyperparameters are transferred to the target neural network.}
\label{tbl:cross_transferability}
\begin{tabular}{c|c|ccc}
\hline
\multirow{2}{*}{$Target$} & \multirow{2}{*}{$Source$} & \multicolumn{3}{c}{Accuracy (\%)} \\ \cline{3-5}
&& \multicolumn{1}{c|}{CIFAR10} & \multicolumn{1}{c|}{CIFAR100} & TinyImageNet \\ \hline
\multirow{3}{*}{VGG16-BN} &
\begin{tabular}[c]{@{}c@{}}VGG16-BN with pruning \end{tabular} &\multicolumn{1}{c|}{$93.61$} & \multicolumn{1}{c|}{$73.36$} & $61.03$ \\ \cline{2-5}
&
\begin{tabular}[c]{@{}c@{}}RestNet56 \end{tabular} &\multicolumn{1}{c|}{$93.64(+0.03)$} & \multicolumn{1}{c|}{$72.31(-1.05)$} & $61.44(+0.41)$ \\ \cline{2-5}
&
\begin{tabular}[c]{@{}c@{}}MobileNetV2 \end{tabular} &\multicolumn{1}{c|}{93.58(-0.03)} & \multicolumn{1}{c|}{72.54(-0.82)} & 60.19(-0.84) \\ \hline
\multirow{3}{*}{MobileNetV2} &
\begin{tabular}[c]{@{}c@{}}MobileNetV2 with pruning \end{tabular} &\multicolumn{1}{c|}{94.54} & \multicolumn{1}{c|}{77.89} & 56.99 \\ \cline{2-5}
&
\begin{tabular}[c]{@{}c@{}}ResNet56 \end{tabular} &\multicolumn{1}{c|}{91.95(-2.92)} & \multicolumn{1}{c|}{75.91(-1.98)} &55.90(-1.09) \\ \cline{2-5}
&
\begin{tabular}[c]{@{}c@{}}VGG16-BN \end{tabular} &\multicolumn{1}{c|}{94.30(-0.57)} & \multicolumn{1}{c|}{76.06(-1.83)} & 56.02(-0.97) \\ \hline
\multirow{3}{*}{RestNet56} &
\begin{tabular}[c]{@{}c@{}}RestNet56 with pruning \end{tabular} &\multicolumn{1}{c|}{93.62} & \multicolumn{1}{c|}{72.31} & 52.29 \\ \cline{2-5}
&
\begin{tabular}[c]{@{}c@{}}MobileNetV2 \end{tabular} &\multicolumn{1}{c|}{93.29(-0.33)} & \multicolumn{1}{c|}{70.82(-1.49)} &50.00(-2.29) \\ \cline{2-5}
&
\begin{tabular}[c]{@{}c@{}}VGG16-BN \end{tabular} &\multicolumn{1}{c|}{93.40(-0.22)} & \multicolumn{1}{c|}{71.77(-0.54)} & 51.50(-0.79) \\ \hline
\end{tabular}
\end{table}
\indent As shown in Table \ref{tbl:family networks}, when the hyperparameters obtained from each family network were transferred, there were small differences of around 0.7\%. This result indicates that the hyperparameter set obtained by performing HPO using family neural networks can achieve suboptimal performance. In other words, it means that suboptimal hyperparameter transfer is possible between Family Neural Networks. Based on this observation, we can think of a simple way to effectively reduce the HPO cost. For example, suppose that a neural network~ ($N_{target}$) with high depth among neural network family that can have very small depth like ResNet family is used for application development. In this case, since the dataset to be used is likely not a commonly used dataset, the HPO is inevitable. Depending on the previously observed fact, if we perform HPO using a family neural network~($N_{small}$) with low depth, we can obtain an approximate optimal hyperparameter range. And if we do HPO by allocating a much smaller budget using the acquired hyperparameter range, we will be able to get the optimal hyperparameter of $N_{target}$ in a short time. However, this step-by-step procedure is more inconvenient than Fig.\ref{fig:HPO Procedures}.(b) and has the disadvantage of having to empirically set up the range of each hyperparameter by using the found suboptimal hyperparameters. On the other hand, if HPO is performed through Fig.\ref{fig:HPO Procedures}.(b), optimal hyperparameter set can be acquired quickly without these drawbacks. \\
\indent From the previous experimental results, we assumed that the hyperparameter transferability was high when the structural similarity was high. In order to support this conjecture, we conducted a hyperparameter transfer experiment between different neural networks rather than family networks~(See Table \ref{tbl:cross_transferability}). The performance of the validation set was measured when the hyperparameter sets obtained by performing HPO using the proxy model of other neural network were used for training of the target neural network. The number in parentheses in each column of Table \ref{tbl:cross_transferability} mean the performance gap that can be obtained through Fig.\ref{fig:HPO Procedures}.(b). The hyperparameter optimizer used in the experiment was TPE~\cite{bergstra2011algorithms}, and the search space pruner was \textit{hyperband}~\cite{li2017hyperband}.\\
\indent From the experimental results, it can be seen that when hyperparameter transfer is performed between VGG16-BN and ResNet56, there is relatively lower performance degradation than the case of hyperparameter transfer between MobileNetV2 and the others~(See Table \ref{tbl:cross_transferability}). On the other hand, looking at the hyperparameter transfer results between MobileNetV2 and the other two neural networks, the performance degradation is relatively high. This trend is consistent across the three datasets. When we look at this trend more closely, the hyperparameter transfer performance gap of ResNet56-MobileNetV2 pair is larger, and VGG16BN-MobileNetV2 pair are smaller than this. Therefore, it can be said that the hyperparameter transferability of VGG16BN-MobileNetV2 pair is higher than that of ResNet56-MobileNetV2 pair. This phenomenon can be attributed to the structural similarity. Consider the basic unit block of VGG16BN and ResNet56. VGG16BN's basic unit block consists of convolution-batchnorm-activation. And the basic unit block of ResNet56 is similar to VGG16BN except an additional convolution-batchnorm-activation block and residual connection. That is, if the two units of the basic block of VGG16BN are connected, there is no difference with the basic block of ResNet56 except for the residual connection. On the other hand, MobileNetV2 not only uses depth-wise convolution, but also the kernel size of the convolution layer is different. Therefore, the pairs including MobileNetV2 showed relatively low hyperparameter transferability. Based on this interpretation for the result of Table \ref{tbl:cross_transferability}, we concluded that it is appropriate to conduct hyperparameter transfer between neural networks with high structural similarity in order to effectively transfer hyperparameters. \\ \\
\begin{figure}[t]
\centering
\begin{center}
\includegraphics[trim=0.0cm 0.0cm 0cm 0.0cm,width=0.8\columnwidth]{Figures/Hyperparameter_distribution_Fig3.png}
\end{center}
\caption{(a), (b) show the range of hyperparameters acquired from typical HPO and from HPO using proxy model, respectively. ResNet56 and CIFAR100 were used as a model and a dataset. Momentum and Gamma represent the momentum of SGD, and learning-rate-decay-term, respectively. Each horizontal line in a box plot represents the mean value.}
\label{fig:HP distributions}
\end{figure}
\indent \textbf{Additional hyperparameters.} We confirmed from Table \ref{tbl:main_results} that when the proxy model is used for HPO, the amount of computation can be significantly reduced and comparable performance can be achieved. Extending this result, we performed additional experiment to confirm that our proposal is still valid even if we optimize more hyperparameters. In this additional experiment, we added the momentum and learning rate decay term of the SGD optimizer as new hyperparameters. We used CIFAR100 and ResNet56 as a dataset and neural network, respectively. We used 5 random seeds in the experiment, and unlike the previous experiments, because the number of hyperparameters increased, the trial was increased to 100 times. And we used TPE~\cite{bergstra2011algorithms} and \textit{hyperband}~\cite{li2017hyperband} as hyperparameter optimizer and search space pruner, respectively.\\
\indent Hyperparameters obtained by performing typical HPO showed accuracy for the validation set of $69.83 \pm 1.33$. And the hyperparameters obtained by HPO using $N_{\mathcal{P}}$ as a proxy model showed accuracy for the validation set of $70.57 \pm 1.03$. These results are similar to the trend of Table \ref{tbl:main_results}, showing that our proposal is still valid even if the number of hyperparameters increases. Meanwhile, we compared the ranges of hyperparameters obtained through Fig.\ref{fig:HPO Procedures} (a) and (b). While the learning rate, weight decay, and learning rate decay term have similar mean value, the batch size and momentum show significant differences with respect to mean values~(See \ref{fig:HP distributions}). Although the two hyperparameters show significant difference, the reason why the hyperparameters obtained through each HPO showed similar performance is because the importance of each hyperparameter is different~\cite{zimmer2021auto}. According to zimmer et al.~\cite{zimmer2021auto}, it can be seen that learning rate, learning rate decay term, and weight decay are important hyperparameters in that order, and the other two hyperparameters are relatively less important. In other words, as the hyperparameter has a wide range of values, its importance is lower. As a result, even if we optimize hyperparameters by using our proposal, the results are consistent with the context of previous study~\cite{zimmer2021auto}.
\section{Introduction}
\label{Intro}
\input{contents/Introduction}
\section{Related Works}
\label{related}
\input{contents/Related_Works}
\section{Hyperparameter Transfer}
\label{HPO}
\input{contents/HPO}
\section{The Variation of Single Shot Pruning}
\label{Pruning}
\input{contents/Pruning}
\section{Experiments}
\label{exp}
\input{contents/Experiments}
\section{Conclusions}
\label{conclusion}
\input{contents/Conclusion}
\bibliographystyle{unsrt}
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{"url":"https:\/\/www.berfinsimsek.com\/publication\/simsek-geometry-2021\/","text":"Geometry of the Loss Landscape in Overparameterized Neural Networks: Symmetries and Invariances\n\nICML 2021\n\nAbstract\n\nWe study how permutation symmetries in overparameterized multi-layer neural networks generate \u2018symmetry-induced\u2019 critical points. Assuming a network with L layers of minimal widths $r_1^\u2217, \\ldots, r_{L-1}^\u2217$ reaches a zero-loss minimum at $r_1^\u2217! \u00b7 \u00b7 \u00b7 r_{L-1}^\u2217!$ isolated points that are permutations of one another, we show that adding one extra neuron to each layer is su\ufb03cient to connect all these previously discrete minima into a single manifold. For a two-layer overparameterized network of width $r^\u2217 + h \u2255 m$ we explicitly describe the manifold of global minima: it consists of $T (r^\u2217, m)$ a\ufb03ne subspaces of dimension at least h that are connected to one another. For a network of width m, we identify the number $G(r, m)$ of a\ufb03ne subspaces containing only symmetry-induced critical points that are related to the critical points of a smaller network of width $r < r^\u2217$. Via a combinatorial analysis, we derive closed-form formulas for $T$ and $G$ and show that the number of symmetry-induced critical subspaces dominates the number of a\ufb03ne subspaces forming the global minima manifold in the mildly overparameterized regime (small $h$) and vice versa in the vastly overparameterized regime ($h \\gg r^\u2217$). Our results provide new insights into the minimization of the nonconvex loss function of overparameterized neural networks.\n\nPublication\narXiv:2105.12221 [cs]","date":"2021-09-20 13:14:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.7853575944900513, \"perplexity\": 622.0666774217856}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057039.7\/warc\/CC-MAIN-20210920131052-20210920161052-00376.warc.gz\"}"} | null | null |
{"url":"https:\/\/codegolf.stackexchange.com\/questions\/194141\/gerrymander-north-carolina\/203212","text":"# The challenge\n\nHow well can you gerrymander North Carolina into 13 voting districts?\n\nIn this challenge, you use the following files to draw different maps for Republicans and Democrats.\n\nFile 1 gives county-level voting data from the 2016 presidential election (based on this and that), while File 2 gives adjacency information between counties (based on this). For File 2, the first entry of each row is adjacent to the other entries in that row. For example, Alexander is adjacent to Alexander, Caldwell, Catawba, Iredell and Wilkes. (Note that every county is adjacent to itself.)\n\nYour code should be written in a way that can gerrymander arbitrary maps. In particular, your code must take each of the following as inputs:\n\n\u2022 A set $$\\C\\$$ of counties and a set $$\\P\\$$ of parties.\n\n\u2022 Weights $$\\(v_{cp})\\$$ that count voters in county $$\\c\\in C\\$$ who vote for party $$\\p\\in P\\$$. (E.g., File 1.)\n\n\u2022 A connected graph $$\\G\\$$ with vertex set $$\\C\\$$. (E.g., File 2.)\n\n\u2022 A number of districts $$\\k\\$$.\n\n\u2022 A party $$\\p^\\star\\in P\\$$ to gerrymander in favor of.\n\nThe output must be an assignment of counties to districts such that the counties in each district induce a connected subgraph of $$\\G\\$$. For example, for the counties in Files 1 and 2 (listed in alphabetical order), the following is one of 13! sequences that specify the partition of counties into districts pictured below:\n\n3,5,5,10,5,5,13,11,10,8,9,12,1,12,13,13,3,9,3,12,13,12,9,8,13,10,13,13,4,4,8,11,11,5,7,9,11,12,11,13,4,11,7,12,12,11,1,13,4,12,7,13,3,13,9,12,12,11,12,2,12,1,1,7,8,11,13,3,13,13,8,13,3,11,9,3,10,10,3,1,9,8,10,1,5,5,12,12,13,10,11,6,11,11,5,8,5,7,5,12\n\n\n(This map is loosely based on the current map of U.S. congressional districts in North Carolina.)\n\n# Scoring\n\nYour score is determined by how your code performs for various problem instances.\n\nGiven a district assignment, let $$\\d\\$$ denote the number of districts for which party $$\\p^\\star\\$$ receives the plurality of votes cast, and let $$\\a\\$$ and $$\\b\\$$ denote the minimum and maximum number of voters in a district, respectively. Then the score for the district assignment is given by\n\n$$\\text{assignment score} = d - \\frac{b}{a}.$$\n\nFor example, if $$\\p^\\star\\$$ is the Republican party, then the district assignment described above has $$\\d=10\\$$, $$\\a=247739\\$$, and $$\\b=527337\\$$, leading to a score of about $$\\7.87\\$$.\n\nTo compute your score, gerrymander North Carolina (as specified in Files 1 and 2) into $$\\k=13\\$$ districts in favor of the Republican party, and then in favor of the Democratic party. Let $$\\S_1\\$$ denote the sum of these two assignment scores. Next, gerrymander 10 different modifications of North Carolina, each obtained by removing one of the following counties from Files 1 and 2:\n\nCleveland,Davidson,Hyde,Johnston,Madison,Montgomery,Pender,Scotland,Warren,Wilkes\n\n\n(Since North Carolina is biconnected, each of these are acceptable problem instances.) For each modification, gerrymander into $$\\k=13\\$$ districts in favor of the Republican party, and then in favor of the Democratic party. Let $$\\S_2\\$$ denote the sum of these 20 assignment scores. Then the score of your submission is given by\n\n$$\\text{submission score} = S_1+\\frac{S_2}{10}.$$\n\nIn addition to your code, you are encouraged to provide an illustration of your two maps that gerrymander all of North Carolina for Republicans and for Democrats. (DRA2020 might be a helpful resource.)\n\n\u2022 Does my solution have to halt before the heat death of the universe? If not, I know a proven perfect algorithm! While we won't know its score, it's known it's perfect anyway. \u2013\u00a0the default. Oct 10 '19 at 15:30\n\u2022 @someone - Feel free to submit your solution when it terminates. ;) \u2013\u00a0Dustin G. Mixon Oct 10 '19 at 16:15\n\u2022 I posit that any given algorithm should complete in no more than 4 years. ;) \u2013\u00a0Draco18s no longer trusts SE Oct 11 '19 at 14:17\n\u2022 I feel this question needs a bounty. \u2013\u00a0user9207 Nov 15 '19 at 20:30\n\u2022 @Anush I'm working on an answer... \u2013\u00a0Oliphaunt - reinstate Monica Nov 24 '19 at 21:47\n\n# C++, score = 33.526203, all test cases in 2min34s\n\n(of course, the score can be improved by running the program for more time; please do not do that to claim a victory or something)\n\nThat score is usually 11 won districts for the Republicans and from 9 to 11 for the Democrats.\n\nIn this revision, the score decreased significantly, because I found some really large bugs in the input parsing. My program also verifies its solution now.\n\n\/\/#define _GLIBCXX_DEBUG\n#include <iostream>\n#include <streambuf>\n#include <iomanip>\n#include <fstream>\n#include <cstring>\n#include <bitset>\n#include <cassert>\n#include <cstdio>\n#include <vector>\n#include <algorithm>\n#include <cmath>\n#include <climits>\n#include <random>\n#include <set>\n#include <map>\n#include <deque>\n#include <string>\n\nconstexpr uint64_t rotl(uint64_t x, char k)\n{\nreturn (x<<k) | (x>>(64-k)); \/\/typo of the day: replace the >> with <<\n}\nuint64_t xs128()\n{\nstatic uint64_t s_0 = 0, s_1 = 1;\nconst uint64_t s0 = s_0;\nuint64_t s1 = s_1;\nuint64_t res = s0 + s1;\ns1 ^= s0;\ns_0 = rotl(s0, 24) ^ s1 ^ (s1 << 16);\ns_1 = rotl(s1, 37);\nreturn res >> 4;\n}\nstd::vector<std::string> getCsvLine(std::istream& str)\n{\nhttps:\/\/stackoverflow.com\/a\/1120224 - saved some time writing a parser...\nstd::vector<std::string> result;\nstd::string line; std::getline(str,line);\nstd::stringstream lineStream(line);\nstd::string cell;\nwhile(std::getline(lineStream,cell, ','))\nresult.push_back(cell);\nreturn result;\n}\nstruct county\n{\nint good = 0;\nint sum = 0;\nstd::vector<int> others;\nstd::string name;\n};\nstruct district\n{\nint good = 0;\nint sum = 0;\nstd::vector<int> others;\nvoid operator += (county& rhs)\n{\ngood += rhs.good; sum += rhs.sum;\nfor(int i = 0; i < others.size(); i++)\nothers[i] += rhs.others[i];\n}\nvoid operator -= (county& rhs)\n{\ngood -= rhs.good; sum -= rhs.sum;\nfor(int i = 0; i < others.size(); i++)\nothers[i] -= rhs.others[i];\n}\n};\n\/\/seems like exactly 100 districts exist, below CHAR_MAX\nstd::vector<std::vector<char>> graph;\nstd::vector<char> curdistrict;\nstd::vector<county> counties;\nstd::vector<district> districts;\nstd::vector<char> marks;\nbool pluralityCheck(const district& d)\n{\nreturn d.good > *std::max_element(d.others.begin(), d.others.end());\n}\nint n, k, excluded;\ndouble getFitness(std::vector<char>& dmap)\n{\nstd::vector<district> districts(k);\nfor(district& d : districts)\nd.others.resize(counties[excluded==0?1:0].others.size());\nfor(int i = 0; i < n; i++)\nif(i != excluded) districts[dmap[i]] += counties[i];\ndouble score = 0;\nint min = INT_MAX, max = INT_MIN;\nfor(int i = 0; i < k; i++)\nscore += pluralityCheck(districts[i]),\nmin = std::min(min, districts[i].sum), max = std::max(max, districts[i].sum);\nscore -= double(max) \/ min;\nreturn score;\n}\nvoid dfs(char v, char color)\n{\nmarks[v] = color;\nfor(int to : graph[v])\nif(marks[to] != color && curdistrict[to] == curdistrict[v])\ndfs(to, color);\n}\nbool dfsFor(char at, char target, char color)\n{\nmarks[at] = color;\nif(at == target) return true;\nif(at == excluded) return false;\nbool ans = false;\nfor(int to : graph[at])\nif(marks[to] != color && curdistrict[to] == curdistrict[at])\nans |= dfsFor(to, target, color);\nreturn ans;\n}\nbool isCutpoint(int v)\n{\nstatic int call = 0;\nchar color = call + 1;\ncall = (call + 1) % 255;\nif(color == 1) for(char& el : marks) el = 0; \/\/reset every 256 calls\nmarks[v] = color;\nint calls = 0;\nfor(char to : graph[v])\n{\nif(curdistrict[to] != curdistrict[v]) continue;\nif(marks[to] == color) continue;\ncalls++;\nif(calls == 2) break;\ndfs(to, color);\n}\nreturn calls != 1; \/\/0 -> kills a district, >=2 -> a cutpoint -> splits a district\n}\nstd::map<std::string, int> dnamemap;\ndouble solve(int n, int k, int target, int excluded)\n{\n::n = n; ::k = k; ::excluded = excluded;\nmarks = std::vector<char>(n);\ncurdistrict = std::vector<char>(n, -1); graph = std::vector<std::vector<char>>(n);\ndistricts = std::vector<district>(k); counties = std::vector<county>(n);\ndnamemap = std::map<std::string, int>();\nstd::ifstream data(\"NC_2016_presidential_election.csv\");\nfor(int i = 0; i < n; i++)\n{\nstd::vector<std::string> row = getCsvLine(data);\ndnamemap[row[0]] = i;\ncounties[i].name = row[0];\nif(i == excluded) continue;\nfor(int j = 1; j < row.size(); j++)\n{\nint v = std::stoi(row[j]);\nif(j - 1 == target)\ncounties[i].good = v;\nelse counties[i].others.push_back(v);\ncounties[i].sum += v;\n}\n\n}\n\/\/NC_county2.csv is obtained by cat NC_county_adjacency.csv | sort | tr -d '\\r'\n\/\/the default counties file caused not one but TWO horrible parsing bugs\n\/\/they silently increased the score.\n\/\/There were \\r\\n line endings instead of \\n, and my program mishandled them.\n\/\/and *the worst*. It seems to be in alphabetic order, but it actually simply isn't.\nfor(int i = 0; i < n; i++)\n{\nstd::vector<std::string> row = getCsvLine(data);\nif(i == excluded) continue;\nfor(int j = 1; j < row.size(); j++)\n{\nif(row[j] == row[0]) continue;\nint to = dnamemap[row[j]];\nif(to == excluded) continue;\ngraph[i].push_back(to);\n}\n}\nfor(district& d : districts)\nd.others.resize(counties[excluded==0?1:0].others.size());\nint di = 0;\nfor(int i = 0; di < k; i++) if(i != excluded) curdistrict[i] = di++;\nfor(int it = n; it --> 0;)\nfor(int i = 0; i < n; i++)\n{\nif(i == excluded || curdistrict[i] != -1) continue;\nfor(int to : graph[i])\n{\nif(to == excluded || curdistrict[to] == -1) continue;\ncurdistrict[i] = curdistrict[to];\nbreak;\n}\n}\nif(excluded >= 0) curdistrict[excluded] = -1;\nfor(int i = 0; i < n; i++) printf(\"%d \", curdistrict[i]);\nprintf(\"%d\\n\", excluded);\nfor(int i = 0; i < n; i++) if(excluded != i) districts[curdistrict[i]] += counties[i];\ndouble temp = 3;\nint its = 2e7;\nstd::vector<char> bestplacement = curdistrict;\nstd::vector<district> bestdistricts = districts;\nint lastimprovement = 0;\ndouble bestscore = getFitness(curdistrict);\nint perblock = 2e4; \/\/max number of iterations without improvement\ndouble delta = 1.01 * temp \/ its;\ndouble curscore = bestscore;\ndouble withoutimprovement = 0;\nfor(int it = 0; it < its; it++)\n{\ntemp -= delta;\nif(it % 1048576 == 0)\nprintf(\"it=%d, cs=%lf \\r\", it, curscore);\nint i = xs128() % n;\nwhile(i == excluded) i = xs128() % n;\nif(isCutpoint(i)) continue; \/\/<- around 50% of program time\nstd::vector<int> nearby;\nbool ok = false;\nfor(int to : graph[i])\nif(curdistrict[to] != curdistrict[i] && curdistrict[to] != -1) ok = true;\nif(!ok) continue;\nint nd = -1;\nwhile(nd == curdistrict[i] || nd == -1)\nnd = curdistrict[graph[i][xs128() % graph[i].size()]];\n\/\/calculate new min size, new max size etc.\n\/\/can be done via std::set and other tree-based stuff, but it's much simpler to use a normal arrays (for k=13)\n\/\/*might* use a segment tree to optimize later, but it's not spending a lot of time here\nchar oldmaj1 = pluralityCheck(districts[curdistrict[i]]);\nchar newmaj1 = pluralityCheck(districts[nd]);\nint maxs = INT_MIN, mins = INT_MAX;\nfor(int i = 0; i < k; i++)\nmaxs = std::max(maxs, districts[i].sum), mins = std::min(mins, districts[i].sum);\ndouble pen1 = double(maxs) \/ mins;\ndistricts[curdistrict[i]] -= counties[i];\ndistricts[nd] += counties[i];\nchar oldmaj2 = pluralityCheck(districts[curdistrict[i]]);\nchar newmaj2 = pluralityCheck(districts[nd]);\nmaxs = INT_MIN, mins = INT_MAX;\nfor(int i = 0; i < k; i++)\nmaxs = std::max(maxs, districts[i].sum), mins = std::min(mins, districts[i].sum);\ndouble pen2 = double(maxs) \/ mins;\ndouble delta = pen1 - pen2 + (oldmaj2 + newmaj2 - oldmaj1 - newmaj1);\nif(delta <= 0 && (temp <= 0 || ldexpf(std::exp(delta \/ temp), 60) < xs128()))\n{\ndistricts[nd] -= counties[i], districts[curdistrict[i]] += counties[i];\n}\nelse\n{\ncurdistrict[i] = nd;\ncurscore += delta;\nif(curscore > bestscore)\n{\nbestscore = curscore; lastimprovement = it;\nbestplacement = curdistrict; withoutimprovement = 0;\nbestdistricts = districts;\n}\n}\nif(curscore < bestscore) withoutimprovement += bestscore - curscore;\nif(it - lastimprovement > perblock || withoutimprovement > 3e4)\n{\n\/\/restart at best score so far\nwithoutimprovement = 0;\nlastimprovement = it;\ncurdistrict = bestplacement;\ndistricts = bestdistricts;\ncurscore = bestscore;\n}\n}\nfor(int i = 0; i < n; i++)\nprintf(\"%d \", curdistrict[i]);\nprintf(\"\\n\");\nfor(district& d : districts)\n{\nprintf(\"%d total, maj=%d, %d good; rest:\", d.sum, (int)pluralityCheck(d), d.good);\nfor(int el : d.others) printf(\" %d\", el);\nprintf(\"\\n\");\n}\nfor(int i = 0; i < n; i++) \/\/verify validness\nfor(int j = i+1; j < n; j++)\n{\nif(i == excluded || j == excluded) continue;\nif(curdistrict[i] != curdistrict[j]) continue;\nfor(char& el : marks) el = 0;\nbool good = dfsFor(i, j, 1);\nif(!good) printf(\"invalid: i=%d, j=%d, d=%d!\\n\", i, j, curdistrict[i]);\n}\ndouble score = getFitness(curdistrict);\nprintf(\"%lf\\n\", score);\nreturn score;\n}\nint main()\/\/int64_t argc, char*argv[])\n{\n\/\/randomly choose and recolor vertices\n\/\/do not recolor a vertex if it is a cutpoint (more known as an articulation point or cut vertex) or alone\n\/\/to check, DFS over same-colored vertices from it. If != 1 calls were done, it's alone or a cutpoint\n\/\/initial state: all counties belong to district 0, but first 13 belong to district i (dumb but why bother)\n\/\/for each district maintain the voters for both parties and total number\n\/\/oh, and don't merely randomly recolor: use Simulated Annealing (TM) (it actually improves a lot!)\nsetbuf(stdout, 0);\nint k = 13, n = 100;\nstd::vector<int> toExclude { -1, 22, 28, 47, 50, 57, 61, 70, 82, 92, 97 }; \/\/the final score computing\n\/\/std::vector<int> toExclude { -1 };\ndouble score = 0;\nfor(int el : toExclude)\n{\ndouble s = 0;\ns += solve(n, k, 0, el);\ns += solve(n, k, 1, el);\nif(el == -1) score += s;\nelse score += s \/ 10;\n}\nprintf(\"%lf\\n\", score);\n}\n\n\nUses (attempts to use) simulated annealing to improve from a dumb assignment by making random changes if they do not ruin things (that is, destroy or split districts). Restarts from the best known assignment when either 20000 iterations passed without an improvement or the accumulated difference between the best score and the current score since the last improvement exceeds 30000. Can be compiled with clang++ hax.cpp -Ofast -march=native -flto -no-pie -o a.out. Assumes files NC_2016_presidential_election.csv and NC_county2.csv are in the current directory, where NC_county2.csv is the NC_county_adjacency.csv, but with its lines actually sorted in alphabetical order, and with Unix (\\n) line endings.\n\nQuoting from the question, \"you are encouraged to provide an illustration\". I am definitely encouraged (I would really like to know whether or not the code has no bugs), but I have no idea how to create an illustration. I also have no idea how to export the output into a format readable by the linked website.\n\nAssignment information on the normal map for the Democrats, then for the Republicans:\n\nbegin assignment for target=0, excluded=-1\nAlamance,4\nAlexander,8\nAlleghany,8\nAnson,12\nAshe,8\nAvery,5\nBeaufort,6\nBertie,6\nBrunswick,9\nBuncombe,5\nBurke,10\nCabarrus,8\nCaldwell,8\nCamden,7\nCarteret,9\nCaswell,0\nCatawba,10\nChatham,12\nCherokee,10\nChowan,7\nClay,10\nCleveland,10\nColumbus,2\nCraven,9\nCumberland,2\nCurrituck,7\nDare,7\nDavidson,12\nDavie,8\nDuplin,6\nDurham,12\nEdgecombe,6\nForsyth,3\nFranklin,7\nGaston,10\nGates,7\nGraham,10\nGranville,0\nGreene,6\nGuilford,4\nHalifax,7\nHarnett,12\nHaywood,10\nHenderson,10\nHertford,7\nHoke,2\nHyde,7\nIredell,8\nJackson,10\nJohnston,12\nJones,6\nLee,12\nLenoir,6\nLincoln,11\nMacon,10\nMartin,6\nMcDowell,5\nMecklenburg,11\nMitchell,5\nMontgomery,12\nMoore,12\nNash,7\nNew Hanover,9\nNorthampton,7\nOnslow,9\nOrange,0\nPamlico,9\nPasquotank,7\nPender,9\nPerquimans,7\nPerson,0\nPitt,6\nPolk,10\nRandolph,4\nRichmond,12\nRobeson,2\nRockingham,0\nRowan,8\nRutherford,10\nSampson,9\nScotland,2\nStanly,8\nStokes,0\nSurry,3\nSwain,10\nTransylvania,10\nTyrrell,7\nUnion,8\nVance,7\nWake,1\nWarren,7\nWashington,6\nWatauga,5\nWayne,9\nWilkes,8\nWilson,6\nYancey,10\nend assignment for target=0, excluded=-1\nbegin assignment for target=1, excluded=-1\nAlamance,0\nAlexander,1\nAlleghany,11\nAnson,7\nAshe,11\nAvery,10\nBeaufort,6\nBertie,6\nBrunswick,12\nBuncombe,10\nBurke,10\nCabarrus,7\nCaldwell,11\nCamden,9\nCarteret,6\nCaswell,0\nCatawba,1\nChatham,2\nCherokee,4\nChowan,9\nClay,4\nCleveland,4\nColumbus,12\nCraven,9\nCumberland,8\nCurrituck,6\nDare,6\nDavidson,0\nDavie,7\nDuplin,9\nDurham,0\nEdgecombe,9\nForsyth,1\nFranklin,6\nGaston,4\nGates,9\nGraham,10\nGranville,6\nGreene,6\nGuilford,11\nHalifax,6\nHarnett,8\nHaywood,10\nHenderson,4\nHertford,9\nHoke,2\nHyde,6\nIredell,7\nJackson,4\nJohnston,8\nJones,9\nLee,2\nLenoir,9\nLincoln,10\nMacon,4\nMartin,9\nMcDowell,10\nMecklenburg,5\nMitchell,10\nMontgomery,2\nMoore,2\nNash,9\nNew Hanover,12\nNorthampton,9\nOnslow,12\nOrange,2\nPamlico,9\nPasquotank,6\nPender,12\nPerquimans,6\nPerson,0\nPitt,6\nPolk,4\nRandolph,0\nRichmond,7\nRobeson,8\nRockingham,0\nRowan,2\nRutherford,4\nSampson,12\nScotland,2\nStanly,2\nStokes,11\nSurry,11\nSwain,10\nTransylvania,4\nTyrrell,6\nUnion,7\nVance,6\nWake,3\nWarren,9\nWashington,6\nWatauga,11\nWayne,9\nWilkes,1\nWilson,9\nend assignment for target=1, excluded=-1\n\u2022 That'd be very nice. Is a CSV with rows like Name,district_number good? \u2013\u00a0the default. 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\section{Insert A head here}
\maketitle
\section{Introduction}
Uranus and Neptune are the outermost planets in the Solar System.
These two planets raise great challenges to planetary scientists in terms of their formation history, evolution path, internal structure and composition, atmospheric dynamics, and many other areas.
Nevertheless, despite so many key questions around these planets, until recently, they have received relatively little attention. \par
Modeling the internal structures of Uranus and Neptune is not simple. They represent a unique planetary class -- it is not possible to simply re-scale models of the terrestrial or gas giant planets, and many of the conclusions inferred on the planets' composition and internal structure surely reflect more the assumptions of the modeler than the reality. In a way, it is still unclear what the most reasonable assumptions should be when modeling these planets.
Today, we know that even the gas giants, Jupiter and Saturn, have far more complex internal structures than had been assumed before the \emph{Juno} and \emph{Cassini} missions (Wahl et al. 2017, Fortney et al. 2019, Stevenson 2020, Mankovich 2020), and it certainly follows that if we now gained new detailed information about Uranus and Neptune, we would be similarly surprised.
The large number of observed exoplanets with masses and radii similar to those of Uranus and Neptune suggests that such planets are very common in the galaxy.
Nevertheless, we still do not know as much about the ``ice giants'' as it is commonly assumed. Before we state that a given exoplanet is similar to Uranus/Neptune, we first need to know what Neptune and Uranus are like. In addition, as we discuss below, it is still unclear how similar Uranus and Neptune are to each other, and we suggest that each planet should be investigated separately accounting for its unique features.
\par
In the last several years efforts for designing space missions dedicated to the exploration of the ice giants have been made (e.g., Arridge et al.~2014, Masters et al.~2014, Mousis et al., 2018, Hofstadter et al.~2019, Fletcher et al.~2020). These efforts are very much ongoing, and we hope that a mission(s) to either or both of the ice giants will become a reality in the relatively near future.
In this paper we focus on the interiors of Uranus and Neptune, and discuss the challenges they impose to the planetary science community. We list the key open questions and discuss the required developments in theory and observations.
Other recent reviews and papers on the topic include Helled et al.~(2020), Guillot (2020), Helled \& Guillot (2018), Fortney \& Nettelmann, (2010), and references therein.
This paper aims to be complementary to these recent publications, with a focus on open questions and what measurements would enable advances in our understanding.
\section{Not "Uranus and Neptune", but "Uranus" and "Neptune"}
It is clear that Uranus and Neptune represent a unique planetary class. They consist of H-He outer envelopes (and atmospheres) of about 10-20\% of their total mass, they are located in the outer part of the solar system, and they have similar masses, radii, and rotation periods.
Therefore, as for Jupiter and Saturn, or Earth and Venus, but perhaps even more so here,
modelers have tended to investigate Uranus and Neptune together.
However, each planet has clearly unique features, which justify a detailed investigation of each individual object. After all, just like the mentioned examples, there are clear differences between Jupiter and Saturn, as well as Venus and Earth, and by grouping them together,
we may miss some of the key features that can reveal more information of the nature of each planet. In addition, always studying the two objects together may erode people's sense of the importance and uniqueness of each planet.
We therefore argue that each planet should be treated separately and uniquely.
\clearpage
Below we briefly discuss some of the key physical quantities of Uranus and Neptune, with a particular focuses on differences between the planets.
Their fundamental physical properties are listed in Table 1.
\begin{table}[h]
{\small
\def.6{1.2}
\centering
\begin{tabular}{lll}
\hline
\hline
{\bf Parameter} & {\bf Uranus} & {\bf Neptune}\\
\hline
Semi-major axis (AU) & 19.201 & 30.047 \\
Mass ($10^{24}$ kg) & 86.8127 $\pm$ 0.0040$^a$& 102.4126 $\pm$ 0.0048$^b$\\
Mean Radius$^*$ (km) & 25362 $\pm$ 7$^c$ & 24622 $\pm$ 19$^c$\\
Mean Density (g cm$^{-3}$) & 1.270 $\pm$ 0.001$^d$& 1.638 $\pm$ 0.004$^d$\\
R$_{\rm ref}$ (km) & 25,559$^a$& 25,225$^b$ \\
J$_2$ ($\times$10$^{6}$) &3510.68 $\pm$ 0.70$^a$& 3408.43 $\pm$ 4.50$^b$\\
J$_4$ ($\times$10$^{6}$) & -34.17 $\pm$ 1.30$^a$ & -33.40 $\pm$ 2.90$^b$\\
Rotation period$^*$ (Voyager) & 17.24 h$^e$& 16.11 h$^f$\\
1-bar Temperature (K) & 76 $\pm$ 2$^g$& 72 $\pm$ 2$^h$\\
Effective Temperature (K) & 59.1 $\pm$ 0.3$^i$ & 59.3 $\pm$ 0.8$^j$ \\
Intrinsic flux (J s$^{-1}$ m$^{-2}$) & 0.042 $\pm$ 0.045$^i$ & 0.433 $\pm$ 0.046$^j$\\
Bond Albedo $A$ & 0.30 $\pm$ 0.049$^h$ & 0.29 $\pm$ 0.067$^h$ \\
Axis tilt & 97.77 & 28.32 \\
\hline
\hline
\end{tabular}
\caption{Basic physical properties of Uranus \& Neptune.
$^a$Jacobson, R.A. 2014. $^b$Jacobson, R.A. 2009.
$^c$Archinal et al. 2018. $^d$Calculated values and associated uncertainty derived from other referenced values and uncertainties in this table. The average density is computed using a volume of a sphere with the listed mean radius.
$^e$Desch et al., 1986 $^f$Warwick et al., 1989. $^g$Lindal et al. 1987.
$^h$Lindal, 1992.
$^i$Pearl et al. 1990 $^j$Pearl \& Conrath, 1991.
$^*$Note that the rotation periods of the planets are not well determined as discussed in detail in Helled et al., 2010.
R$_{\rm ref}$ is the reference equatorial radius in respect to the measured gravitational harmonics J$_2$ and J$_4$. }
}
\label{tab:1}
\end{table}
\subsection{Basic Physical Properties}
Uranus' mass is slightly smaller than Neptune's, but its radius is a bit larger. As a result, they differ in their mean densities which are 1.270 g cm$^{-3}$ and 1.638 g cm$^{-3}$ for Uranus and Neptune, respectively. This difference can already hint at different bulk compositions. In addition, the inferred normalized moment of inertia (MoI) value of Uranus ($\sim$ 0.22) is smaller than that of Neptune ($\sim$ 0.24) , suggesting that Uranus is more centrally concentrated than Neptune (see Podolak \& Helled, 2012 and Nettelmann et al., 2013 for details).
While the rotation periods of the planets are not well-determined, the \emph{Voyager} rotation periods suggest a difference in rotation period of 7\%. \emph{Voyager 2} measurements of periodic variations in their radio signals and of fits to the magnetic fields of Uranus and Neptune imply rotation periods of 17.24 h and 16.11 h, respectively. However, see Helled et al. (2010) for a discussion of why these values may be incorrect. These differences in the most fundamental planetary properties already suggest that Uranus and Neptune should not be considered as "twin planets".
\subsection{Heliocentric Distance}
Uranus is located at $\sim$ 19 AU while Neptune is at $\sim$ 30 AU. While both of these locations are far from the sun, and represent the outer regions of the solar system, they have nearly 10 AU between them. For comparison, this is about the distance between Mercury and Saturn! The factor of 2.5 difference in incident solar flux is larger than the difference between either Earth and Venus or Earth and Mars.
While at both of these radial distances it is clear that the two planets formed beyond the water and the CO$_2$ ice lines, 30 AU is closer to the nominal CO ice line (e.g., {\"O}berg et al., 2011). A difference of 10 AU could lead to substantial differences in the heavy element enrichments of the planets (see Mousis et al., 2020 and references therein).
In addition, since the solid-surface densities at the two locations is different, the planetary growth history is is also expected to differ (e.g., Helled \& Bodenheimer, 2014).
It is possible that Uranus and Neptune formed much closer to the Sun, and even switched positions\footnote{With Neptune originally being closer to the Sun than Uranus, which could be supported by the fact that Neptune is slightly more massive than Uranus.} (e.g., Thommes et al. 1999; Tsiganis et al. 2005). This can partially resolve the formation timescale problem of Uranus and Neptune (e.g., Helled \& Bodenheimer, 2014). However, a calculation that includes the planetary growth, accounting for the heavy-elements self-consistently, and that can lead to the predicted H-He-to-heavy elements ratios of either planet is still missing.
\subsection{Axis tilt \& Satellite systems}
A distinct feature of Uranus is its axial tilt.
This has typically been thought to be a result of a giant impact (Safronov 1966, Stevenson 1986, Podolak \& Helled, 2012, Kegerreis et al., 2018, 2019, Kurosaki \& Inutsuka, 2018, Reinhardt et al. 2020) although the tilt could also be a result of a spin-orbit resonance (Kubo-Oka \& Nakazawa 1995, Bou{\'e} \& Laskar 2010, Rogoszinski \& Hamilton 2020).
Giant impacts might also be responsible for some of the observed differences of the planets such as their heat fluxes and internal structure\footnote{In fact, the small difference in masses could be a result of the impacts if indeed Neptune's impact was head-on, leading to the absorption of the entire impactor's mass, unlike an oblique impact on Uranus.} (Stevenson, 1986; Reinhardt et al., 2019). In any case, no matter what the origin of the tilt is, it is clear that the seasons and temperature variations on the two planets are different, which affects the connection between the atmosphere and the deep interior and therefore the characterization of the planets (e.g., Guillot, 2020, Hueso et al., 2020).
Uranus has regular satellites, suggesting they formed from a circumplanetary disk. It is yet to be determined whether the circumplanetary disk was formed as a result of a giant impact (e.g., Kegerreis et al., 2018, Reinhardt et al., 2020, Ida et al., 2020) or a result of the gas accretion (e.g., Canup \& Ward, 2006).
Unlike Uranus' regular satellites, the orbit of
Neptune's largest moon, Triton, is retrograde, implying that Triton was captured (e.g., Nesvorný et al., 2007), which perhaps destroyed any original regular satellite system.
While the differences in the axis tilt and satellite system could be a result of giant impacts with different conditions with the planets being more similar to each other shortly after formation, it would suggest different evolution histories. In addition, the origin of the moons of Uranus and Neptune is still being investigated, as well as the cause for the differences between the two planets. It is therefore clear that the unique features of each planet should be accounted for in their modeling.
\subsection{Heat Flux and Albedo}
A strong indication for the dichotomy between Uranus and Neptune comes from the far different energy balances of Uranus and Neptune. Uranus's intrinsic power, as determined from the Voyager IRIS instrument is $42 \pm 47$ erg s$^{-1}$ cm$^{-2}$ (essentially a non-detection), while the value for Neptune is $433 \pm 46$, a value $\sim$5-10$\times$ larger (at $1\sigma$) (Pearl et al., 1990, Pearl \& Conrath 1991). Uranus appears to be in equilibrium with solar insolation.
Both planets have a similar Bond albedo, as determined by \emph{Voyager} data (Pearl \& Conrath 1991). We discuss this in light of new data on Jupiter's Bond albedo in a later section.
\subsection{Atmosphere: Activity, Depth of Winds, Composition}
Images of Uranus and Neptune imply that Uranus' atmosphere has only very few features while Neptune's atmosphere seems more active and complex including storms and vortexes. This could either be a result of different internal heat fluxes, the different orbital properties, as well as the different seasons due to the different axial tilt. It is desirable to understand the origin of the different atmospheric activity.
In terms of atmospheric winds, the two planets seem to have a similar wind profile characterized by a strong eastward jet at the equator. The wind velocities are referenced in comparison to the underlying assumed rotation periods which are likely more uncertainly than typically appreciated. It was shown in Helled et al.~(2010) that the inferred flattening of the planets are inconsistent with the \emph{Voyager} radio periods, and new rotation periods which are more consistent with the data have been suggested. With these modified periods (of 16.58 h for Uranus and 17.46 h for Neptune) the wind velocities on the two planets are much more similar and are slower than previously thought. As a result, the wind speeds are not well known but are expected to be of the order of a couple 100 m/s. The penetration depths of these winds are unknown, but are thought to be as deep as 1100 km for both planets, as estimated from their gravity data (Hubbard et al. 1991, Kaspi et al., 2013).
The atmospheres of both planets are so cold that most volatile species have condensed into clouds far below the visible atmosphere. In addition, the noble gases would only be detectable via an entry probe with a mass spectrometer. The atmospheres of Uranus and Neptune are mostly H-He (by particle numbers) with smaller fractions of heavier elements. The only heavy element with a well-determined composition is carbon, in the form of methane, CH$_4$. Even this measurement is problematic, as CH$_4$ partially condenses at such cool temperatures. Recent assessments of the CH$_4$ abundances are Karkoschka \& Tomasko (2011) for Uranus and Sromovsky et al. (2011) for Neptune. See Atreya et al. (2020) for a detailed review and discussion of how latitude-dependent condensation effects these measurements.
Compared to the solar carbon abundance, these carbon values are $85\pm 15$ for Uranus and $89 \pm 22$ for Neptune. For comparison, the methane abundance in Saturn from \emph{Cassini} spectroscopy is $9.9 \pm 0.4$ (Fletcher et al., 2009), and for Jupiter it is $4.4 \pm 1.1$ from the \emph{Galileo} entry probe (Wong et al., 2004). This may suggest that there is a strong anti-correlation between atmospheric metallicity and giant planet mass. However, the expected very different formation locations of the four planets make this direct interpretation difficult.
The previously mentioned condensation of CH$_4$, and other volatiles like H$_2$S and H$_2$O, is important for reasons beyond understanding composition. Condensation into clouds removes these molecules from the gas phase, which alters the mean molecular weight of the atmosphere across the relatively narrow thickness of the cloud. This may lead to a number of regions of the middle and deep atmosphere becoming superadiabatic, as a steeper temperature gradient would be needed to drive convective motion. The connection with the planetary interior is that these atmospheric superadiabatic regions would lead the deep interior to be hotter than previous simple estimates. Much additional discussion can be found in Guillot (1995), Guillot (2020), Fletcher et al., (2020), Leconte et al., (2017), and Friedson \& Gonzalez (2017).
\section{Interior Models}
The basic idea of planetary modeling is as follows: given the measured physical properties of a planet (mass, radius, rotation rate, gravity field, etc.) a structure model is developed to reproduce the observed properties. The density profiles that fit the data can teach us about the planetary composition and its depth dependence.
The more measurements we have, and the more accurate they are, the better the internal structure is determined.
However, accurate measurements are insufficient to \emph{uniquely infer} the planetary structure and composition, given the degenerate nature of the problem. Even if we precisely knew all the fundamental properties of a planet, such as its mass, shape, gravitational and magnetic fields, there would still be ambiguity in determining the composition and internal structure. This is because there is more than one solution for the planetary density profile that can satisfy all the observational constraints.
In addition, structure models suffer from "theoretical uncertainties" that are linked to the EOS, composition, energy transport mechanism, and structure assumed by the modeler.
\par
Nevertheless, the available data can be used to exclude certain solutions, and identify solutions that seem to be consistent with complementary knowledge about the planets such as their formation location (far from the sun, beyond certain ice lines) and predicted composition, magnetic fields (e.g., the need for convection and high-enough electrical conductivity), and the behaviour of elements at high pressures/temperatures that can guide structure models (see Helled et al., 2020 and references therein for details).
However, at the moment, some key observed properties of Uranus and Neptune are not well determined.
This is in particular notable now when the two planets are compared to Jupiter and Saturn which were explored by various spacecraft through the years including the recent visits by \emph{Juno} and \emph{Cassini} which provided unprecedented accurate measurements of their gravity fields.
For Uranus and Neptune the gravitational moments are determined only up to fourth degree ($J_2, J_4$), with a relatively large uncertainty, and detailed information on their atmospheric composition, planetary shape, and rotation periods is missing (e.g., Helled et al. 2010). It should be noted that a recent study provided a new estimate for Neptune's second gravitational moment, $J_2 = 3409.1 \pm 2.9 \times 10^{-6}$ (Brozović et al., 2020). This value is consistent with the value of $3408.43 \pm 4.5 \times 10^{-6}$ inferred by Jacobson (2009).
\begin{table}[h!]
{\small
\def.6{1.3}
\centering
\begin{tabular}{lllll}
\hline
\hline
& {\bf Jupiter} & {\bf Saturn} &{\bf Uranus} & {\bf Neptune}\\
\hline
J$_2$ old ($\times$10$^{6}$) & 14696.43 $\pm$ 0.21 &
16290.71 $\pm$ 0.27 & 3516 $\pm$ 3.2 & 3539 $\pm$ 10 \\
J$_2$ new ($\times$10$^{6}$) & 14696.572 $\pm$ 0.014 & 16290.557 $\pm$ 0.028 & 3510.68 $\pm$ 0.70 & 3408.43 $\pm$ 4.50 \\
J$_4$ old ($\times$10$^{6}$) & -587.14 $\pm$ 1.68 & -935.8 $\pm$ 2.8 & -35.4 $\pm$ 4.1 & -28 $\pm$ 22 \\
J$_4$ new ($\times$10$^{6}$) & -586.609 $\pm$ 0.004 & -935.318 $\pm$0.044 & -30.44 $\pm$ 1.02 & -33.40 $\pm$ 2.90\\
\hline
\hline
\end{tabular}
\caption{Gravity data from Figure 1. The old and new gravity data for Jupiter are taken from Jacobson (2003) and Iess et al.~(2018), respectively. For Saturn old gravity data corresponds to Jacobson (2006) and the updated one from Iess et al.~(2019). The gravity data for Uranus are from French et al.~(1988, "old") and Jacobson (2014, "new"), and for Neptune from Tyler et al.~(1989,"old") and Jacobson (2009, "new"), respectively. Note that "old" data correspond to the latest data before the most recent, i.e., "new" data.}
}
\label{tab:1}
\end{table}
Figure 1 shows the gravity measurement ($J_2$, $J_4$) uncertainties of the outer planets. Shown are the improvements of new data in comparison to older ones. The values of the gravitational moments used for this figure are listed in Table 2. Thanks to the \emph{Juno} and \emph{Cassini} Grand Finale measurements, the uncertainties in the gravity data decreased dramatically for Jupiter and Saturn. Such accurate data push structure models to the next-level of complexity (see Helled, 2018 for review). Unfortunately, it is clear that the quality of the data for Uranus and Neptune is orders of magnitude worse. As a result, the compositions and internal structures of these planets are poorly determined.
It is clear that measuring the gravity fields of Uranus and Neptune accurately is desirable; such accurate measurements can only be done via a Juno-like spacecraft which orbits the plant several times, with the orbits being polar, covering different regions, and reach close to the planet. We therefore strongly support orbiter missions to the ice giants (e.g., Fletcher et al., 2020 and references therein).
Uranus and Neptune are special from a theoretical modeling perspective because they are clearly different from the terrestrial planets and the gas giants. Therefore, it is not clear what the best approach to take is when modeling their interiors.
Contrary to most published models, and corresponding artists conceptions, it is unlikely that they are fully differentiated objects with distinct layers. Although it is often assumed that they are water-rich, their composition is poorly constrained as discussed in Helled et al.~(2011, 2020). Also, since they are not H-He dominated, the chosen materials and their distribution significantly affects the inferred composition. Planets in this mass regime are in fact most sensitive to the assumed planetary internal structure and the equation of state (EOS) used by the modeler (e.g., Baraffe et al., 2008, Vazan et al., 2013). \\
Planet formation models suggest that deep interiors of gaseous planets are expected to consist of composition gradients, although this is just started to be included in current structure models.
During the planetary growth phase the accreted heavy elements (pebbles and/or planetesimals) are dissolved in the envelope already when the core mass reaches a few Earth masses (M$_{\oplus}$). An example for such a gradient has been presented in Helled \& Stevenson (2017) who also note that the more gradual structure is associated with a lower solid surface density. As a result, the primordial interior of Uranus and Neptune are expected to have a large region with composition gradients. Figure 2 shows the heavy-element mass fraction as a function of planetary mass during the planetary growth when assuming two different solid surface densities.
These curves are not meant to represent proto-Uranus/Neptune but to demonstrate the sensitivity of the inferred primordial structure to the assumed formation environment. As a result, a better understanding of the evolution and internal structure of Uranus and Neptune, could help us to better understand their origin (see Helled et al.~2020 for further discussions).
The heavy-element accretion rate is affected by the (assumed) solid surface density $\sigma$, and this in return affects the subsequent growth and the gas accretion rate. As a result, not only do the planets form in different timescales but they also have very different final compositions. It is clear that for the case with lower $\sigma$ the planet is more H-He rich. In both cases, the planets are expected to have primordial internal structures with composition gradients (see Helled \& Stevenson, 2017 and Helled et al.~2020 for further discussions).
Such primordial composition gradients are expected to evolve on a timescale of 10$^9$ and convective mixing could lead to an outer region that is convective. We suggest that future studies should investigate different formation paths that can lead to planets that are similar to Uranus and Neptune.
Since both Uranus and Neptune are located in the outer parts of the Solar System where the solid-surface density is low, even if they migrated outward they are likely to have formed further than Jupiter and Saturn, and therefore, have composition gradients. Composition gradients are also predicted for the gas giants. There is compelling seismic evidence for a deep composition gradient in Saturn (Fuller 2014) and interesting (but more indirect, since it is from the gravity field) for Jupiter (Wahl et al., 2017, Vazan et al., 2018, Debras \& Chabrier, 2019).
Interior models of Uranus and Neptune assuming adiabatic temperature profiles with distinct layers have been presented by Nettelmann et al.~(2013). In these models the planets are assumed to consist of a rocky core surrounded by a water layer, and a H-He atmosphere with a given metallicity. These models use physical EOSs to model different materials. While these models are in some way "more physical" they are rather sensitive to the model assumptions. Perhaps most troubling is that such distinct-layer models tend to predict extremely high water-to-rock ratios for both planets. The inferred Uranus and Neptune models of Nettelann et al.~(2013) predict that the water-to-rock ratio is 19-35 times for Uranus, and 4-15 times for Neptune, with the total H-He mass is typically 2 and 3 M$_{\oplus}$ for Uranus and Neptune, respectively. The exact estimates are highly model-dependent, and are sensitive to the assumed composition, thermal structure (which depends on the assumed heat transport mechanism), and rotation rate.
It is important to note that such structure models with distinct layers of different compositions are likely unrealistic since rock and water can be mixed as well as water and hydrogen (Soubiran \& Militzer, 2015, Soubiran et al. 2017). In addition, the extremely high water-to-rock ratios are not observed in any solar-system object, and as discussed above there is a strong indication from formation and evolution models that the planets consist of composition gradients. Indeed, models where rock is more gradually distributed within the planet go in the direction of alleviating the high water-to-rock ratio problem. As a result, such 3-layer models are unlikely to properly represent the interiors Uranus and Neptune.
Another approach for interior modeling is to take a more unbiased view on the internal structures of the planets. This is by producing empirical density profiles (e.g., Marley et al., 1995, Podolak et al., 2000, Helled et al., 2011).
In that case the planetary density profile is represented by a series of random steps in density or a mathematical function, and all the density profiles that match the observed properties are inferred. These can then be interpreted using physical EOS.
While the inferred density profiles are not based on knowledge of the behavior of elements at high pressures and temperatures, and can therefore might be viewed as "less physical", they can probe solutions that are missed by the standard models, in particular, solutions that represent more complex interiors (e.g., composition gradients) with various temperature profiles (e.g., sub- and super- adiabatic).
\par
For example, empirical models of Uranus and Neptune using 6th order polynomials to represent the density profile suggest that both planets can have a continuous density profile in which there is a gradual increase of the heavier material toward the center (Helled et al.,~2011), and that the overall metals mass fraction of the planets is 0.75-0.92 and 0.76-0.9 for Uranus and Neptune, respectively.
It was also shown that the planetary interiors are not necessarily water-rich, and that the measured gravitational field can be reproduced also if the planets are assumed to be rock-dominated. Finally, it was shown that the inferred densities are consistent with compositions gradients, and therefore with non-adiabatic temperature profiles (see Helled et al., 2020 for discussion).
Representative density profiles of Uranus (left) and Neptune (right) inferred by various studies are presented in Figure 3.
\par
Recently, it was shown that Uranus' low luminosity can be a result of a combination of primordial composition gradients that inhibit convection and a low planetary luminosity post-formation (Vazan \& Helled, 2020).
In this scenario the deep interior of Uranus can be very hot, which also suggests that the planet could consist of more refractory materials, since with higher internal temperatures one is able to include more rocks within a given model (since the high temperature makes the rocks lower density).
In addition, a stable (to convection) composition gradient implies that Uranus' current-state internal structure has not significantly evolved, and could be used to guide planet formation models.
Although these evolution models are not designed to fit the gravity data exactly, they can reproduce the basic measured planetary properties.
Given that a wide range of structures are possible to fit the gravity fields of each planet, this naturally begs the question regarding how large the possible range of solutions really is? Movshovitz et al. (2020a) presented a Bayesian MCMC-driven approach to exploring the full range of interior models for a giant planet, given the gravity field. The first application was to Saturn, but in Figure \ref{fig:naor} we show preliminary work from Moshovitz et al.~(2020b) that shows all the density profiles that fit the current gravity fields of Uranus and Neptune. These models use an 8th order polynomial for most of the planet, which allows for relatively steep changes in density vs.~radius, if such changes are needed to fit the gravity field constraints. As was previously demonstrated (Marley et al. 1995, Helled et al. 2011), for the interiors of Uranus and Neptune, there is no need not be distinct layers.
Currently, the gravity field of Uranus is better constrained than Neptune (see Table 1). This manifests itself as wider range of allowed interior structures in Neptune in Figure \ref{fig:naor}. This can be seen most clearly from 0.3 to 1.0 planetary radii (see in particular the insets) where Uranus is better-constrained. This is because
the behavior of the contribution functions of the gravitational moments, which describe how the different layers in the planetary interiors contribute to the gravitational moments. For both Uranus and Neptune the innermost 30\% of the planets is not well sampled by the gravitational harmonics. As a result, conclusions about the innermost part of the planets must be inferred indirectly (see Helled et al., 2011 for details).
In addition, the error bar on $J_6$ on either planet is so large that it provides no real additional constraint.
As the atmospheric composition of Uranus and Neptune can be viewed as windows to their deep interiors, improved understanding of their atmospheres can further constrain their formation path, thermal evolution, and internal structures (see Helled et al., 2020 for details).
For instance, the pollution of the protoplanetary atmospheres with heavy elements such as water, ammonia, and methane can significantly affect the cooling of the growing planet and therefore its formation and evolution paths (Kurosaki \& Ikoma, 2017). In addition, the elemental abundances can reveal information on the formation locations of the planets and/or the composition of their building blocks.
Therefore, it is clear that measuring the atmospheric composition of Uranus and Neptune is also desirable.
\section{Energy Balance and Evolution}
Models that aim to understand the amount of thermal flux coming from the planetary interior today are an important complement to models of a planet's current density structure. In principle, one should aim for a coherent picture of interior structure and thermal evolution over time, tied to the planet's formation.
Significant thought and modeling work have gone into trying to explain the dramatic difference in the heat flux between the two planets. The ``standard story'' (e.g., Fortney et al. 2011) has been that Neptune's flux value is basically what one would expect for the cooling of a 3-layer model with an adiabatic interior, while the value for Uranus is far too low. Investigations have shown that either assumed barriers to convection (Nettelmann et al. 2016) or composition gradients suggested by formation models (Vazan \& Helled, 2020) can radically alter Uranus's cooling history, leading to a low intrinsic flux today, along with a hot interior that his unable to efficiently cool off.
At the same time, it has been seen that the standard story may not be the correct one. Recently, due to \emph{Cassini} Mission Jupiter fly-by data, Jupiter's Bond albedo and intrinsic flux determination were significantly updated (Li et al., 2018) -- the Bond albedo increased from 0.343 to 0.503 (a 46\% increase), and the intrinsic flux determination increasing by 38\%, based on a combination of the new Bond albedo data and improved measurements of the planet's total thermal emission. This at least brings about the possibility that \emph{Voyager}-derived energy balance values for the other 3 giant planets (Pearl \& Conrath, 1991) could be in need of substantial revision. Figure \ref{fig:alb} shows the Bond albedo values for the planets. We note that the Bond albedo can only be determined by a space mission that observes scattered sunlight over a wide range of wavelengths and phase angles.
Such energy balance revisions can radically alter the standard Uranus/Neptune picture. Furthermore, the accuracy of input physics is always improving, and it is essential to revisit models as physics improves. Scheibe et al. (2019), assuming the \emph{Voyager} Bond albedos for each planet and a time-evolving solar luminosity, find that Neptune is actually \emph{overluminous} compared to the expectation of an adiabatic cooling model, and Uranus is still underluminous, but not be as much as previously thought. A modestly higher Bond albedo for Uranus (from 0.3 up to 0.4) would make Uranus's model adiabatic cooling history fit with observations. Neptune would appear to be overluminous, with \emph{any} Bond albedo. This is a "flip" from previous work, marking Neptune, rather than Uranus, as the "odd" planet.
Given this wide variety of evidence, it is now becoming ever clearer that Uranus and Neptune should not be modeled assuming adiabatic interiors with distinct layers, and the field is certainly moving in that direction (Nettelmann et al. 2016, Podolak et al., 2019, Scheibe et al.~2019, Vazan \& Helled, 2020). An important path forward is expected to come from detailed models of planet formation, and then evolving such models over 4.5 Gyr of time, to assess their current structure and heat flux today, compared to observations.
\section{Should we keep calling Uranus and Neptune the "ice giants"?}
Often Uranus and Neptune are referred to as the "ice giants." The origin of the name is probably linked to the mean densities of the planets which are comparable to the density of water, and due to the fact that they are located at large radial distances where volatile materials can condense to form ices (water, ammonia, methane).
However, we actually do not know if the compositions of Uranus and Neptune are dominated by these materials, and even Pluto seems to consist of more rock than ice (McKinnon et al., 2017). Indeed it was shown that the observed properties of the planets can be fit also with a rock-dominated composition (Helled et al., 2011), and recently, it has been suggested that Neptune could be a "rock-giant" based on measured atmospheric abundances (see Teanby et al., 2020 and references therein for further details).
Also, although the argument that the planets must consist of large fractions of water to have high enough electrical conductivities (ionic/super-ionic water) to generate their magnetic fields is convincing (e.g., Redmer et al. 2011), it is yet to be determined how much water is required and whether other materials could contribute to the ionic interior.
In addition, even if the planets have substantial amounts of volatile materials (e.g., water), in their deep interiors, the physical state of the material would not be in a solid state, and therefore it is inappropriate to describe the materials as "ices" since they would be in the liquid (fluid) state. This is in fact also true for Jupiter and Saturn which are called the "gas giants", because their composition is dominated by hydrogen (H), although the material in their deep interiors is not in the gaseous phase.
Similarly, we suggest that calling Uranus and Neptune ``ice giants" is rather misleading. This name biases the community to think of these planets as being water- (volatiles) dominated and also gives the wrong impression for the physical state of the material in their deep interiors.
We therefore propose that naming Uranus and Neptune "sub-giants" or "outer-giants" instead of "ice giants" is more appropriate\footnote{We note that the earliest recorded use of "ice-giant" we can find is in the introduction to a 1978 NASA report about the Mariner 10 mission to Mercury (Dunne \& Burgess, 1978).}.
\section{Summary \& Future Plans}
Uranus and Neptune remain mysterious planets and it is clear that further exploration of these planets theoretically and observationally is needed.
Key fundamental questions regarding Uranus and Neptune remain open, such as:
Key questions regarding Uranus and Neptune include:
\begin{itemize}
\item[+] How do planets like Uranus and Neptune form?
\item[+] How do these planets evolve?
\item[+] What are the compositions and internal structures of Uranus and Neptune?
\item[+] Are Uranus and Neptune volatile-rich?
\item[+] What parts of each planet are superadiabatic?
\item[+] How different are Uranus and Neptune? What is the origin of these differences?
\item[+] How is the magnetic field generated?
\end{itemize}
The collection of these important open questions ensures that investigating Uranus and Neptune in the near and far future will be extremely rewarding.
Although the development of new theoretical models is crucial, it is clear that significant progress in our knowledge cannot be achieved without more data.
We argue that missions to Uranus and Neptune are essential. In particular we emphasize the importance of accurate measurements of the planets' {\bf gravitational fields} (preferable with several \emph{Juno}-like polar orbits).
Better gravity data will allow us to exclude certain solutions for the interiors, and can assist us in understanding the differences between Uranus and Neptune.
We will then be able to reduce the parameter space of possible internal structures.
Better determining the structure and potential variability of the planetary {\bf magnetic fields} will allow us to further constrain the planetary structure via the required conditions to sustain a dynamo (electrically conducting material + convection), as discussed in detail in the complementary chapter by Soderlund \& Stanley (2020) in this issue.
At the same time, measuring the
{\bf atmospheric composition} from an entry probe can be used to better understand the connection between the atmosphere and the deep interior (e.g., Guillot, 2020) as well as the origin of the planets (e.g., Mousis et al., 2020).
In addition, determining the {\bf rotation rates} of the planets helps to tighten constraints on interior models. Better establishing each planet's {\bf Bond albedo} and {\bf thermal fluxes} can be used to further constrain structure and evolution models, respectively.
Due to the complex nature of planets, it is clear that having one measurement, even if very accurate, is insufficient to break the degeneracy of structure models and reveal the true nature of the planets.
What is needed is a comprehensive investigation of each of the two planets using different measurements so we can slowly put the different pieces of the puzzle together until we better understand Uranus and Neptune.
In the nearer future, before potential space missions, some progress is envision in the following fonts.
Further improvements in EOS calculations and experiments of volatile materials such as water, ammonia, and methane, their mixtures, as well as their mixtures with rock or with hydrogen (and helium), are essential. This could build on recent advances (e.g., Bethkenhagen et al., 2017, Millot et al., 2019).
Understanding the behaviour of the materials, and their mixtures, we think exist in Uranus and Neptune at high pressures and temperatures, will allow us to exclude certain compositions, and can guide us in terms of the model assumptions: i.e., what materials are likely to be mixed vs. differentiated and what are the important chemical interactions that should be considered.
A longer-term investment, that could start now, would be further investigations towards gains that could be made by seismology of Uranus and Neptune. Building on important work for Jupiter and Saturn over the past decade (Gaulme et al., 2011, Fuller 2014, Mankovich et al., 2019), further theoretical and observation-driven work could be imagined. This includes assessing the potential of ``ring seismology of Uranus" using detailed observations of its ring system (Hedman et al., 2014), or also long time baseline photometry (Rowe et al., 2017).
Detailed and accurate measurements of atmospheric abundances of giant and intermediate-mass exoplanets will also be valuable. An overview of the variation in atmospheric composition of gaseous exoplanets in the mass/size range of Uranus and Neptune, and its connection to the host star's properties, in concert with determinations of the planetary mean density will allow us to understand the nature of gaseous-rich intermediate mass planets.
\par
\section*{Acknowledgments}
We thank the two anonymous referees for valuable comments. RH thanks D.~Soyuer and B.~Neuenschwander for technical support. RH also acknowledges support from SNSF grant 200020\_188460. JJF acknowledges support from NASA grant NNX16AI43G and NSF-AAG grant 1908615.
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}
\clearpage
\begin{figure}
\centering
\includegraphics[angle=0,height=7.5cm]{fig_deltaJs.pdf}
\caption{
\footnotesize{
{\bf Gravity measurement uncertainties of the outer planets.}
Shown are relative observational uncertainties in $J_2$ and $J_4$ of the four giant planets. The triangles and circles represent "old" and "new" data, respectively.
"Old" data correspond to the latest data before
the most recent, i.e., "new" data. The used numbers are summarized in Table 2.
}
}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[angle=0,height=7.15cm]{HS_fig.pdf}
\caption{
\footnotesize{
{\bf Heavy element mass fraction $Z$ vs.~planetary mass up to a mass of 20 M$_{\oplus}$.}
The solid-black and dotted-blue curves correspond to formation models assuming solid surface densities of 10 g cm$^{-2}$ and 6 g cm$^{-2}$, respectively. This demonstrates the dependence of the planetary composition and primordial internal structure on the relative accretion rates (see Helled \& Stevenson 2017 for details).
}}
\label{fig:1}
\end{figure}
\newpage
\begin{figure}
\includegraphics[angle=0,height=6.5cm]{ura_r.pdf}
\includegraphics[angle=0,height=6.5cm]{nep_r.pdf}
\caption{
\footnotesize{{\bf Density profiles of Uranus and Neptune.} The colors correspond to different studies. For Uranus, the solid and dashed orange curves are models U1 and U2 from Nettelmann et al.~(2013), the solid red is from Helled et al.~(2011), and the solid and dashed blue lines are models V2 and V3 from Vazan \& Helled, 2020, respectively. For Neptune, the solid and dashed orange lines are models N1 and N2b from Nettelmann et al.~(2013), and the solid red is from Helled et al.~(2011).}}
\label{fig:1}
\end{figure}
\begin{figure*}
\includegraphics[angle=0,height=6cm]{Uranus_J24_posterior_average_w_inset.pdf}
\includegraphics[angle=0,height=6cm]{Neptune_J24_posterior_average_w_inset.pdf}
\caption{
\footnotesize{{\bf Empirical density profiles of Uranus (left) and Neptune (right) derived from 8th-order polynomials.} Visualization of the posterior probability distribution of Uranus and Neptune interior density profiles. The thick black line is the sample-median of density on each level surface. The dashed lines mark the the 16th and 84th percentiles and the dotted
lines mark the 2nd and 98th percentiles; between the lines percentile value is color-coded. Inset shows a zoom-in in outer layers. Uranus is better-constrained than Neptune}.}
\label{fig:naor}
\end{figure*}
\begin{figure}
\centering
\includegraphics[angle=0,height=6.5cm]{AlbCompare.pdf}
\caption{
\footnotesize{{\bf Bond albedo determinations of the giant planets.}
Shown are the Bond albedos of Jupiter, Saturn, Uranus, and Neptune, from left to right. Values from Voyager data are shown in blue with $1 \sigma$ error bars. The data for Jupiter, Saturn, Uranus and Neptune are based on Hanel et al., (1981), Hanel et al., (1983), Peal et al.~(1990), and Pearl \& Conrath, (1991), respectively.
The red value for Jupiter is from \emph{Cassini} fly-by data (Li et al., 2018).
}}
\label{fig:alb}
\end{figure}
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 513 |
Art listings, exhibits
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Lots of sunshine. High around 90F. Winds WSW at 5 to 10 mph..
ETHAN WEINER
West Caldwell man accused of having images showing sex abuse of children
Ethan Weiner, 29, of West Caldwell was accused of possessing more than 1,000 items depicting the sexual abuse and exploitation of children, Acting Essex County Prosecutor Theodore Stephens II said Friday, June 21.
Weiner was charged with second-degree endangering the welfare of a child.
On Thursday, June 20, members of the Essex County Prosecutor's Office Special Victims Unit, New Jersey State Police, Homeland Security Investigations and West Caldwell police executed a search warrant at his home.
Devices and digital storage media were seized for forensic examination and analysis.
Weiner was identified during an investigation initiated by the Essex County Prosecutor's Office Internet Crimes Against Children Task Force into the distribution of images and videos depicting the sexual exploitation and abuse of children through file-sharing networks.
Weiner's brother Adam, 32, who resides at the same address in West Caldwell, was charged with possession of more than 50 grams of marijuana, possession with intent to distribute and possession with intent to distribute within 1,000 feet of school property.
Ethan Weiner was released on pre-trial monitoring. Adam Weiner was charged on a summons.
West Caldwell
Adam Weiner
Endangering The Welfare Of A Child
Sexual Abuse And Exploitation
Watchung Hills grad awarded National Merit Scholarship
Rockaway law firm opens InfoCenter on unsolved lynching
Florham Park Borough Attorney opens InfoCenter on unsolved lynching
Former county clerk opens online center on unsolved lynching
Backpack, school supplies giveaway July 21 at Warren, Gillette Verizon stores
See It. Click It. ... Fix It! | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,954 |
Gli Uffici di sanità marittima, aerea e di frontiera (noti anche con l'acronimo: USMAF), sono uffici periferici del Ministero della salute che si occupano del controllo sanitario su passeggeri e merci che transitano attraverso i punti d'ingresso transfrontalieri.
Funzioni
Gli Uffici sono situati nei maggiori porti ed aeroporti nazionali al fine di prevenire il rischio di importazione di malattie dall'estero. Anche se presso questi uffici si espletano attività sanitarie diverse (ad es. prestazioni di medicina legale per i naviganti, vaccino-profilassi per i viaggiatori) il principale compito istituzionale degli USMAF è la vigilanza igienico-sanitaria su mezzi, merci e persone di provenienza extra-UE.
Organizzazione
Gli USMAF sono 12, suddivisi a loro volta in varie Unità territoriali, da cui dipendono e che coprono tutto il territorio nazionale. Questa l'attuale dislocazione:
Genova, con le seguenti Unità territoriali: Savona, Imperia, La Spezia;
Milano Malpensa, con le seguenti Unità territoriali: Torino Caselle, Bergamo Orio al Serio;
Trieste, con la seguente Unità territoriale: Venezia;
Bologna, con la seguente Unità territoriale: Ravenna;
Livorno, con la seguente Unità territoriale: Pisa;
Fiumicino, con le seguenti Unità territoriali: Civitavecchia, Roma;
Napoli, con le seguenti Unità territoriali: Salerno, Cagliari, Porto Torres;
Pescara, con la seguente Unità territoriale: Ancona;
Bari, con le seguenti Unità territoriale: Manfredonia, Taranto;
Brindisi;
Catania, con le seguenti Unità territoriali: Messina, Siracusa, Augusta, Comiso, Reggio Calabria, Gioia Tauro;
Palermo, con le seguenti Unità territoriali: Porto Empedocle, Trapani.
Voci correlate
Dogana
PIF
Punti di entrata per i vegetali
Uffici veterinari per gli adempimenti degli obblighi comunitari
SASN
Ministero della salute
Tecnico della Prevenzione
Collegamenti esterni
Ministero della salute | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 577 |
Q: Javascript Date to string with ISO format (with timezone) I've been using MomentJS a lot but I'm starting a new project and I don't want to include this library since I play with dates only a couple of times.
So what I'm trying to do is to get the string representation of a date, in ISO-like format ('YYYY-MM-DDZHH:mm:ss' or 'YYYY-MM-DD HH:mm:ss'). I don't want it in UTC: I want it in a given timezone (that I can provide programatically).
E.g the representation for right now would be "2017-04-11 11:20:00" (French timezone - eq to "2017-04-11 09:22:00Z".)
I want native Javascript. I've been playing with toLocaleString with no success.
Thanks
[edit]
In a perfect world, I'm looking for a function that takes a date format, a timezone, and return the string I want. Like:
function magicDateFormatter(format, tz) {
/* ... */
}
var now = new Date();
console.log(magicDateFormatter('YYYY-MM-DD HH:mm:ss', 'Europe/Paris'));
// print "2017-04-11 11:20:00"
A: const dt = new Date().toLocaleString("sv-SE");
A: Something like this might work for you:
function formatDateWithZone(date, tz) {
var s = date.toLocaleString('en-GB', { timeZone: tz });
var a = s.split(/\D/);
return a[2] + '-' + a[1] + '-' + a[0] + ' ' + a[4] + ':' + a[5] + ':' + a[6];
}
Usage:
formatDateWithZone(new Date(), 'Europe/Paris') // "2017-04-12 03:37:59"
This will work in environments that have implemented time zone support via ECMA-402. The compatibility table here will show you which support them, by expanding the DateTimeFormat section, and looking at the row labeled accepts IANA timezone names.
A: Something like this might be what you need:
First format the timestamp as you wish, including timezone offset and then change the position of the numeric items as you need it, usign the replace function with regex:
const timeStamp = new Date().toLocaleString('de-DE', {
timeZone: 'Europe/Berlin',
hour12: false,
year: "numeric",
month: "2-digit",
day: "2-digit",
hour: "2-digit",
minute: "2-digit",
second: "2-digit",
}).replace(/(\d+)\.(\d+)\.(\d+),\s(\d+):(\d+):(\d+)/, "$3$2$1_$4$5$6");
A: You can't do what you're asking using built-in methods as ECMAScript implementations aren't required to store historical data for all (or even any) time zones. They are only required to have access to the current time zone offset of the host.
Also, the ECMA-402 Internationalisation API isn't sufficient either. While it allows the timezone to be specified, not all implementations support ECMA-402 and of those that do, not all support time zones values other than "UTC" (which is the only one ECMA-402 requires support for, others are optional). And it's very difficult to impossible to exactly specify the format using only ECMA-402.
I think you're much better off to use an existing library like moment.js with Moment-timezone.js. The download isn't that big relative to the general size of web pages (which seem to be 1MB at least lately). If you try to write something yourself you'll just end up with something very moment-like anyway.
To reduce the size of the download, you can get just the time zone data for 2012 to 2022.
If you want to use your simplified call, then a small magicDateFormatter can leverage the moment.js stuff:
/* Return date string for a date in the required timezone and format
** @param {string} timezone: IANA time zone designator, e.g. Asia/Sakhalin
** @param {Date} [date] : date to use, default is current date
** @param {string} [format]: output format using Moment.js tokens,
** default is 'YYYY-MM-DDTHH:mm:ssZZ'
** @returns {string} : formatted string
**
** magicDateFormatter(timezone[, date[, format]])
*/
function magicDateFormatter(timezone, date, format) {
// Validate that the time zone is supported
if (!moment.tz.zone(timezone)) {
return 'Unknown time zone: "' + timezone + '"';
}
// Use current date if not supplied
date = date || new Date();
// Use default format if not supplied
format = format || 'YYYY-MM-DDTHH:mm:ssZZ'
return moment(date).tz(timezone).format(format)
}
console.log(magicDateFormatter('Europe/Paris'));
console.log(magicDateFormatter('Asia/Sakhalin', new Date(2016,1,29), 'Do MMMM, YYYY'));
console.log(magicDateFormatter('foo/bar'));
<script src="https://cdnjs.cloudflare.com/ajax/libs/moment.js/2.18.1/moment.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/moment-timezone/0.5.13/moment-timezone-with-data.min.js"></script>
Of course you could also specify the format in the call if you want.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 311 |
Q: Can't parse a JSON string in Java so I have a JSON string:
{
"dontSend" : 0,
"dontSendSMS" : 0,
"dontSendPush" : 0,
"daysAvail" : [1, 1, 1, 1, 1, 1, 1],
"allAreas" : ["Dublin 1", "Dublin 2"]
}
Here's my code:
JSONObject obj = new JSONObject(str); //str as above
int[] daysAvail = (int[]) obj.get(obj.get("daysAvail")); //won't work!
I get this error:
java.lang.String cannot be cast to int[]
I also tried:
JSONObject obj = new JSONObject(str); //str as above
JSONArray daysAvail = obj.getJSONArray("daysAvail"); //won't work either!
I get this error:
java.lang.String cannot be cast to org.json.JSONArray
Anyone have any ideas?
The string is definitely as above. It's as if the Java parser doesn't recognise the [1,1,1,1,1,1] sequence.
A: int[] daysAvail = (int[]) obj.get(obj.get("daysAvail"));
you are using the returned value of obj.get("daysAvail"), a JSONArray in your case, as key for the same JSONObject, and casting its return value to int[]. I would have expected something simpler (If I am not missing something obvious):
JSONArray array = obj.optJSONArray("daysAvail");
if (array != null) {
int[] daysAvail = new int[array.length()];
for(int i=0; i<array.length(); i++){
daysAvail[i] = array.optInt(i);
}
}
A: seems daysAvail is not jsonarray but string
you can do this
JSONObject obj = new JSONObject(str); //str as above
String str=obj.getString("daysAvail");
str=str.substring(1, str.length()-1);
String[] array=str.split(",");
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 764 |
{"url":"http:\/\/e-maxx-eng.appspot.com\/graph\/depth-first-search.html","text":"# Depth First Search\n\nDepth First Search is one of the main graph algorithms.\n\nDepth First Search finds the lexicographical first path in the graph from a source vertex $u$ to each vertex. Depth First Search will also find the shortest paths on a tree, but on general graphs this is not the case.\n\nThe algorithm works in $O(m + n)$ time where $n$ is the number of vertices and $m$ is the number of edges.\n\n## Description of the algorithm\n\nThe idea behind DFS is to go as deep into the graph as possible, and backtrack once you are at a vertex without any unvisited adjacent vertices.\n\nIt is very easy to describe \/ implement the algorithm recursively: We start the search at one vertex. After visiting a vertex, we further perform a DFS for each adjacent vertex that we haven't visited before. This way we visit all vertices that are reachable from the starting vertex.\n\nFor more details check out the implementation.\n\n## Applications of Depth First Search\n\n\u2022 Find any path in the graph from source vertex $u$ to all vertices.\n\n\u2022 Find lexicographical first path in the graph from source $u$ to all vertices.\n\n\u2022 Check if a vertex in a tree is an ancestor of some other vertex:\n\nAt the beginning and end of each search call we remember the entry and exit \"time\" of each vertex. Now you can find the answer for any pair of vertices $(i, j)$ in $O(1)$: vertex $i$ is an ancestor of vertex $j$ if and only if $\\text{entry}[i] < \\text{entry}[j]$ and $\\text{exit}[i] > \\text{exit}[j]$.\n\n\u2022 Find the lowest common ancestor (LCA) of two vertices.\n\n\u2022 Topological sorting:\n\nRun a series of depth first searches so as to visit each vertex exactly once in $O(n + m)$ time. The required topological ordering will be the vertices sorted in descending order of exit time.\n\n\u2022 Check whether a given graph is acyclic and find cycles in a graph.\n\n\u2022 Find strongly connected components in a directed graph:\n\nFirst do a topological sorting of the graph. Then transpose the graph and run another series of depth first searches in the order defined by the topological sort. For each DFS call the component created by it is a strongly connected component.\n\n\u2022 Find bridges in an undirected graph:\n\nFirst convert the given graph into a directed graph by running a series of depth first searches and making each edge directed as we go through it, in the direction we went. Second, find the strongly connected components in this directed graph. Bridges are the edges whose ends belong to different strongly connected components.\n\n## Implementation\n\nvector<vector<int>> adj; \/\/ graph represented as an adjacency list\nint n; \/\/ number of vertices\n\nvector<bool> visited;\n\nvoid dfs(int v) {\nvisited[v] = true;\nif (!visited[u])\ndfs(u);\n}\n\n\nThis is the most simple implementation of Depth First Search. As described int the applications it might be useful to also compute the entry and exit times and vertex color. We will color all vertices with the color 0, if we haven't visited them, with the color 1 if we visited them, and with the color 2, if we already exited the vertex.\n\nHere is a generic implementation that additionally computes those:\n\nvector<vector<int>> adj; \/\/ graph represented as an adjacency list\nint n; \/\/ number of vertices\n\nvector<int> color;\n\nvector<int> time_in, time_out;\nint dfs_timer = 0;\n\nvoid dfs(int v) {\ntime_in[v] = dfs_timer++;\ncolor[v] = 1;","date":"2018-02-24 09:37:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3358115553855896, \"perplexity\": 619.7379946010573}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-09\/segments\/1518891815544.79\/warc\/CC-MAIN-20180224092906-20180224112906-00448.warc.gz\"}"} | null | null |
Hidden Gems in Rome – The Unfrequented Spots
Trigger Warning: Descriptions of Violence
There's a certain pang in my chest whenever people say they only want to visit Rome for a day or two. It's not enough to tour the must-see tourist attractions of Rome, because there are many hidden gems. If you're a first-timer in Rome, you'll want to visit the obvious and famous hotspots. Places like the Colosseum, the Pantheon, or the Trevi Fountain. But as much as they are an outstanding part of the country's history, you'll eventually realise that you'll want to see more. The crowd in these usual landmarks will make it almost impossible for you to take in the true beauty of the Eternal City. Especially if you're a returning visitor and you've already been to these places.
There are secret spots in Rome that really make the city rich in history and culture. I would consider these jewels as off the beaten tracks. The world doesn't call Rome "The Eternal City" for nothing. The ancient Romans always believed that no matter what happens to the world, Rome would go on forever. And these spots are an important part of Italy's culture and somewhat dark history. After centuries and decades, visitors and locals are still able to visit them and bask in the antiquity of the city.
Quartiere Coppedè
Source: www.romeing.it
This architectural piece of Rome located in the whimsical neighbourhood of Piazza Mincio is definitely an often-missed landmark. Built in 1915, architect and sculptor Gino Coppede had circuitously designed this quarter, now residential compound, with dainty towers, cornices, galleries and columns, human and animal figures, clocks and crests, frescoes and inscriptions: every decorative element in the book and then some. The materials he used are stone and brick, marble, terracotta, wood glass, and wrought iron. It's a mixture of baroque and gothic style. A famous director even shot and filmed a couple of his horror films here!
It is arguably one of the best hidden gems in Rome. It's bang on in the middle of the city, yet people tend to ignore it. You will have to settle for a picture from the outside, though. You wouldn't be able to enter, as it's a private residential building. So relish in the luring beauty of this architecture and observe its intricate design from afar. Follow the the wide arch and you will reach a two-tiered fountain decorated with frogs. Perfect for a selfie snap!
Source: www.romecentral.com
Head over to the Trastevere neighborhood and visit the ancient pharmacy of Santa Maria della Scala. The apothecary was built in the 16th century and had been treating patients up to 1954. It is so hidden that you won't even notice the tiny entrance door. It is located on the first floor of the ancient monastery in Piazza della Scala. Back in the day, it was a friar who would welcome and treat patients. The friars introduced the patients and visitors to herbs, potions, unguents, oils, and clays. Basically, the friars concocted any natural and unnatural chemistry products they believed could heal a human being's illness. They offered these to the oncoming cure-seekers.
As soon as you step into the dark and eerie threshold, you are instantly taken back to centuries ago. You can still smell the potions and lotions created by the Catholic chemists hundreds of years ago. The walls, displayed bottles of medicine, potions and antidotes all tell a story. The friars of the Discalced Carmelite were experts in the study of botany, chemistry, and medicine. So much so that they were hailed as the "Pharmacy of the Popes". Because back then, all the Popes would visit and seek help from the friars.
Santo Stefano Rotondo
Source: yarenciftci.wordpress.com
If you're visiting Italy, then you already know about the churches. You can't visit the country and not go into one. But Rome, in particular, is a city of churches. You've heard of St. Peter's and Santa Maria Maggiore, but I bet you haven't heard of Santo Stefano Rotondo. If you're an intrepid traveller ready to look beyond the most Instagrammed churches and the tourist books, visit this church as soon as possible. It is definitely one of the best hidden gems in Rome. It is the largest circular church in Rome. Entering it is like going into heaven's light. Its high ceiling boasts a beautifully lit altar flooding in from the windows. In fact, it's a popular local choice for wedding venues.
But don't let this beauty fool you. If you look at the painting and art in and around the church, you'll see how gory and violent the décor really is. From bearded men being boiled, fried, grilled, crimped, singed, and eaten by wild beasts to boys being buried alive, torn apart, and chopped up. You will also see women having their breasts torn with iron pinches, tongues cut out, ears screwed off, jaws broken, and bodies stretched upon the rack. Anything violent that you can think of, name it, and it's here. Yes, it is disturbing. But also, it's part of Italy's dark history. And in some sick, twisted way, it's actually beguiling. It's like a scary horror film that you continue to watch and even end up doing a standing ovation in the end.
Source: www.whatalifetours.com
The Appain Way is one of the first and strategically most important roman roads of the ancient republic. The road allowed trade and access to other countries like Greece. It is still very much visible today and open to visitors, but not well-known, which makes it one of the great hidden gems in Rome. Many significant tombs and architecture line its borders, so there's so much more to see than the road itself. Some of the sights to see during your walk are the ruined Roman monuments, two Christian catacombs, and a church marking where Peter had a vision of Jesus.
If you're a history nerd, you'll know that Spartacus' army was divided and hemmed in by Crassus' eight legions. The Gauls and Germans were defeated and Spartacus himself fell while in battle. Pompey's army intercepted and killed hundreds, if not thousands, of slaves who were escaping. Adding to all this, 6,000 prisoners were crucified by Crassus along the Appian Way. There is literally and metaphorically blood on these streets. These roads in Rome have a rich and dark history that you shouldn't miss.
Catacombe di Roma
Source: www.contexttour.com
I saved the best for last. I'm a huge horror buff, and visiting the Catacombs of Rome brought me that scare and thrill I'd been looking for. But it's not the sole fright that intrigued me, it was the stories of the many skulls and bones – pieces of once living humans. Now they are just decorations, they complete the art of the catacombs. It's eerily beautiful and sinister at the same time. Every skull has a history to tell. I could stay there for hours and hours and I wouldn't run out of things to see. In fact, it didn't feel right for me to take photos inside because I wanted to respect the deceased.
Although the catacombs are most famous for Christian burials, there are also Jews and adherents of a variety of Pagans buried here. The townspeople buried the rich and the poor alongside each other. It is a true testament that underneath it all, we really are all the same. Humans buried here were once friars, farmers, rich aristocrats, and other citizens of the city. All of them stripped away of everything else and reduced to skeletal beings – a bunch of bones. There are 40 known catacombs in Rome (many more undiscovered) but only 5 are open to the public. So, research your way into one and learn about the lives of the dead. I promise you it's worth it.
Rachel Anne Galvan
Rachel is an intrepid world traveller, lifestyle connoisseur, and Her Adventures' resident beauty advisor. She is a lover of the ocean, flowers, cupcakes, karaoke, makeup & skincare, and journalism. When she is not busy with life, she writes stories of her globetrots as well as tips & tricks on how to make the best out of travelling.
Rachel is an intrepid world traveller, lifestyle connoisseur, and Her Adventures' resident beauty advisor. She is a lover of the ocean, flowers, cupcakes, karaoke, makeup & skincare, and journalism. When she is not busy with life, she writes stories of her globetrots as well as tips & tricks on how to make the best out of travelling.
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Important Reasons to See a Travel Doctor
Things to do in Haunted Chicago
Whidbey Island, the Pearl of the Pacific NW
A Marvelous London Trip – How to Best Spend 36 Hours | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,859 |
Q: Turn Quaternions into set of 3 vectors (base vectors). I am very new in this topic.
My question is the following:
I have a set of quaternions that represent the attitude of a satellite.
I need to turn these into a set of vectors representing the coordinate system on this satellite.
The satellite is located in the origin of this system.
I know how to convert the quaternions into a rotationmatrix.
Since the satellite is located in the origin of this system, do I get the satellites attitude now by simply multiplying the base vectors
([1,0,0], [0,1,0], [0,0,1])
to my rotationmatrix?
The thing I don't understand, I would always get only the diagonal entries.
Or is the rotationmatrix in this case my attitude matrix, since the satellite is already located in the origin of this frame?
Thank you so much fpr your help in advance.
| {
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} | 9,198 |
<!doctype html>
<html i18n-values="dir:textdirection;lang:language">
<head>
<meta charset="utf-8">
<title>WebRTC Internals</title>
<link rel="stylesheet" href="webrtc_internals.css">
<script src="chrome://resources/js/util.js"></script>
<script src="webrtc_internals.js"></script>
</head>
<body>
<p id='content-root'>
</p>
</body>
</html>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,724 |
Національна ліга В 1964—1965 — 18-й чемпіонат Швейцарії з хокею (Національна ліга В), чемпіоном став клуб Ла Шо-де-Фон.
Груповий етап
Група Захід
Група Схід
Фінал
ХК «Ла Шо-де-Фон» — Амбрі-Піотта в серії переміг ХК «Ла Шо-де-Фон».
Джерела
Сезон 1965
Чемпіонат Швейцарії з хокею
Швейцарія
Швейцарія | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,499 |
Kristina Kapralova (born 4 January 1995) is a Kazakhstani handball player. She plays for the club Kazygurt Handball and is member of the Kazakhstani national team. She competed at the 2015 World Women's Handball Championship in Denmark.
References
1995 births
Living people
Kazakhstani female handball players
Handball players at the 2014 Asian Games
Handball players at the 2018 Asian Games
Asian Games bronze medalists for Kazakhstan
Asian Games medalists in handball
Medalists at the 2014 Asian Games
21st-century Kazakhstani women
20th-century Kazakhstani women | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,768 |
Olax benthamiana är en tvåhjärtbladig växtart som beskrevs av Friedrich Anton Wilhelm Miquel. Olax benthamiana ingår i släktet Olax och familjen Olacaceae. Inga underarter finns listade i Catalogue of Life.
Källor
Sandelträdsordningen
benthamiana | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,724 |
Q: Passing 'require' to haml guard Manually compiling haml to html with requirement of external file is performed like so
haml --require .\stuff.rb --require .\const.rb .\pages\exit.haml .\pages\exit.html
However, I wanted to switch to haml guard. Unfortunately I am unable to find correct flag to pass for it to work. I need something along the lines of:
guard :haml, haml_options: {require: './stuff.rb ./const.rb'} do
watch(/^.+(\.haml)$/)
end
The result:
14:19:19 - ERROR - HAML compilation of pages/exit.haml failed!
[#] Error: undefined method `html_safe' for nil:NilClass
which shows that neither the method nor constants were included.
Any ideas?
edit:
I'm using Ruby 2.3.1p112 (2016-04-26 revision 54768) [x64-mingw32] (Win10), Haml 4.0.7.
Minimized example:
test.haml
!!!
%html
%header
%body
%p
=$BT_OK.html_safe
const.rb
$BT_OK = "
".html_safe
helpers.rb
class String
def html_safe?
defined?(@html_safe) && @html_safe
end
def html_safe
@html_safe = true
self
end
end
require 'haml/helpers/xss_mods'
module Haml::Helpers
include Haml::Helpers::XssMods
end
Output using command line haml .\debug\test.haml .\debug\test.html -r .\const.rb -r .\helpers.rb
<!DOCTYPE html>
<html>
<header></header>
<body>
<p>
</p>
</body>
</html>
Error when using matt's solution:
08:02:06 - ERROR - Invalid Guardfile, original error is:
> [#]
> [#] undefined method `html_escape' for module `Haml::Helpers',
> [#] backtrace:
> [#] (dsl)> C
> [#] (dsl)> C
> [#] (dsl)> C
> [#] (dsl)> C
A: guard-haml runs Haml "in process", so in order to have those files available in your Haml script you need to require them in your Guardfile. You'll also need to require Haml first too, since your helpers refer to some on Haml's modules:
require 'haml'
require './helpers'
require './const'
guard :haml do
watch(/^.+(\.haml)$/)
end
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,210 |
Two Lexington High School Seniors Attend Marine Energy Research Conference
Annie Yadi Wang and Daniel Xu
Jaime Yu and Anjali Asthagiri, two seniors at Lexington High School, recently participated in a marine energy research conference in Portland, Ore. The University Marine Energy Research Community (UMERC)...
The LHS Math Team Looks to MAML for Wins and Accessibility for All
Angela Tang and Jolene Cai
In October, thousands of high school students across Massachusetts took the Massachusetts Association of Mathematics League Competition (MAML). The competition has two levels. Level one consists of...
Carolyn Bertozzi '84 Becomes First Lexington High School Graduate to Win the Nobel Prize
William Tang and Seiya Saneyoshi
Carolyn Bertozzi, a Lexington High School graduate and Professor of Chemistry at Stanford University, was one of three scientists awarded the 2022 Nobel Prize in Chemistry for her advancements in . She... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,029 |
Today's Special for Diwali Sweet Series is Chum Chum, a specialty from Bengali. This is again same as how one makes Ras malai, Rasa gulla. Though this is served as a individual piece, I thought of serving along with the Malai. Traditionally it is served with the malai topped on it.
I must say it tasted great, though I never really bothered to eat before. You must surely attempt at making it for it's delicious taste. I adapted this recipe from my recipe collection. But guess the recipe is pretty much the same how everybody makes.
Chum chum is served with malai and Pistachios on top.
Once the paneer is ready, knead it with 2 tsp of flour to a soft dough. Pinch into small balls and shape them into oval shape.
Heat the sugar in a pan with enough water to make a thin syrup. Add the saffron strands and kesar colour for the chum chum to absorb.
Add the paneer and cook for 5 mins. When you see that it's done, allow it to cool.
Once the cham cham are cooled, remove from the sugar syrup.
Though traditionally chum chum is served as such, I served with it dipped in condensed milk.
You can make your own condensed milk by boiling milk and reducing it further down. That will give you that grainy texture.
Other toppings that are common on a Chum Chum would be fine coconut mixed with sugar. rabri etc.
Yum yum cham cham looks super rich..
Chum chum is one of my favorite sweet. Yours looks absolutely delicious. Must try.
Wow……such lovely looking Chum chum ……love it.
Looks so tempting and delicious ,one of my fav. Bengali sweet.
wow awesome and perfectly shaped chum chum….
If you want you can link this recipe to my Diwali Event….
Yum, that looks wonderful, beautifully done. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,910 |
Jacques Dufilho, född 19 februari 1914 i Bègles, död 28 augusti 2005 i Lectoure, var en fransk skådespelare och komiker.
Han gjorde rösterna till Dupondtarna i spelfilmen Tintin i piraternas våld från 1961. Karaktärerna själva spelas av de spanska tvillingbröderna Gamonal, som krediteras som "inkognito" (som betyder anonyma).
Privatliv
Oktober 1947 gifte sig Dufilho med Colette Coras. Tillsammans fick de dottern Colette Dufilho-Legendre (född 1954).
Referenser
Män
Födda 1914
Avlidna 2005
Franska skådespelare under 1900-talet
Personer från Bègles | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,034 |
I'm linking up with Joyce for the Wednesday Hodgepodge (a linky where you can get to know other bloggers).
It's hard to pick just 1 memory of my mom, but I'll try! I remember coming home from college, sitting on my bed, and talking to her; really talking. Until I met CH, she was my best friend.
My mom and her sisters were nurses. There is no doubt that being a nurse came in pretty handy with 10 kids!
4. Do you have a desk? Is it organized? If so, share your secret to keeping it that way. If you don't have a desk, where in your home do you take care of family paperwork and business? Where do you normally sit to blog?
Weeeeell, I have a desk now that my dad refinished. It's actually a "secretary"; it's big and heavy (just ask the guys who offered to move it into my house for me!). I spent a day a couple of weeks ago organizing it.
I'm not going to add a picture of what it looks like at this minute! I used to do my blogging either on the couch or at the kitchen table; now I have my own quiet space.
I was going into 4th grade in a brand spanking new elementary school. We thought we were the schiznet!
I don't know about you, but I like this shorter format! Whoever suggested it, THANK YOU! I know it has to save Joyce from time and stress!!!
I like the shorter format, too. But it still took me the same amount of time to think through my answers. LOL One of Granddaddy's brothers married a nurse and they had 9 kids so, yes, it was quite handy having a nurse in their family. :) They also took in 3 of the cousins when they lost their parents. I still can't remember a dang thing about being 9. Must have been the blandest year of my life. LOL Have a great day!
I think I prefer fewer questions too. Coming up with seven was time consuming, but so was answering them and then reading all the entries. Happy Mother's Day to you!
How sweet that your mother was your best friend. You chicken on a croissant sounds perfect. Have a lovely Mother's Day.
Your desk is sooooooo organized! I'm jealous!!
It's fun reading memories of moms today! I like fewer questions too. Good all the way around.
It was such fun reading your memories today. I am impressed with the desk shot and I love chicken salad on a croissant. We have a coffee shop here who makes amazing ones. | {
"redpajama_set_name": "RedPajamaC4"
} | 318 |
using System.IO;
using Microsoft.AspNetCore.Hosting;
using Microsoft.Extensions.Configuration;
namespace silverneedleweb
{
public class Program
{
public static void Main(string[] args)
{
var host = new WebHostBuilder()
.UseKestrel()
.UseContentRoot(Directory.GetCurrentDirectory())
.UseIISIntegration()
.UseStartup<Startup>()
.Build();
host.Run();
}
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,859 |
package quickfix
import (
"errors"
"strconv"
)
const (
//-
asciiMinus = 45
//ascii numbers 0-9
ascii0 = 48
ascii9 = 57
)
//atoi is similar to the function in strconv, but is tuned for ints appearing in FIX field types.
func atoi(d []byte) (int, error) {
if d[0] == asciiMinus {
n, err := parseUInt(d[1:])
return (-1) * n, err
}
return parseUInt(d)
}
//parseUInt is similar to the function in strconv, but is tuned for ints appearing in FIX field types.
func parseUInt(d []byte) (n int, err error) {
if len(d) == 0 {
err = errors.New("empty bytes")
return
}
for _, dec := range d {
if dec < ascii0 || dec > ascii9 {
err = errors.New("invalid format")
return
}
n = n*10 + (int(dec) - ascii0)
}
return
}
//FIXInt is a FIX Int Value, implements FieldValue
type FIXInt int
//Int converts the FIXInt value to int
func (f FIXInt) Int() int { return int(f) }
func (f *FIXInt) Read(bytes []byte) error {
i, err := atoi(bytes)
if err != nil {
return err
}
*f = FIXInt(i)
return nil
}
func (f FIXInt) Write() []byte {
return strconv.AppendInt(nil, int64(f), 10)
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,411 |
Planar Hall effect in magnetite (100) films
SHVETS, IGOR
X. S Jin, R. Ramos, Y. Zhou, C. McEvoy, I. V. Shvets 'Planar Hall effect in magnetite (100) films' in Journal of Applied Physics, 99, (8), 2006, 08C509
JAP_99_08C509_2006.pdf (published (publisher copy) peer-reviewed) 200.4Kb
Giant planar Hall effect (GPHE) has been observed in epitaxial magnetite (100) films grown on MgO substrates. The effect is manifested as jumps in the transverse resistivity when the film is subjected to a swept, in-plane magnetic field. The jumps are two orders of magnitude higher than previously observed in metallic ferromagnets. Recently, the same effect has been reported for other materials, but unlike our results, they present GPHE at low temperature only. The magnitude of the GPHE observed at room temperature has potential applications such as magnetic sensors and nonvolatile memory elements.
Science Foundation Ireland
http://people.tcd.ie/ivchvets
Author: SHVETS, IGOR | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,808 |
Q: Is the Radon Nikodym derivative conditioned on a filtration monotonically increasing? Let $(\Omega, \mathcal F, (\mathcal F_t)_{t\geq 0},\mathbb P)$ a filtered probability space satisfying the usual conditions (especially $\mathcal F_0$ contains all null sets) and $\mathbb Q$ a further probability measure with $\mathbb Q\ll\mathbb P$. If we now restict the measures for each $t$ to the sub-$\sigma$-algebras $\mathcal F_t$ of the filtration, the property $$\mathbb Q|_{\mathcal F_t}\ll\mathbb P|_{\mathcal F_t}$$ is conserved. Thus the Radon Nikodym derivatives
$$S_t:=\frac{\mathrm d\mathbb Q|_{\mathcal F_t}}{\mathrm d\mathbb P|_{\mathcal F_t}}=\mathbb E_{\mathbb P}\left(\left.\frac{\mathrm d\mathbb Q}{\mathrm d\mathbb P}\right|\mathcal F_t\right)
$$
exist and form a martingale. As the Radon Nikodym derivative is only unique up to a $\mathbb P$-null set, I'm asking myself:
1. Question: Can the stochastic process $S:=(S_t)_{t\geq0}$ be choosen monotonically increasing?
The background of this question is: As $S$ is a martingale converging a.s. and in $L^1$ to $S_\infty:=\frac{\mathrm d\mathbb Q}{\mathrm d\mathbb P}$, I want to show that for a monotonacally decreasing convex function $\phi$ the process $\phi(S)$ is a submartingale converging in $L^1$ to $\phi(S_\infty)$. But to deduce that $\phi(S)$ is a submartingale I need $\phi(S)$ to be integrable (as a requirement for the Jensen inequality).
2. Question: Can be shown without any further assumptions that $\phi(S)$ is integrable?
In the final step I want to use something like the monotone convergence theorem to deduce the $L^1$-convergence of $S$.
A: 1. No, outside the trivial case $\Bbb Q=\Bbb P$. Because $S$ is a martingale, $\Bbb E_{\Bbb P}[S_t]=\Bbb E_{\Bbb P}[S_u]$ for all $0\le t<u$. If it were true that $S_t\le S_u$, then you would have $\Bbb P[S_t=S_u]=1$.
2. If $\phi$ is understood to be a decreasing convex function mapping $[0,\infty)$ to $\Bbb R$, then the integrability of each $\phi(S_t)$ is assured. On the one hand, $\phi(x)\le\phi(0)$ for each $x\ge 0$, so $\phi(S_t)$ is bounded above. In the other direction, by convexity there are reals $a$ and $b$ such that $\phi(x)\ge a+bx$ for all $x\ge 0$. Thus, $\phi(S_t)\ge a+bS_t$. Because $S_t$ is integrable, these two bounds ensure the integrability of $\phi(S_t)$, which is therefore a submartingale. In addition, you have the bounds
$$
a+bS_t-\phi(0)\le\phi(S_t)-\phi(S_\infty)\le \phi(0)-a-bS_\infty,
$$
which together with the uniform integrability of $(S_t)_{0\le t\le\infty}$ is enough to guarantee that $\phi(S_t)$ converges to $\phi(S_\infty)$ in $L^1$.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,213 |
\section{Introduction}
\noindent Rare signal searches, such as those performed for direct dark matter detection (\cite{futuredarkmatter,DDReview,chepelReview} and references therein) and neutrino-less double beta decay experiments (\cite{doublebeta} and references therein), are typically carried out in deep underground laboratories. The rock over-burden of such facilities removes or dramatically reduces many of the background signals that would be present if the experiments were conducted in surface laboratories. As improved sensitivity is achieved, the need to characterise and mitigate remaining backgrounds becomes ever more important. One of the most problematic backgrounds that still remains is that of cosmic-ray muon-induced neutrons, which may become a limiting factor in some next-generation rare event searches. This specific type of background already shows its impact in current dark matter experiments, with XENON100 reporting it to be the dominant contributor to their nuclear recoil background expectation~\cite{xenon100}.
Neutrons arising from radioactive decays, for example in a fission process or produced in ($\alpha$,n) reactions following \mbox {$\alpha$-decay} of trace contaminations of heavy radio-isotopes, have energies limited to a few MeV. In contrast, neutrons produced through interaction of high energy cosmic-ray muons with matter can reach energies of several GeV. Consequently, while radioactivity neutrons may be effectively controlled by appropriate shielding constructions and selection of radio-pure building materials, removing cosmic-ray induced neutrons is more difficult, with the most effective solution being to go deep underground where the muon flux is reduced by several orders of magnitude compared to that at the surface. Further mitigation of this background involves large muon vetoes, such as instrumented water tanks, to efficiently detect muon tracks far away from the detector.
The neutron production cross-section for high energy muons is very large in high-A materials. Yet, several rare event search projects utilise large amounts of lead to provide shielding against ambient $\gamma$-rays. Thus, the accurate knowledge of the production rate of neutrons by cosmic-ray muons in this material is very important for assessing and planning the capability of these projects, present and future.
A penetrating cosmic-ray muon may produce neutrons via four main processes:
(i) muon spallation --- muon-nuclear interaction via the exchange of a virtual photon, resulting in nuclear disintegration,
(ii) muon capture (only dominant for shallow depths, $\lesssim$100~m~w.e.),
(iii) photo-nuclear interactions in muon-triggered electromagnetic showers, and
(iv) hadron-production in hadronic cascades initiated by the muon.
These secondary cascades make up most of the muon-induced neutron production in deep sites. Specifically, neutrons are predominantly created by photo-nuclear interactions of $\gamma$-rays produced in electromagnetic showers, neutron inelastic scattering, pion spallation and pion absorption at rest. The rate of neutron production by direct muon nuclear interaction is significantly smaller than for the other processes listed~\cite{kozlov,Lindote,MUSUN,muonsim,wang}.
The non-trivial task of measuring the cosmic-ray muon induced neutron yield has been pursued by a number of underground experiments (see~\cite{MUSUN,muonsim,wang,mei} for a compilation of such results). Most recently, the KamLAND collaboration has presented muon-induced neutron rates for a number of
target isotopes~\cite{kamland}. Additional work from other groups is ongoing~\cite{kozlov}. While for low-A targets agreement between the different measurements and simulation toolkits (GEANT4~\cite{geant4}, FLUKA~\cite{fluka1,fluka2}) is reasonable, studies of heavy targets are somewhat controversial and inconsistent~\cite{Kudryavtsev}. Older measurements for Pb targets~\cite{gorshkov,bergamasco}, including beam measurements~\cite{na55}, without Monte Carlo simulations of neutron production, transport and detection, show much larger neutron yields than expected from simulations~\cite{MUSUN,muonsim,wang,marino}. On the contrary, measurements with the veto of the ZEPLIN--II experiment at the Boulby Underground Laboratory showed an over-production in the simulation by $\sim$80$\%$~\cite{muonmeasurement}.
Here we present a new measurement of the muon-induced neutron yield in lead using the data accrued by a highly segmented anti-coincidence detector installed around the ZEPLIN--III dark matter instrument. The measurement was conducted in parallel to the 319-day long second science run of the experiment in 2010/11.
\section{Experimental apparatus}
\label{sec:experimental}
\noindent The ZEPLIN--III instrument \cite{zeplindesign,zeplinsim} is a dual phase liquid/gas xenon detector, built to observe low energy nuclear recoils resulting from elastic scattering of weakly interacting massive particles (WIMPs). The final scientific exploitation of the long-running ZEPLIN project at the Boulby Underground Laboratory (at 2850~m~w.e.) achieved cross-section limits for scalar and spin-dependent WIMP--neutron channels of 3.9~$\times$~10$^{-8}$~pb and 8.0~$\times$~10$^{-3}$~pb near 50~GeV/c$^{2}$ (90$\%$ confidence), respectively~\cite{FSR, SSR}. For the second science run the detector was upgraded with a new array of photomultiplier tubes (PMTs), decreasing internal background sources significantly~\cite{backgrounds}, and a 1-tonne plastic scintillator anti-coincidence detector system (the `veto'), mounted around the main instrument~\cite{veto-design}. Designed for rejecting background events in the WIMP target, the veto also helped to decrease systematic uncertainties in the estimation of background rates due to independent measurements of $\gamma$-ray and neutron rates in the vicinity of the dark matter detector~\cite{veto-performance}. In addition, the veto provided an independent measurement of the high energy cosmic-ray muon flux and its spallation products. The performance of the instrument is well understood through data and validated Monte Carlo simulations~\cite{backgrounds, veto-performance}.
The veto system consists of 52 individual plastic scintillator modules (51 were active during the second science run) surrounding a 15~cm thick Gd-loaded polypropylene shielding, which encircles the ZEPLIN--III instrument. This entire structure is then enclosed in a 20~cm thick lead castle. A CAD rendering of the full setup is shown in Fig.~\ref{crosszep}. For a detailed description of the design and performance of each individual component see Ref.~\cite{veto-design}. Here, only a short summary is presented.
\begin{figure}
\begin{center}
\includegraphics[width=3.0in]{figure_1_veto}
\caption{(Colour online) CAD rendering of the veto system surrounding the ZEPLIN--III dark matter detection instrument. The veto barrel consists of 32 vertical Gd-loaded polypropylene pieces (white) surrounded by the same number of active scintillator modules (black), with PMTs housed in cups and recessed into the lower polypropylene structure. The roof of the veto detector is composed of 20 scintillator modules, which are placed on top of a roof plug. The lower polypropylene structure contains no Gd and rests on a copper and lead base. Finally, a lead castle (only the first few lead blocks are shown on the sides facing the back) envelops the entire assembly ($\sim$2.3~m in length and $\sim$2.4~m in height). For display purposes only, a quarter of the scintillator bars from the barrel are not drawn to reveal the ZEPLIN--III detector.}
\label{crosszep}
\end{center}
\end{figure}
The structure formed by assembling the individual modules can be described by two main geometrical shapes: a circular barrel composed of 32 vertical scintillator bars and a roof constructed from 20 individual scintillator blocks. Each barrel bar has a trapezoidal cross-section with parallel sides of length 15~cm and 12~cm and a height of 15~cm. The length of the barrel scintillators is 1~m. The roof sections are of four different lengths (80, 75, 67 and 51~cm) oriented to form a pseudo-circular shape divided into quadrants and are of rectangular cross-section with side lengths of 15~cm $\times$ 16~cm. The individual detector bars are made from polystyrene-based plastic scintillator UPS-923A (p-terphenyl 2$\%$, POPOP 0.02$\%$), produced by Amcrys-H, Kharkov, Ukraine~\cite{Amcrys-Hwebsite}. A single PMT (ETEL-9302~KB) is optically coupled to one end of each individual scintillator bar. Additionally, all bars have been wrapped in PTFE sheet of high diffuse reflectivity, and a highly-specular reflective aluminised Mylar film is located at the end opposite to the PMT to increase light collection.
The polypropylene shielding inside the scintillator construction is loaded with $\sim$0.4$\%$ Gd by weight \cite{veto-performance}. Thus, many neutrons, moderated to thermal energies, undergo radiative capture on $^{157}$Gd due to its very high capture cross-section of 2.4~$\times$~10$^{5}$~barn~\cite{157Gdcrosssec}. This is of great advantage for detecting and identifying radioactivity neutrons from internal detector components. These are detected with high efficiency through the emission of 3--4~$\gamma$-rays (totalling $\sim$8~MeV) with a mean delay of only $\sim$11~$\mu$s~\cite{veto-performance}. For the more energetic muon-induced neutrons, which are mostly produced externally, a slower capture on hydrogen in the plastic scintillator is expected.
Data were accrued with a dedicated data acquisition system (CAEN model V1724), digitising waveforms with 14-bit resolution, an input range of 2.25 V, 40~MHz bandwidth and a sampling rate of 10~MS/s. The waveforms of recorded events were 320~$\mu$s in length; they were parameterised using a bespoke data reduction software (`RaVen') adapted from that developed for the ZEPLIN--III instrument~\cite{ze3ra}.
The veto detector was operated in `slave' and `master' mode simultaneously. In slave mode the veto acquisition system was triggered by an external signal generated by ZEPLIN--III. The trigger point and timeline lengths were tailored to enable quasi dead time free recording of coincident events. The master mode allowed for independent triggering of the veto system when certain requirements were met. One of these conditions was the sum of simultaneously occurring pulses in the roof modules exceeding a set threshold (summed in a dedicated hardware unit). At this depth, most cosmic-ray muons have an arrival direction which is close to vertical, and thus such a trigger condition provides a high efficiency for detection of cosmic-ray muons, but adds little to the total data storage or rate implied for the experiment.
Critical for a long running experiment is the stability of the detector system over time. Thus, a number of parameters, including electronic gains (measured with the single photoelectron response of the PMTs), coincidence rates, background rates, tagging efficiencies of electron recoil events and environmental parameters, were monitored throughout the course of the experiment. Additionally, a dedicated calibration run was performed on a weekly basis, with a pulsed blue LED, coupled via fibre optic cable to each individual scintillator bar at the end opposite to the PMT. Monitoring of the mean of the single photoelectron peak and of the centroid of the LED-generated peak over the duration of the experiment confirmed the system's stability~\cite{veto-performance}.
\section{Monte Carlo simulations}
\label{sim_general}
\noindent Simulated primary muon energy spectra and angular distributions were obtained by propagation of atmospheric muons from the Earth's surface through an appropriate depth of rock using the MUSIC code~\cite{MUSIC,musicmusun}; this distribution was then sampled with the MUSUN code~\cite{MUSUN,musicmusun}. The energy, momentum, position and charge of each muon was recorded at the point where it intersected the surface of a cuboid fully enclosing the main cavern of the laboratory. The cuboid included an extra 5~m of rock on each side, except for the top which enclosed a total of 7~m of additional rock. The mean energy of the muon distribution was $\sim$260~GeV and 20 million of these muons were generated. The equivalent live-time of the final simulation for the present study amounts to $\sim$3.1~years.
The comprehensive simulation that was developed for the ZEPLIN--III experiment has already been well established in previous studies~\cite{zeplinsim, backgrounds, Leff}. Complementary investigations of the veto detector have also been performed~\cite{veto-design, veto-performance}. This simulation was updated to run with version 9.5 (patch 01) of GEANT4 for this work.
To model the physical processes for this setup the modular physics list {\tt Shielding}, currently recommended for shielding applications at high energies, was implemented. It uses the Fritiof string model (FTF) and the Bertini cascade (BERT) for the high and low energy ranges (up to 5~GeV), respectively, similar to the {\tt FTFP$\_$BERT} reference list but with different neutron cross-section data (JENDL-HE-2007~\cite{JENDL} up to 3~GeV and evaluated cross-sections~\cite{Barashenkov} above 3~GeV)~\cite{shielding}. Neutron interactions below 20~MeV are described by high-precision data-driven models with data obtained from the ENDF/B-VII library~\cite{endf}. Additionally, thermal scattering off chemically bound atoms was implemented for neutron energies below 4~eV, which is especially important to model thermalisation in the plastics~\cite{thermal}.
Secondary particle production thresholds (`cuts') were set to 0.1~mm for $\gamma$-rays and e$^{-}$/e$^{+}$ which, in lead, translate to $\sim$30~keV and $\sim$250~keV, respectively. This is safely below photo- and electro-nuclear reaction thresholds.
The output generated by the simulation has been designed to recreate that of the experiment, {\em i.e.}~a waveform-like readout with a resolution of 0.1~$\mu$s for all 52 individual channels separately. Thus, direct comparison to data as well as the use of similar analysis cuts for experimental and simulated data is possible.
\section{Event selection}
\noindent During the second science run, it was required that the veto be maximally sensitive to the low energy deposits expected from multiply scattering radioactivity neutrons and $\gamma$-rays. Consequently, bias voltages for each PMT were adjusted to deliver a dynamic range in the region of~1--70~photoelectrons (phe), corresponding to approximately 20--1300 keV at the far end of the scintillator. A minimum-ionising muon crossing the full thickness of a scintillator bar deposits at least $\sim$20~MeV and, thus, muon signals, and a greater number of MeV energy deposits from ambient \mbox{$\gamma$-ray} background, result in heavily saturated pulses. Given a single range data acquisition, recording of non-saturated muon events simultaneously with the signal expected from captured neutrons would not be possible. Selection of muons from this data set is therefore non-trivial, but can be achieved by searching for coincident saturated signals in roof and barrel scintillators, due to the optical separation of the modules.
Figure~\ref{data_muon_select} shows the greatest energy deposition observed in a roof module plotted against the largest corresponding (coincident) signal in a barrel module for each event, occurring within $\pm$0.2~$\mu$s around the trigger point. This is similar to the prompt coincidence window used for tagging $\gamma$-ray events in \mbox{ZEPLIN--III}~\cite{veto-performance}. A well separated population is observed, with a graphical selection criterion indicated. Here, the photoelectron scale is defined by using a constant conversion factor between the pulse area and the pulse height parameter, which was utilised for the single photoelectron calibration. Given that the pulse area is less affected by saturation than the pulse height (due to the abrupt cutoff in the latter), the impact of saturation can be pushed to higher energies ($\gtrsim$100~phe), and so improve separation of event populations.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_2_muons_data}
\caption{(Colour online) Highest energy deposition observed in one of the roof scintillator modules versus the time-coincident highest energy deposition in a barrel scintillator module. The dashed line indicates the graphical cut used to select the muons. Note that energy depositions $\gtrsim$100~phe are saturated.}
\label{data_muon_select}
\end{center}
\end{figure}
Selection of muons in the simulation followed a very similar procedure. Firstly, events with a minimum energy deposition observed from the summed signal of the veto roof, analogous to the trigger function of the veto detector, were selected. Additionally, as in the data, a cut on time coincidence (0-0.4~$\mu$s) between roof and barrel was applied. In Fig.~\ref{muon_eff}, the Monte Carlo data are plotted as a function of the largest energy deposition in the (coincident) barrel module only. Separate curves are shown for all events satisfying the coincidence condition, and for only those events corresponding to energy depositions directly resulting from muon traversal of scintillator modules. The difference between the two curves is predominantly due to the energy depositions from particles generated in showers as muons pass nearby. A simple cut at the position indicated by the dashed line selects a population which is composed of~$\sim$93$\%$ muon energy depositions with the required coincidence, {\em i.e}~at least one roof and one barrel module firing within the defined coincidence window.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_3_muons_simulation}
\caption{(Colour online) The plot shows the highest energy deposition observed in a barrel scintillator module when measured in coincidence with a roof module for all prompt energy depositions in the simulation (black solid). The red dashed spectrum shows the same but for muon hits only, {\em i.e.}~the muon crosses both the roof and the coincident barrel module. The cut used to select muon events is indicated by the thick dashed vertical line. $\sim$88$\%$ of muons with a roof -- barrel coincidence have energies above this threshold.}
\label{muon_eff}
\end{center}
\end{figure}
Confirmation that the identified region in the experimental data corresponds to the muon event region in the Monte Carlo is provided by comparing the event distributions between pairs of roof modules and barrel modules (scaled to the overall observed muon rate as measured from the experimental data), as shown in Fig.~\ref{coinc_channels}. Here, the two upper panels show the distribution of roof modules (numbered 32--51) registering a coincidence with a specific barrel module (modules 3 and 19, as indicated). Similarly, the lower panels show which barrel modules (numbered 0--31) are in coincidence with which roof modules (39 and 46). The inactive module is one of the central roof scintillator bars (number 50) featuring a length of 80~cm. The combination of the relative orientations of the modules with respect to each other, their individual response functions, and the asymmetric impact of the surrounding laboratory geometry, results in a complex distribution of coincidences between modules. However, the Monte Carlo reproduces the experimental data reasonably well (the average reduced $\chi^{2}$ value of these 51 coincidence contributions is $\sim$1.9), confirming that the selected experimental data correspond to cosmic-ray muon events.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{figure_4_muons_channel}
\caption{(Colour online) Sample of coincident channels from two barrel slabs (top, module 3 and 19) and two roof slabs (bottom, module 39 and 46) with all modules (channels) of the roof and the barrel, respectively. The simulation, scaled to the total muon rate observed in the data, is shown by the red dashed hatched histogram in comparison to the data (black histogram).}
\label{coinc_channels}
\end{center}
\end{figure}
An overall efficiency for pure muon events, including previously mentioned effects and the (geometric) requirement for coincidence between barrel and roof is 36.8$\pm$0.6$\%$, where the error includes uncertainties due to the precise choice of the location of the selection cuts.
A total number of 7979 muons was selected from the full dataset translating to a rate of 32.3$\pm$0.4~muons/day.
By comparing the measured rate to the Monte Carlo prediction, using the normalised flux through a sphere in the simulation in a similar way to \cite{muonmeasurement}, we deduce a muon flux of (3.75$\pm$0.09)~$\times$~10$^{-8}$~muons/s/cm$^{2}$. This result is in excellent agreement with the last reported value for the muon flux in the Boulby Underground Laboratory of (3.79$\pm$0.15)~$\times$~10$^{-8}$~muons/s/cm$^{2}$ \cite{muonmeasurement}, measured in the cavern hosting both the ZEPLIN--II and ZEPLIN--III detectors, and $\sim$8$\%$ lower than the value obtained for another cavern in Boulby reported in~\cite{Robinson}.
\section{Muon-induced neutron yield}
\noindent The vast majority of detected neutrons produced by muons in this set-up originates in the $\sim$60-tonne lead shield, which protects the experiment from ambient $\gamma$-rays. To determine the muon-induced neutron yield in lead from the present data we count the number of neutrons captured in the veto following a recorded muon event. This is compared with simulations performed using the same analysis cuts. We note that a data set with single photoelectron resolution is a real asset: at the expense of a small increase in background rate, the low threshold analysis increases the number of detected neutrons substantially in comparison with previous works.
\subsection{Experiment}
\label{exp_neutrons}
\noindent As described previously, neutrons are identified through signals occurring in one or more of the 51 scintillators as a result of the $\gamma$-rays emitted following their capture. These signals are delayed relative to the muon's passage due to the time for thermalisation and capture to occur. Ideally, the data would be searched for the signatures of neutron captures over the entire period in which these signals may arrive. However, the PMT response to a large energy deposition is such that the timelines become at first heavily saturated, and then exhibit a large signal overshoot. For extreme energy depositions the overshoots persisted for up to 40~$\mu$s. A sample waveform of a heavily saturated signal from an energy deposition of a muon passing through a roof scintillator bar is given in Fig.~\ref{waveform}. The effect of these `dead' waveform periods can be seen in Fig.~\ref{timeline}, showing a significantly reduced pulse rate for the first $\sim$40~$\mu$s after the muon trigger. Thus, the timeline for detecting delayed neutrons was restricted to the region of 40--300~$\mu$s relative to the observed muon. An efficiency of $\sim$47$\%$ was retained from this timeline selection cut (calculated from simulations). Furthermore, the maximum number of recorded pulses was restricted to 300 entries per event (an equivalent cut was implemented in the analysis of the simulation). The impact of this restriction is discussed in Section~\ref{neutron_sim}.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_5_muon_waveform_example}
\caption{Sample waveform showing a highly saturated pulse ($\sim$20~$\mu$s) from an energy deposition of a muon passing through a roof scintillator bar. The saturated pulse is followed by significant overshoot. Single photoelectron-like pulses are visible where the pulse overshoot starts to recover (between 40--50~$\mu$s). These are suppressed by the pulse-finding algorithm as these pulses are below the baseline of the waveform. At $\sim$185~$\mu$s the delayed signal from an accepted muon-induced neutron event (this particular capture signal was observed in 5 scintillator bars simultaneously) is shown and another one at $\sim$260~$\mu$s. In this case the signal, with a size of 4~phe, is part of a channel multiplicity 2 event.}
\label{waveform}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_6_data_time_distribution}
\caption{(Colour online) Time delay distribution for all recorded pulses above given thresholds relative to the muon.}
\label{timeline}
\end{center}
\end{figure}
For the detection of muon-induced neutrons, as compared to internally-generated radioactivity neutrons, one expects an increased importance of neutrons capturing on hydrogen. In analysing veto data to support the dark matter search, neutrons will have scattered within the \mbox{ZEPLIN--III} instrument and thus have a high geometrical probability of being captured in the Gd-loaded polypropylene shielding immediately surrounding the target. Most muon-induced neutrons come from outside of the setup and will more likely be captured in the hydrocarbon scintillator material surrounding the Gd-loaded shielding. A single $\sim$2.2~MeV $\gamma$-ray is emitted following capture on hydrogen, and therefore signals observed in a single scintillator module are more likely to occur -- in contrast to the several $\gamma$-ray signature from Gd capture, which can be recorded simultaneously in several scintillator modules. Due to the relatively long capture times of neutrons on hydrogen in comparison to captures on gadolinium, the rejection of the first 40~$\mu$s of the waveforms reduces the probability of detection with single scintillator signals by only $\sim$26$\%$.
Single scintillator events are more exposed to backgrounds, and careful consideration of thresholds and a good knowledge of those backgrounds are required. Since available statistics of the limited pre-trigger timeline fraction are very scarce, an additional data set from the same run with similar trigger conditions was used to estimate the background correctly. A dataset of synchronised (with ZEPLIN--III) `slave' triggered veto events (see Section~\ref{sec:experimental}) was considered to calculate the background. As shown from the analysis of the dark matter search data, these events are fully consistent with $\gamma$-ray background~\cite{SSR}. If tagged by the veto, a prompt signal occurs within a time window of 0.4~$\mu$s~\cite{veto-performance}. Any signals recorded in the waveforms of the veto some $\mu$s away from the trigger time are due to uncorrelated background events only. Optimisation of the number of neutron captures observed in the muon triggered data, with respect to the number of false events due to background, results in a threshold of $\geqslant$10~phe being chosen for single scintillator events (the present results were shown to be largely insensitive to the precise threshold). Figure~\ref{const_background} shows the rate of background events with a threshold of $\geqslant$10~phe applied. The rate is approximately constant, {\em i.e.}~it is independent of the time since the trigger occurred.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_7_background}
\caption{(Colour online) Time delay distribution of channel multiplicity one events (threshold of $\geqslant$10~phe) in ZEPLIN--III coincident background data. The dashed (red) line indicates the constant fit to the background.}
\label{const_background}
\end{center}
\end{figure}
Following the methodology used in the analysis of the dark matter search data~\cite{SSR}, coincident signals in multiple scintillators can also be searched for, detecting multiple scatters and $\gamma$-rays following neutron capture on gadolinium at later times. Coincidences are defined as occurring within $\pm$0.2~$\mu$s of each other. To optimise efficiency, different signal size thresholds have been required depending on the number of signals in coincidence, balanced against the rate of false signals arising from non-neutron related sources (background and induced noise). Noise from the PMTs can be intrinsic, {\em i.e.}~from thermionic emission and internal radioactive decays, or directly induced. Especially after larger signals, such as resulting from muons, positive ions generated from ionisation of residual gases in the PMTs lead to secondary signals, creating afterpulses at short time scales of up to several $\mu$s dependent on the ion transit time (see~\cite{Birks,Campbell} and references therein). In the present data afterpulses of small amplitudes are suppressed at short decay times due to the large pulse overshoots observed following a muon energy deposit (see Fig.~\ref{timeline}). A second noise component observed at longer time scales, visible in Fig.~\ref{timeline} between $\sim$30--50~$\mu$s after the start of the muon signal, with sizes of 4~phe and below, may be attributed to the organic scintillator. Luminescence with long time constants is expected from phosphorescence and delayed fluorescence processes in the plastic scintillator (see {\em e.g.}~\cite{Birks,Kawada,Marvin}).
These additional signals could lead to false coincidences between scintillators, generating spurious neutron detections. Based on the event rates, the probability of false coincidences can be calculated. It was found that for neutron capture events with a channel multiplicity of two, {\em i.e.}~two scintillator bars firing within $\pm$0.2~$\mu$s of each other, a signal size requirement of threshold $\geqslant$4~phe (in each pulse) was sufficient to remove afterpulses. For three-fold coincidences between scintillators, a threshold of $\geqslant$2~phe per signal was found to be appropriate, and for four or more scintillators, a threshold at the level of a single photoelectron was sufficient. For consistency, a global requirement was set that regardless of the number of scintillators fired in coincidence, all events must have a total signal size of at least 8~phe. Despite the lower threshold for multiple scintillator events, accidental rates arising from background are a lot smaller due to the required coincidence of pulses. The same dataset used earlier to estimate the background rate in the single scintillator case has thence been utilised to calculate the contribution from background to the yields of neutron captures found from the multiple scintillator requirements. Background rates found are at the level of statistical uncertainties.
Table~\ref{data_neutron_rates} summarises the results. Each instance in which the designated criteria were met is interpreted as indicating a neutron capture. Most muon-induced neutron captures are observed through events seen in single scintillators only, despite the higher threshold required. However, a significant number also generate energy depositions observed in coincidence in several scintillators. Overall, a mean of 0.346$\pm$0.007~neutrons (including background corrections) are observed for every muon detected.
\begin{table}
\centering
\caption{Measured number of neutrons per muon from the data in comparison to neutron rates extracted from simulations using the same requirements and cuts as in the experimental data analysis. Background rates, for correction of the data, are listed individually for the different channel multiplicities, with their required thresholds detailed in the text. The errors given for the data are the sum of statistical errors and the rate coming from random accidental coincidences of pulses calculated from the average observed pulse rate for a given threshold. Errors of simulated rates are statistical only.}
\vspace{2mm}
\label{data_neutron_rates}
\begin{tabular}{p{1.0cm} p{1.45cm} p{1.45cm}p{1.45cm}|p{1.45cm}}
\hline
\multicolumn{5}{c}{\hspace{33mm}Data\hspace{29mm}Simulation}\\
\hline
Channel mult. & Events/ muon & Background rate & n/muon (bkg.corr.) & n/muon\\
\hline
\hline
1 & 0.216(5) & 0.019(1) & 0.197(5) & 0.145(2)\\
2 & 0.088(3) & 0.0049(5) & 0.083(3) & 0.076(1)\\
3 & 0.039(2) & 0.0019(3) & 0.037(2) & 0.0321(9)\\
$\geqslant$4 & 0.029(2) & 0.0008(2) & 0.028(2) & 0.0231(8)\\
\hline
\hline
Total & 0.372(7) & 0.026(1) & 0.346(7) & 0.275(3)\\
\hline
\end{tabular}
\end{table}
\subsection{Comparison with simulations}
\label{neutron_sim}
\noindent Monte Carlo simulations of the experiment have been performed as described in Section~\ref{sim_general}. The dimensions and parameters of the apparatus have been previously measured and documented~\cite{veto-design,Leff,qfpaper}, including signal gains and attenuation lengths of the scintillators so that photoelectron spectra can be generated. This allows Monte Carlo pseudo-data to be analysed using identical routines as used for the real data. The overall agreement on the rate of detected neutrons between data and simulation obtained in this work is good, at the level of 25$\%$. For the initial discussion the total number of neutrons has been normalised to the data.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_8_neutrons_time_distribution}
\caption{(Colour online) Time delay distribution of detected captured neutrons from experimental data (black solid histogram) and simulations (red dashed hatched histogram). The constant background has been subtracted from the data histogram. Results from simulation are normalised to the total number of neutrons observed in the data.}
\label{neutron_time_delay}
\end{center}
\end{figure}
Figure~\ref{neutron_time_delay} compares the time delay distributions for detected neutrons from data (solid black) and simulation (red dashed). Excellent agreement between the two distributions is found. Moreover, the module (channel) multiplicity per neutron event can be used as an additional consistency check, beyond the initial muon identification, taking advantage of the segmented nature of the detector. Figure~\ref{neutron_channel_mult} shows the number of channels with coincident signals involved in each individual neutron event. The data are corrected for the contributions from background coincidences, as given in Table~\ref{data_neutron_rates}. Again, excellent agreement, over the full range of channel multiplicities, is demonstrated.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_9_neutrons_channel_multiplicity}
\caption{(Colour online) Comparison of channel (scintillator module) multiplicities per detected neutron in the data (black solid) to simulations (red dashed hatched histogram). The data are background corrected according to Table~\ref{data_neutron_rates}. Results from simulation are normalised to the total number of neutrons observed in the data.}
\label{neutron_channel_mult}
\end{center}
\end{figure}
In Fig.~\ref{neutron_edep} the energy depositions associated with the observed (captured) neutrons are given in the region before the onset of saturation, with excellent agreement between simulation and data obtained. Although the energy calibration is only known to within 10$\%$ due to, amongst other factors, the saturation of the data (see also previous studies with the same instrument~\cite{veto-performance,qfpaper}), tests in varying the energy scale by this amount resulted in only small neutron rate differences and are considered in the systematic error of the simulated rate.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_10_neutrons_energy_depositions}
\caption{(Colour online) Energy depositions of detected neutrons from background corrected data (black solid) and simulations (red dashed hatched histogram) below the saturation point in the data, {\em i.e.}~the energy scale is given in absolute number of photoelectrons (1~phe~$\simeq$~20~keV). Results from simulation are normalised to the total number of neutrons observed in the data.}
\label{neutron_edep}
\end{center}
\end{figure}
A muon may produce more than one fast neutron in a cascade, resulting in several neutron capture signals at different times and in different locations in the veto. Figure~\ref{neutron_neutron_mult} shows the relative fraction of observed neutrons per muon for data and simulation. When exploring neutron multiplicities, rather than scaling the simulation to the total number of neutrons observed in the data, a simple normalisation to the number of detected muons has been applied. Background corrections assume even distribution of background events. As such, most non-neutron signals occur in one of the empty waveforms following a muon trigger, making up almost 90$\%$ of all observed muon events. Generally, good agreement is observed.
\begin{figure}
\begin{center}
\includegraphics[width=0.45\textwidth]{figure_11_neutrons_neutron_multiplicity}
\caption{(Colour online) Relative fraction of neutron multiplicities per muon, {\em i.e.}~the number of delayed signals observed after a muon trigger in the defined time window, for background corrected data (black solid) and simulation (red dashed hatched histogram) normalised to the total number of observed muons in each case. 66$\%$ of neutron capture signals associated with muon events which registered a single detected neutron only are observed in a single scintillator.}
\label{neutron_neutron_mult}
\end{center}
\end{figure}
When scaled to the number of neutrons detected, the simulations reproduce well the time distributions, the energy depositions and the number of scintillators involved in each event. The absolute numbers of neutrons expected to be observed per muon, as determined from the simulation for each individual channel multiplicity, are also summarised in Table~\ref{data_neutron_rates}, showing an overall reduced neutron rate from simulations of $\sim$20$\%$ ({\em i.e.}~the total yield from the data exceeds the simulation by $\sim$26$\%$). Discrepancies are largest for single scintillator events. At higher multiplicities absolute agreement between simulation and data is of order 10--20$\%$ (cf.~$\sim$36$\%$ for single scintillator events).
The expected total muon-induced neutron rate calculated from simulations is 0.275$\pm$0.003~(stat.)~$^{+0.004}_{-0.007}$~(syst.)~neutrons/muon. Systematic errors are calculated from the variability in the energy calibration.
To assess the greater discrepancy for single channel neutron events between data and simulation, tests in raising the detection threshold and limiting the time window to search for neutron capture signals (further away from the trigger) were performed. No significant differences are observed. Additionally, exclusion of the two channels with the highest energy depositions from the traversing muon (used prior to the muon-induced neutron analysis for the initial selection of muon events) from the final analysis reduces the number of neutron captures by almost the same factor in data and simulations. For neutron captures giving a signal in one scintillator only, the difference in the reduction factors is practically the same as for higher channel multiplicities. For the whole sample of neutron captures the slight difference in the reduction factors between data and simulations introduces a systematic uncertainty of 4$\%$. Thus, the overall detected muon-induced neutron rate from experimental data results in 0.346$\pm$0.007~(stat.)~$^{+0.000}_{-0.014}$~(syst.)~neutrons/muon.
As previously mentioned, a restriction on the maximum number of recorded pulses was applied to the data, and similarly in the analysis of the simulation. Importantly, the number of selected muon events, in both data and simulation, affected by this limitation is in excellent agreement, further supporting the performance of the Monte Carlo simulation. In the data 11 events were found which were affected by this cut; the number of events associated with more than 300 pulses in the simulation (scaled to the data) amounts to 11$\pm$2. When including all energy depositions in the simulation a higher absolute neutron rate is observed ($\sim$30$\%$). This increase is associated with only a few muon events (approximately 1 in 1700) featuring exceptionally high neutron multiplicities. It is worth noting that these high multiplicity events are less significant for dark matter searches due to the generally high veto efficiencies expected for these.
Table~\ref{neutrons_produced} shows the relative production of neutrons in different materials for all neutrons generated in the simulation and for detected neutrons only. As expected, nearly all neutrons are produced in the rock cavern of the underground laboratory, reflecting that the simulation included sufficient volume to remove edge effects. Importantly, less than 1.5$\%$ of detected neutrons are produced in the rock, confirming the effectiveness of the shielding setup of the ZEPLIN--III detector. On the other hand, the lead component of the shielding enclosure provides an effective target for neutron production by high energy comic-ray muons, with $\sim$95$\%$ of all neutrons created there.
Table~\ref{neutron_capture_element} lists the specific elements involved in the capture of the neutrons, both for detected neutrons and for all the neutrons in the simulation. As previously mentioned, the vast majority of detected muon-induced neutrons are captured on hydrogen, emphasising the importance of measuring the single $\sim$2.2~MeV $\gamma$-ray from this process. The captures on Gd amount to $\sim$7.0$\%$ for this configuration at these neutron energies (being much more effective for internal radioactivity neutrons due to the detector geometry described in Section~\ref{sec:experimental}).
\begin{table}
\centering
\caption{Fractions of neutrons produced in different materials for all generated neutrons in the simulation and for detected neutrons only.}
\vspace{2mm}
\label{neutrons_produced}
\begin{tabular}{p{2.5cm} p{2.5cm} p{2.5cm}}
\hline
\multicolumn{3}{c}{\hspace{23mm}Production material of}\\
Material & all neutrons & detected neutrons\\
\hline
\hline
Lead & 0.2$\%$ & 95.0$\%$\\
Rock & 99.8$\%$ & 1.4$\%$\\
Steel & - & 1.2$\%$\\
C$_{8}$H$_{8}$ & - & 0.9$\%$\\
Copper & - & 0.8$\%$\\
CH$_{2}$ & - & 0.5$\%$\\
Gd-epoxy & - & 0.1$\%$\\
Liquid Xe & - & 0.1$\%$\\
\hline
\end{tabular}
\end{table}
\begin{table}
\centering
\caption{Fractions of neutrons captured on different elements for all and for detected neutrons only.}
\vspace{2mm}
\label{neutron_capture_element}
\begin{tabular}{p{2.5cm} p{2.5cm} p{2.5cm}}
\hline
\multicolumn{3}{c}{\hspace{23mm}Capture element of}\\
Element & all neutrons & detected neutrons\\
\hline
\hline
H & - & 71.1$\%$\\
Fe & - & 11.5$\%$\\
Cl & 94$\%$ & 7.0$\%$\\
Gd & - & 7.0$\%$\\
Pb & - & 1.3$\%$\\
C & - & 1.1$\%$\\
Cu & - & 0.6$\%$\\
Na & 6$\%$ & 0.2$\%$\\
Mn & - & 0.2$\%$\\
\hline
\end{tabular}
\end{table}
\subsection{Muon-induced neutron yield in lead}
\noindent As shown in Table~\ref{neutrons_produced}, the detected neutrons have predominantly been produced in lead. Thus, the observed neutron rate may be used to derive an absolute neutron production yield in this material. The methodology used follows that of Refs.~\cite{Lindote,muonmeasurement}, and is essentially to scale an idealised simulation of neutron production by a mono-energetic beam of muons in pure lead by the ratio in rate observed between the present data and the full detector simulation (assuming that the fraction of detected neutrons produced in lead ($\sim$95$\%$) is well described by the simulation).
The simulation of a mono-energetic muon beam in lead was conducted as follows. Neutron production was recorded for a mono-energetic 260~GeV $\mu^{-}$~beam (mean muon energy at Boulby), incident on the centre of a lead block of 3200~g/cm$^{2}$ thickness. Figure~\ref{neutron_yield_in_lead} shows the differential energy spectrum of neutrons produced. Only neutrons from the central half length of the lead block were considered to avoid surface/edge effects. To prevent double counting in neutron inelastic processes, the first neutron produced in each reaction was dismissed independently of its energy. As summarised in Table~\ref{neutron_yield_table}, a production rate of (4.594$\pm$0.004)~$\times$~10$^{-3}$~neutrons/muon/(g/cm$^{2}$) was obtained for the physics list and version of GEANT4 used throughout this work ({\tt Shielding} with version 9.5). However, since the experimental muon-induced neutron rate was found to be a factor of 1.26$\pm$0.03~(stat.)~$^{+0.03}_{-0.05}$~(syst.) higher, our results suggest a true production rate by 260~GeV muons of (5.78$\pm$0.13~(stat.)~$^{+0.16}_{-0.25}$~(syst.))~$\times$~10$^{-3}$~neutrons/muon/(g/cm$^{2}$), assuming neutron transport and detection are modelled accurately.
\begin{figure}
\begin{center}
\includegraphics[width=0.36\textwidth]{figure_12_neutron_yield}
\caption{Differential energy spectrum of muon-induced neutrons produced in lead from $\mu^{-}$ of 260 ~GeV.}
\label{neutron_yield_in_lead}
\end{center}
\end{figure}
In this analysis, uncorrelated arrival of muons is assumed, as opposed to muon bundles produced together by primary cosmic-rays in the atmosphere. A study based on a simple approximation to find the survival probability of muons at a given depth showed the effect to be negligible and the error to be very small for the measurement performed with the \mbox{ZEPLIN--III} veto detector.
A similar comparison between simulation and experiment was performed for the ZEPLIN-II anti-coincidence system~\cite{Lindote,muonmeasurement}. In that work a muon-induced neutron yield in lead of (1.31$\pm$0.06)~$\times$~10$^{-3}$~neutrons/muon/(g/cm$^{2}$) was reported. Here we have revisited the simulation, now using GEANT4 version 9.5 and the {\tt Shielding} physics list, including thermal scattering cross-sections, and a significantly larger sample of primary muons. This resulted in a new estimate for the neutron yield in lead of (3.4$\pm$0.1)~$\times$~10$^{-3}$~neutrons/muon/(g/cm$^{2}$) in that setup. While it is clear that a significant contribution of the newly obtained ZEPLIN--II yield comes from the updated simulation, there remains a significant discrepancy with the present result. One possible explanation for this is that the angular distribution of emitted neutrons may not be accurately modelled. The ZEPLIN-III veto scintillators are predominantly sensitive to neutrons produced in lead above and around the scintillators. The ZEPLIN-II veto system detected neutrons produced in lead below and around the liquid scintillator vessel. This and other differences in configuration coupled to possible inaccuracies in GEANT4 modelling of the angular distribution of neutron emission may explain the observed discrepancy between the two results.
We have also explored the evolution of the neutron production yield with successive versions of GEANT4. To do this further simulations of a mono-energetic $\mu^{-}$-beam focused on a lead block have been performed. Table~\ref{neutron_yield_table} summarises the results, including the yield obtained with version 8.2 from Ref.~\cite{Lindote}. In addition, combination of different physics lists and GEANT4 versions are listed, also linking the custom list used in~\cite{Lindote} to the current high energy reference lists. The bespoke physics list is very similar to {\tt QGSP$\_$BIC$\_$HP}, featuring the Quark-Gluon String (QGS) theoretical model at high energies coupled to nuclear de-excitation with a pre-compound model, the intra-nuclear Binary Cascade (BIC) model below 6~GeV and the data driven high precision neutron package ({\tt NeutronHP}) to transport neutrons below 20~MeV down to thermal energies. Reasonable variation of change-over energies between the BIC and QGS models in the custom physics list in comparison to the reference one has little impact ($<$3$\%$) on the overall neutron yield. A steady increase with every new version of GEANT4 is demonstrated.
\begin{table}
\centering
\caption{Muon-induced production yields for neutrons for different versions of GEANT4 and physics lists (for 260~GeV muons). The neutron yield from version 8.2 is based on the value reported in Ref.~\cite{Lindote}. A small modification has been applied to correct for a previously unaccounted error in the rejection of neutrons produced in neutron inelastic processes to avoid double counting (referred to as `stars' in that work).}
\vspace{2mm}
\label{neutron_yield_table}
\begin{tabular}{ p{1.4cm}p{2cm}p{4cm}}
\hline
GEANT4 version & physics list & muon-induced~neutron yield [neutrons/muon/(g/cm$^{2}$)]\\
\hline
\hline
8.2 & custom list & (2.846$\pm$0.006) $\times$~10$^{-3}$ \\
9.4 & custom list & (3.304$\pm$0.003)~$\times$~10$^{-3}$ \\
9.4 & {\tt QGSP$\_$BIC$\_$HP} & (3.376$\pm$0.003)~$\times$~10$^{-3}$ \\
9.4 & {\tt Shielding} & (3.682$\pm$0.003)~$\times$~10$^{-3}$ \\
9.5 & {\tt QGSP$\_$BIC$\_$HP} & (3.993$\pm$0.004)~$\times$~10$^{-3}$ \\
9.5 & {\tt QGSP$\_$BERT$\_$HP} & (4.369$\pm$0.004)~$\times$~10$^{-3}$\\
9.5 & {\tt FTFP$\_$BERT} & (4.467$\pm$0.004)~$\times$~10$^{-3}$\\
9.5 & {\tt Shielding} & (4.594$\pm$0.004)~$\times$~10$^{-3}$ \\
\hline
\end{tabular}
\end{table}
The {\tt Shielding} physics list shows not only the largest muon-induced neutron production yield in comparison to other reference lists, but is also subject to the highest increase in going from version 9.4 to 9.5. This is explored in detail in Fig.~\ref{neutron_yield_in_lead_process}, showing the individual contributions from the most important neutron creation processes for muons in lead. The main difference lies in the increased neutron production in inelastic scattering of hadrons and in particular neutrons. A $\sim$38$\%$ higher production yield for this process is observed.
Part of the increase observed between versions 9.4 and 9.5 of the toolkit (applicable to all standard lists used in this study) can be attributed to the muon-nucleus interaction model ({\tt G4VDMuonNuclearModel}); as in previous versions, this still relies on the Kokoulin mu-nuclear cross-sections~\cite{muoninteraction}, but the final state of the hadronic vertex is now replaced by a $\pi^{0}$ interacting further through the Bertini intra-nuclear cascade. The previous model ({\tt G4MuNuclearInteraction}) replaced the virtual photon with $\pi^{+/-}$ instead, which would then interact through the low/high energy parameterised models (LEP/HEP) --- these are known to yield fewer neutrons. There has also been increased neutron production in the FTF model, which may account for some of the enhanced yields in the {\tt Shielding} and {\tt FTFP$\_$BERT} lists; The addition of the Reggeon cascade~\cite{fritiof}, which can cause more nucleon secondaries, is a possible explanation, but further study is required~\cite{wright}.
\begin{figure}
\begin{center}
\includegraphics[width=0.36\textwidth]{figure_13_neutron_processes}
\caption{(Colour online) Absolute neutron yields of the most important production processes for muon-induced neutrons generated from firing 260~GeV $\mu^{-}$ on lead using the {\tt Shielding} physics list and GEANT4 version 9.5 (black histogram) and version 9.4 (red dashed histogram). The neutron creation processes are: 1: photo-nuclear interaction of $\gamma$-rays ($\gamma$ $\rightarrow$ N), 2: neutron inelastic scattering (n $\rightarrow$ N), 3: pion spallation ($\pi$ $\rightarrow$ N), 4: muon spallation ($\mu$ $\rightarrow$N), 5: proton spallation (p $\rightarrow$ N), 6: pion absorption ($\pi^{-}$ abs) and 7: all other neutron production processes.}
\label{neutron_yield_in_lead_process}
\end{center}
\end{figure}
\section{Conclusion}
\noindent For the development of future rare-event searches, especially in the context of direct dark matter experiments, accurate data on muon-induced neutron yields in several materials is of great importance, as is the ability to simulate these processes using modern Monte Carlo toolkits. Complex models inform the design of large and expensive shielding and veto systems around these experiments, as well as the interpretation of their data (background expectations). There exists significant uncertainty in the simulated muon-induced neutron rate, as evidenced by the steady variation in the total neutron yield with every new version of GEANT4 and physics list; experimental measurements have been likewise uncertain.
In this study, a dataset from 319 days of operation of the \mbox{ZEPLIN--III} anti-coincidence detector has been analysed for high energy cosmic-ray muons. The number of muon-induced neutrons has been evaluated by detecting delayed \mbox{$\gamma$-ray} signals following radiative captures. A muon flux in the Boulby Underground Laboratory of (3.75$\pm$0.09)~$\times$~10$^{-8}$~muons/s/cm$^{2}$ has been determined, consistent with and improving upon previous measurements. The muon-induced neutron detection rate was measured to be 0.346$^{+0.007}_{-0.016}$~neutrons/muon (quadratically combined statistical and systematic errors) traversing the ZEPLIN--III scintillator veto. Monte Carlo simulations, using GEANT4 (version 9.5) and the {\tt Shielding} physics list with the same cuts and thresholds applied as used for the analysis of the data, resulted in a neutron capture rate of 0.275$^{+0.005}_{-0.008}$~neutrons/muon, which is $\sim$20$\%$ lower than the experimentally measured value. However, absolute rates aside, the simulation reproduced very well all tested parameters, strengthening confidence in the results. The ratio of neutron rates between data and simulation have been used to evaluate a muon-induced neutron yield in pure lead of (5.78$^{+0.21}_{-0.28}$)~$\times$~10$^{-3}$~neutrons/muon/(g/cm$^{2}$) for a mean muon energy of 260~GeV. Additional simulations exploring previous versions of the GEANT4 simulation package confirm the trend of an increasing neutron production rate in lead with every successive distribution of GEANT4 (also shown in other simulation studies~\cite{muonsim,vito}).
Finally, our results confirm the very significant contribution of lead to the production of muon-induced neutrons. As such, the use of lead-based shielding to prevent $\gamma$-rays from the environment to propagate into the sensitive volume of the detector should be carefully assessed for any future rare event search. Alternative shielding compositions, such as large water tanks surrounding the detectors, are already being used in some current dark matter searches~\cite{lux,DEAP,XMASS}, as well as discussed for near future next generation experiments~\cite{LZ,eureca}.
\vspace{10mm}
\newproof{ack}{Acknowledgement}
\begin{ack}
The UK groups acknowledge the support of the Science \& Technology Facilities Council (STFC) for the ZEPLIN--III project (in particular, the analysis work presented here was supported by STFC grants ST/K006436/1,~ST/K003178/1,~ST/K003208/1,~ST/K006428/1, ST/K003186/1, ST/K006444/1 and ST/K006770/1) and for maintenance and operation of the underground Palmer Laboratory which is hosted by Cleveland Potash Ltd (CPL) at Boulby Mine, UK. The project would not have been possible without the co-operation of the management and staff of CPL. Additionally, we want to thank the Boulby science facility team for their support during underground aspects of this work. We also acknowledge support from a Joint International Project award, held at ITEP and Imperial College, from the Russian Foundation of Basic Research (08-02-91851 KO\_a) and the Royal Society. LIP--Coimbra acknowledges financial support from Funda\c c\~ao para a Ci\^encia e Tecnologia (FCT) through the project-grants CERN/FP/109320/2009 and /116374/2010, and postdoctoral grants SFRH/BPD/27054/2006, /47320/2008, /63096/2009 and /73676/2010. Furthermore, we acknowledge the Edinburgh Compute and Data Facilities for accommodating the heavy usage of the Edinburgh computer cluster `Eddie' for this analysis.
This work was supported in part by SC Rosatom, contract contract $\#$H.4e.45.90.11.1059 from 10.03.2011. The University of Edinburgh is a charitable body, registered in Scotland, with registration number SC005336.
\newline
\end{ack}
\bibliographystyle{elsarticle-num}
\biboptions{sort&compress}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,081 |
\section{Introduction}
\indent Since the observation of the Gravitational Waves (GW) event GW150914 emanating from the inward spiral and merger of two stellar masses black holes \citep{GW150914}, a new era in astronomy began. GWs have become the newest astronomical messenger joining electromagnetic waves, cosmic rays and neutrinos.\\
\indent The H.E.S.S. Imaging Atmospheric Cherenkov Telescope array is composed of one 28-m and four 12-m telescopes with field of views of 3.2$^{\circ}$ and 5$^{\circ}$. It is sensitive to gamma rays in the range of 50 GeV to 100 TeV, and is capable of detecting a point source at a 5 $\sigma$ level with a similar strength to the Crab in less than one minute \citep{H.E.S.S.sensitivity}. The preparation for the GW alerts from the advanced configurations of the LIGO and VIRGO interferometers within the H.E.S.S. collaboration started in summer 2016. After several technical and commissioning runs used to optimize the response to GW alerts, the first observation campaigns were conducted in August 2017.\\
\indent On August $1^{st}$ 2017 the Advanced Virgo interferometer joined the two LIGO interferometers in their second Observation Run O2. On August $14^{th}$ 2017, a GW signal was detected by the interferometers at 10h30m43s UTC. This signal was produced by the merger of two stellar masses black holes at a distance of $540^{+130}_{-210}$ Mpc, which corresponds to a redshift of z=$0.11^{+0.03}_{-0.04}$. The initial masses of the black holes were $30.5^{+5.7}_{-3.0}$ \(\textup{M}_\odot\) and $25.3^{+2.8}_{-4.2}$ \(\textup{M}_\odot\) \citep{GW170814}. This was the first detection by the three observatories and the added independent baselines from Virgo reduced the localisation uncertainty significantly. Three days later, on August 17, the coalescence of two neutron stars was detected for the first time \citep{GW170817} followed by a GRB detection by Fermi's GBM \citep{GRB}. Information from the three interferometers was used to compute a localisation area of ~30 deg$^2$.\\
\indent We here present for the first time the analysis of the H.E.S.S. data obtained during follow-up observations of GW170814. We also briefly summarize the results obtained during the prompt H.E.S.S. campaign on GW170817.
\section{GW170814}
\begin{figure}[h]
\begin{center}
\includegraphics[width=65mm]{HESS_OBS.png}
\end{center}
\caption{\footnotesize
H.E.S.S. follow-up observations of GW170814. The white contours are the probability contours from the initial LALInference probability map which was used for the H.E.S.S. observation scheduling. The filled contour represents 90\% of the final LALInference map. The rings indicate the field of view of the H.E.S.S. observation runs throughout the three nights.}
\label{H.E.S.S.Obs}
\end{figure}
When reconstructing the localization of the GW emitter, there is a trade-off between latency and accuracy, which is represented by two different algorithms: the BAYESTAR algorithm \citep{BAYESTAR} is for rapid localisation and LALInference \citep{LALInference}, which is scanning a larger parameter space and marginalizing over calibration uncertainties is for a more precise but slower approach. The sky map used to schedule the H.E.S.S. observations was a LALInference based map and was issued on August $15^{th}$, 2017 at 20h17m51s UTC. It had an uncertainty region of 190 deg$^2$ due to the use of LIGO and Virgo data. However, after noise removal and offline detector calibration, a full parameter estimation constrained the 90\% credible region to a slightly shifted area of 60 deg$^2$. This is the final probability region that was published several weeks later \citep{GW170814}. In Fig. \ref{H.E.S.S.Obs} we show the different probability contours (in white) for the map used for the H.E.S.S. scheduling. Quoted values correspond to the \% containment probability region for the GW event localisation. The filled contour encloses a region of the updated (final) LALInference map with a 90\% probability of containing the event. We note that even thought the observation runs obtained by H.E.S.S. where scheduled using the initial LALInference map, it covers most of the final 90\% uncertainty region and the systematic shifts in the GW data don't affect the H.E.S.S. capabilities to cover the region.\\
\indent During the GW170814 follow-up, the 28-m H.E.S.S. telescope was used alongside three 12-m telescopes. Observations started on August $17^{th}$ at 00h10m UTC and a total of 11 runs of 28 minutes each were obtained during three consecutive nights covering approximately 90\% of the final localisation region.
\begin{figure}[h]
\begin{center}
\includegraphics[width=65mm]{HESS_SIGNIFICANCE.png}
\end{center}
\caption{\footnotesize VHE gamma-ray significance map in $\sigma$ units obtained from H.E.S.S. observations of GW170814.}
\label{H.E.S.S._SIGNIFICANCE}
\end{figure}
\begin{figure}[h]
\begin{center}
\includegraphics[width=65mm]{HESS_UL.png}
\end{center}
\caption{\footnotesize Integral upper limits map in erg cm$^{-2}$ s$^{-1}$ derived from H.E.S.S. follow-up of GW170814 valid in the energy range 250 GeV $<$ E $<$ 20 TeV. The white contours are the probability contours from the initial LALInference probability map.}
\label{H.E.S.S._UL}
\end{figure}
\indent Fig. \ref{H.E.S.S._SIGNIFICANCE} shows the gamma-ray significance map which results from the analysis of the data obtained following the pointing pattern presented in Fig \ref{H.E.S.S.Obs}. No significant gamma-ray emission is found during this follow-up campaign.
\indent Combining all observations obtained with H.E.S.S. during the follow-up campaign of GW170814, we derive in Fig. \ref{H.E.S.S._UL} a sky map showing the integral upper limits to constrain the emission in the region for 250 GeV $<$ E $<$ 20 TeV assuming a generic $E^{-2}$ spectrum. Induced by the radially decreasing acceptance of the telescope, the obtained limits are less constraining when approaching the border of the field-of-view. This map can be used to constrain the emission of known gamma-ray sources in our region of interest but more importantly it allows for the first time a constraint to be placed on the level of very-high energy emission from the merger of two stellar mass black holes detected by both LIGO and Virgo detectors.
\section{GW170817}
\indent Extensive details on the GW170817 follow-up with H.E.S.S. can be found in \citep{H.E.S.S._GW170817}. Observations started 5.3 hours after the NS merger. The H.E.S.S. semi-automatic reaction and the implemented observation strategies allowed to react promptly and get on target 5 minutes after the release of the LALInference sky localisation. This allowed to get the first observations on the NS merger from a ground based observatory. Although no significant detection was reported, it allowed to place for the first time stringent upper limits for non-thermal emission for NS-NS mergers in the VHE gamma-ray domain (270 GeV $<$ E $<$ 8.55 TeV). Fig. \ref{H.E.S.S._GW170817} summarizes the follow-up campaign of GW170817 with H.E.S.S.
\begin{strip}
\InsertBoxC{
\includegraphics[scale =0.40]{HESS_GW170817.png}}
\captionof{figure}{Timeline of the observations following the detection of GW170817 with a focus on the high-energy, non-thermal domain.}
\label{H.E.S.S._GW170817}
\end{strip}
\section{Conclusion}
\indent In this contribution we summarized the most interesting H.E.S.S. observations of GW events during O2. Even though no significant VHE gamma-ray emission was detected, we are able to constrain the non-thermal emission from these events by computing integral upper limits on the non-thermal emission of the remnants. These successful observation campaigns provided also important feedback for the next observation run O3. With the approach of O3, new follow-up strategies that further increase the speed of the H.E.S.S. reaction to GW events have been fully implemented and tested within the H.E.S.S. experiment and are used to prepare the multi-messenger program of the upcoming future Cherenkov Telescope Array (CTA).
\bibliographystyle{aa}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,732 |
\section{Introduction}
In the last few decades, we have witnessed an exponential growth of the demand for wireless communication systems that provide reliable communications and ensure ubiquitous coverage, high spectral efficiency and low latency \cite{giordani2020toward}. To meet these requirements, several new technologies have been incorporated in $5$G communication standards, which include Massive multiple-input multiple-output (MIMO) \cite{Chien2017a}, millimeter-wave communications \cite{rappaport2017overview}, and network densification \cite{Bjornson2016c}. Among them, Massive MIMO has gained significant attention since it can offer a good service to many users in the network. Moreover, the net throughput offered by a Massive MIMO system is close to the Shannon capacity, in many scenarios, by only employing simple linear processing techniques, such as maximum ratio (MR) or zero forcing (ZF) processing. Since the net throughput can be computed in a closed-form expression that only depends on the channel statistics, the optimization solutions are applicable for a long period of time \cite{van2020power}. The colocated Massive MIMO architecture has the advantage of low backhaul requirements since the base station antennas are installed in a compact array. Conventional cellular networks, however, are impaired by intercell interference. In particular, the users at the cell boundaries are impaired by high intracell interference and path loss, and hence, they may experience insufficient performance. More advanced signal processing methods are necessary to overcome the inherent intercell interference that characterizes conventional cellular network deployments.
Cell-Free Massive MIMO has recently been introduced to reduce the intercell interference of colocated Massive MIMO architectures. Cell-Free Massive MIMO is a network deployment where a large number of access points (APs) are located in a given coverage area to serve a small number of users \cite{ngo2017cell,9136914}. All APs collaborate with each other via a backhaul network and serve all the users in the absence of cell boundaries. The system performance is enhanced in Cell-Free Massive MIMO systems because they inherit the benefits of the distributed MIMO and network MIMO architectures, but the users are also close to the APs. When each AP is equipped with a single antenna, MR processing results in a good net throughput for every user, while ensuring a low computational complexity and offering a distributed implementation that is convenient for scalability purposes \cite{9064545}. However, Cell-Free Massive MIMO cannot guarantee a good quality of service under harsh propagation conditions, such as in the presence of poor scattering environments or high attenuation due to large obstacles.
Reconfigurable intelligent surface (RIS) is an emerging technology that is capable of shaping the radio waves at the electromagnetic level without applying digital signal processing methods and without requiring power amplifiers \cite{wu2019intelligent,le2020robust, 9140329}. Each element of the RIS scatters (e.g., reflects) the incident signal without using radio frequency chains and power amplification \cite{9326394}. Integrating an RIS into wireless networks introduces digitally controllable links that scale up with the number of scattering elements of the RIS, whose estimation is, however, challenged by the lack of digital signal processing units at the RIS \cite{9198125,9200578,abrardo2020intelligent,8937491,wei2021channel}. For simplicity, the main attention has so far been concentrated on designing the phase shifts with perfect channel state information (CSI) \cite{9198125,perovic2021achievable} and the references therein. In \cite{wei2021channel}, the authors have recently discussed the fundamental issues of performing channel estimation in RIS-assisted wireless systems. The impact of channel estimation overhead and reporting on the spectral efficiency, energy efficiency, and their tradeoff has recently been investigated in \cite{9200578}. In \cite{9198125} and \cite{abrardo2020intelligent}, to reduce the impact of channel estimation overhead, the authors have investigated the design of RIS-assisted communications in the presence of statistical CSI. As far as the integration of Cell-Free Massive MIMO and RIS is concerned, recent works have formulated and solved optimization problems with different communication objectives under the assumption of perfect (and instantaneous) CSI \cite{zhou2020achievable, zhang2020capacity, bashar2020performance, 9286726, zhang2021beyond}. Recent results in the context of single-input single-output (SISO) and multi-user MIMO systems have, however, shown that designs for the phase shifts of the RIS elements that are based on statistical CSI may be of practical interest and provide good performance \cite{abrardo2020intelligent,9195523,hou2020reconfigurable,van2021outage}.
In the depicted context, no prior work has analyzed the performance of RIS-assisted Cell-Free Massive MIMO systems in the presence of spatially-correlated channels. In this work, motivated by these considerations, we introduce an analytical framework for analyzing and optimizing the uplink and downlink transmission of RIS-assisted Cell-Free Massive MIMO systems under spatially correlated channels
and in the presence of direct links subject to the presence of blockages. In particular, the main contributions made by this paper can be summarized as follows:
\begin{itemize}
\item We consider an RIS-assisted Cell-Free Massive MIMO under spatially correlated channels. All APs estimate the instantaneous channels in the uplink pilot training phase. We exploit a channel estimation scheme that estimates the aggregated channels including both the direct and indirect links, instead of every individual channel coefficient as in previous works \cite{bashar2020performance,wei2021channel}. For generality, the pilot contamination is assumed to originate from an arbitrary pilot reuse pattern.
\item We analytically show that, even by using a low complexity MR technique, the non-coherent interference, small-scale fading effects, and additive noise are averaged out when the number of APs and RIS elements increases. The received signal includes, hence, only the desired signal and the coherent interference. Besides, the indirect links become dominant if the number of phase shifts increases.
\item We derive a closed-form expression of the net throughput for both the uplink and downlink data transmissions. The impact of the array gain, coherent joint transmission, channel estimation errors, pilot contamination, spatial correlation, and phase shifts of the RIS, which determine the system performance, are explicitly observable in the obtained analytical expressions.
\item With the aid of numerical simulations, we verify the effectiveness of the proposed channel estimation scheme and the accuracy of the closed-form expressions of the net throughput. The obtained numerical results show that the use of RISs enhance the net throughput per user significantly, especially when the direct links are blocked with high probability.
\end{itemize}
The rest of this paper is organized as follows: Section~\ref{Sec:SysModel} presents the system model, the channel model, and the channel estimation protocol. The uplink data transmission protocol and the asymptotic analysis by assuming a very large number of APs and phase shifts of the RIS are discussed in Section~\ref{Sec:UL}. A similar analysis for the downlink data transmission is reported in Section~\ref{Sec:Downlink}. Finally, Section~\ref{Sec:NumRes} illustrates several numerical results, while the main conclusions are drawn in Section~\ref{Sec:Conclusion}.
\textit{Notation}: Upper and lower bold letters are used to denote matrices and vectors, respectively. The identity matrix of size $N \times N$ is denoted by $\mathbf{I}_N$. The imaginary unit of a complex number is denoted by $j$ with $\sqrt{j} = -1$. The superscripts $(\cdot)^{\ast},$ $(\cdot)^T,$ and $(\cdot)^H$ denote the complex conjugate, transpose, and Hermitian transpose, respectively. $\mathbb{E}\{ \cdot\}$ and $\mathsf{Var} \{ \cdot \}$ denote the expectation and variance of a random variable. The circularly symmetric Gaussian distribution is denoted by $\mathcal{CN}(\cdot, \cdot)$ and $\mathrm{diag} (\mathbf{x})$ is the diagonal matrix whose main diagonal is given by $\mathbf{x}$. $\mathrm{tr}(\cdot)$ is the trace operator. The Euclidean norm of vector $\mathbf{x}$ is $\| \mathbf{x}\|$, and $\| \mathbf{X} \|$ is the spectral norm of matrix $\mathbf{X}$. Finally, $\mathrm{mod}(\cdot,\cdot)$ is the modulus operation and $\lfloor \cdot \rfloor$ denotes the truncated argument.
\begin{figure}[t]
\centering
\includegraphics[trim=3.0cm 2.4cm 5.7cm 6.8cm, clip=true, width=3.2in]{FigSysModelIRSCellFreeV1} \vspace*{-0.2cm}
\caption{An RIS-assisted Cell-Free Massive MIMO system where $M$ APs collaborate with each other to serve $K$ distant users.}
\label{FigSysModel}
\vspace*{-0.5cm}
\end{figure}
\vspace*{-0.3cm}
\section{System Model, Channel Estimation, and RIS Phase Shift Control} \label{Sec:SysModel}
\vspace*{-0.15cm}
We consider an RIS-assisted Cell-Free Massive MIMO system, where $M$ APs connected to a central processing unit (CPU) serve $K$ users on the same time and frequency resource, as schematically illustrated in Fig.~\ref{FigSysModel}. All APs and users are equipped with a single antenna and they are randomly located in the coverage area. Since the considered users are far away from the APs, the communication is assisted by an RIS that comprises $N$ scattering elements that can modify the phases of the incident signals.\footnote{In general, there are also users that are close to the APs. However, we aim at improving the
data rates of the users who are far away from the APs, and, hence, have very weak
direct channels to the APs. The users near the APs are not considered.} The matrix of phase shifts of the RIS is denoted by $\pmb{\Phi} = \mathrm{diag} \left( [ e^{j\theta_1}, \ldots, e^{j\theta_N}]^T \right)$, where $\theta_n \in [-\pi, \pi]$ is the phase shift applied by the $n$-th element of the RIS. The phase shifts are adjusted by a controller which exchanges information with the APs via a backhaul link (see Fig.~\ref{FigSysModel}). As a canonical form of Cell-Free Massive MIMO systems, we assume that the system operates in time-division duplexing (TDD) mode. Thus, we assume that channel reciprocity holds in the consisted system model.
\vspace*{-0.5cm}
\subsection{Channel Model}
\vspace*{-0.15cm}
We assume a quasi-static block fading model where the channels are static and frequency flat in each coherence interval comprising $\tau_c$ symbols. We assume that the APs have knowledge of only the channel statistics instead of the instantaneous channel realizations. Also, $\tau_p$ symbols ($\tau_p < \tau_c$) in each coherence interval are dedicated to the channel estimation and the remaining $(\tau_c - \tau_p)$ symbols are utilized for the uplink and downlink data transmissions.
The following notation is used: $g_{mk}$ is the channel between the user~$k$ and the AP~$m$, which is the direct link \cite{wu2019intelligent}; $\mathbf{h}_m \in \mathbb{C}^{N}$ is the channel between the AP~$m$ and the RIS; and $\mathbf{z}_{mk} \in \mathbb{C}^{N}$ is the channel between the RIS and the user~$k$. Each pair of the cascaded channels $\mathbf{h}_m$ and $\mathbf{z}_{mk}$ results in an indirect link (virtual line-of-sight link), which enhances the communication reliability between the AP~$m$ and the user~$k$ \cite{di2019smart}. The majority of existing works assume that the wireless channels undergo uncorrelated Rayleigh fading. In this paper, we consider a more realistic channel model by taking into account the spatial correlation among the scattering elements of the RIS, which is due to their sub-wavelength size, sub-wavelength inter-distance, and geometric layout.
In an isotropic propagation environment, in particular, $g_{mk}$, $\mathbf{h}_{m}$, and $\mathbf{z}_{mk}$ can be modeled as follows
\begin{equation} \label{eq:Channels}
g_{mk} \sim \mathcal{CN}(0, \beta_{mk}), \mathbf{h}_{m} \sim \mathcal{CN}(\mathbf{0}, \mathbf{R}_{m}), \mathbf{z}_{mk} \sim \mathcal{CN}(\mathbf{0}, \widetilde{\mathbf{R}}_{mk}),
\end{equation}
where $\beta_{mk}$ is the large-scale fading coefficien
; $\mathbf{R}_{m} \in \mathbb{C}^{N \times N}$ and $\widetilde{\mathbf{R}}_{mk} \in \mathbb{C}^{N \times N}$ are the covariance matrices that characterize the spatial correlation among the channels of the RIS elements. The covariance matrices in \eqref{eq:Channels} correspond to a general model, which can be further particularized for application to typical RIS designs and propagation environments. For example, a simple exponential model was used to describe the spatial correlation among the scattering elements of the RIS in \cite{alwazani2020intelligent}. Another recent model that is applicable to isotropic scattering with uniformly distributed multipath components in the half-space in front of the RIS was recently reported in \cite{bjornson2020rayleigh}, whose covariance matrices are
\begin{equation}\label{eq:CovarMa}
\mathbf{R}_m = \alpha_{m} d_Hd_V\mathbf{R} \mbox{ and } \widetilde{\mathbf{R}}_{mk} = \tilde{\alpha}_{mk} d_Hd_V\mathbf{R},
\end{equation}
where $\alpha_{m}, \tilde{\alpha}_{mk} \in \mathbb{C}$ are the large-scale channel coefficients. The matrices in \eqref{eq:CovarMa} assume that the size of each element of the RIS is $d_H \times d_V$, with $d_H$ being the horizontal width and $d_V$ being the vertical height of each RIS element. In particular, the $(m',n')-$th element of the spatial correlation matrix $\mathbf{R} \in \mathbb{C}^{N \times N }$ in \eqref{eq:CovarMa} is
\begin{equation}
[\mathbf{R}]_{m'n'}= \mathrm{sinc} (2 \|\mathbf{u}_{m'} - \mathbf{u}_{n'} \|/ \lambda),
\end{equation}
where $\lambda$ is the wavelength and $\mathrm{sinc}(x) = \sin(\pi x) / (\pi x)$ is the $\mathrm{sinc}$ function. The vector $\mathbf{u}_{x}, x \in \{ m',n'\}$ is given by
\begin{equation}
\mathbf{u}_{x} = [0, \mod(x-1,N_H)d_H, \lfloor (x-1)/N_H\rfloor d_V]^T,
\end{equation}
where $N_H$ and $N_V$ denote the total number of RIS elements in each row and column, respectively. The channel model in \eqref{eq:Channels} is significantly distinct from related works since the small-scale fading and the spatial correlation matrices are included in both links of the virtual line-of-sight link that comprises the RIS. In \cite{alwazani2020intelligent}, by contrast, the channels between the transmitters and the RIS are assumed to be deterministic, for analytical tractability.
\vspace*{-0.5cm}
\subsection{Uplink Pilot Training Phase}
\vspace*{-0.15cm}
The channels are independently estimated from the $\tau_p$ pilot sequences transmitted by the $K$ users. All the users share the same $\tau_p$ pilot sequences. In particular, $\pmb{\phi}_k \in \mathbb{C}^{\tau_p}$ with $\| \pmb{\phi}_k \|^2 = 1$ is defined as the pilot sequence allocated to the user~$k$. We denote by $\mathcal{P}_k$ the set of indices of the users (including the user $k$) that share the same pilot sequence as the user $k$. The pilot sequences are assumed to be mutually orthogonal such that the pilot reuse pattern is
\begin{equation}
\pmb{\phi}_{k'}^H \pmb{\phi}_k = \begin{cases}
1, & \mbox{if } k' \in \mathcal{P}_k,\\
0, & \mbox{if } k' \notin \mathcal{P}_k.
\end{cases}
\end{equation}
During the pilot training phase, all the $K$ users transmit the pilot sequences to the $M$ APs simultaneously. In particular, the user~$k$ transmits the pilot sequence $\sqrt{\tau_p} \pmb{\phi}_k$. The received training signal at the AP~$m$, $\mathbf{y}_{pm} \in \mathbb{C}^{\tau_p}$, can be written as
\begin{equation} \label{eq:ReceivedPilot}
\mathbf{y}_{pm} = \sum_{k=1}^K \sqrt{p \tau_p} g_{mk} \pmb{\phi}_k + \sum_{k=1}^K \sqrt{p \tau_p} \mathbf{h}_{m}^H \pmb{\Phi} \mathbf{z}_{mk} \pmb{\phi}_k + \mathbf{w}_{pm},
\end{equation}
where $p$ is the normalized signal-to-noise ratio (SNR) of each pilot symbol, and $\mathbf{w}_{pm} \in \mathbb{C}^{\tau_p}$ is the additive noise at the AP~$m$, which is distributed as $\mathbf{w}_{pm} \sim \mathcal{CN} (\mathbf{0}, \mathbf{I}_{\tau_p})$. In order for the AP~$m$ to estimate the desired channels from the user~$k$, the received training signal in \eqref{eq:ReceivedPilot} is projected on $\pmb{\phi}_k^H$ as
\begin{equation} \label{eq:ReceivedPilotv1}
y_{pmk} = \pmb{\phi}_k^H \mathbf{y}_{pm} = \sqrt{p \tau_p} \left(g_{mk} + \mathbf{h}_{m}^H \pmb{\Phi} \mathbf{z}_{mk} \right) + \sum_{k' \in \mathcal{P}_k \setminus \{k\} } \sqrt{p\tau_p} \left(g_{mk'} + \mathbf{h}_{m}^H \pmb{\Phi} \mathbf{z}_{mk'} \right) + w_{pmk},
\end{equation}
where $w_{pmk} = \pmb{\phi}_k^H \mathbf{w}_{pm} \sim \mathcal{CN}(0, 1)$. We emphasize that the co-existence of direct and indirect channels due to the presence of the RIS results in a complicated channel estimation process. In particular, the cascaded channel in \eqref{eq:ReceivedPilotv1} results in a nontrivial procedure to apply the minimum mean-square error (MMSE) estimation method, as reported in previous works, for processing the projected signals \cite{ngo2017cell,9136914}. Based on specific signal structure in \eqref{eq:ReceivedPilotv1}, we denote the channel between the AP~$m$ and the user~$k$ through the RIS as
\begin{equation} \label{eq:umk}
u_{mk} = g_{mk} + \mathbf{h}_{m}^H \pmb{\Phi} \mathbf{z}_{mk},
\end{equation}
which is referred to as the \textit{aggregated channel} that comprises the direct and indirect link between the user~$k$ and the AP~$m$. By capitalizing on the definition of the aggregated channel in \eqref{eq:umk}, the required channels can be estimated in an effective manner even in the presence of the RIS. In particular, the aggregated channel in \eqref{eq:umk} is given by the product of weighted complex Gaussian and spatially correlated random variables, as given in \eqref{eq:Channels}. Despite the complex analytical form, the following lemma gives information on the statistics of the aggregated channel $u_{mk}, \forall m,k$.
\begin{lemma} \label{lemma:ChannelProperty}
The second and fourth moments of the aggregated channel $u_{mk}$ can be formulated as follows
\begin{align}
\mathbb{E} \{ |u_{mk}|^2 \} &= \beta_{mk} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big), \label{eq:2Order} \\
\mathbb{E} \{ |u_{mk}|^4 \} &= 2 \left(\beta_{\mathrm{mk}} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) \right)^2 + 2 \mathrm{tr} \left( \big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)^2 \right).\label{eq:4Order}
\end{align}
\end{lemma}
\begin{proof}
See Appendix~\ref{appendix:ChannelProperty}.
\end{proof}
The moments in Lemma~\ref{lemma:ChannelProperty} are employed next for channel estimation and for analyzing the net throughput. We note, in addition, that the odd moments of $u_{mk}$, e.g., the first and third moments, are equal to zero. Conditioned on the phase shifts, we employ the linear MMSE method for estimating $u_{mk}$ at the AP. In spite of the complex structure of the RIS-assisted cascaded channel, Lemma~\ref{lemma:ChannelEst} provides analytical expressions of the estimated channels.
\begin{lemma} \label{lemma:ChannelEst}
By assuming that the AP~$m$ employs the linear MMSE estimation method based on the observation in \eqref{eq:ReceivedPilotv1}, the estimate of the aggregate channel ${u}_{mk}$ can be formulated as
\begin{equation} \label{eq:ChannelEst}
\hat{u}_{mk} = \big(\mathbb{E}\{ y_{pmk}^\ast u_{mk} \} y_{pmk} \big)/ \mathbb{E} \{ | y_{pmk} |^2 \} = c_{mk} y_{pmk},
\end{equation}
where $c_{mk} = \mathbb{E}\{ y_{pmk}^\ast u_{mk} \} / \mathbb{E} \{ | y_{pmk} |^2 \}$ has the following closed-form expression
\begin{equation}
c_{mk} = \frac{\sqrt{p\tau_p} \big( \beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)\big) }{p\tau_p \sum_{k' \in \mathcal{P}_k} \big( \beta_{mk'} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big)\big) + 1}.
\end{equation}
The estimated channel in \eqref{eq:ChannelEst} has zero mean and variance $\gamma_{mk}$ equal to
\begin{equation} \label{eq:gammamk}
\gamma_{mk} = \mathbb{E} \{ |\hat{u}_{mk}|^2 \} = \sqrt{p\tau_p} \big(\beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) \big)c_{mk}.
\end{equation}
Also, the channel estimation error $e_{mk} = u_{mk} - \hat{u}_{mk}$ and the channel estimate $\hat{u}_{mk}$ are uncorrelated. Furthermore, the channel estimation error has zero mean and variance equal to
\begin{equation} \label{eq:EstError}
\mathbb{E}\big\{ |e_{mk} |^2 \big\} = \beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) - \gamma_{mk}.
\end{equation}
\end{lemma}
\begin{proof}
It is similar to the proof in \cite{Kay1993a}, and is obtained by applying similar analytical steps to the received signal in \eqref{eq:ReceivedPilotv1} and by taking into account the structure of the RIS-assisted cascaded channel and the spatial correlation matrices in \eqref{eq:Channels}.
\end{proof}
Lemma~\ref{lemma:ChannelEst} shows that, by assuming $\pmb{\Phi}$ fixed, the aggregated channel in \eqref{eq:umk} can be estimated without increasing the pilot training overhead, as compared to a conventional Cell-Free Massive MIMO system. The obtained channel estimate in \eqref{eq:ChannelEst} unveils the relation $\hat{u}_{mk'} = \frac{c_{mk'}}{c_{mk}}\hat{u}_{mk}
$ if the user~$k'$ uses the same pilot sequence as the user~$k$. This implies that, because of pilot contamination, it may be difficult to distinguish the signals of these two users. In that regard it is worth noting that, to get rid of pilot contamination, one can assign mutually orthogonal pilot signals to all the users in the network (if the coherence time is long enough so that $\tau_p \geq K$). Under mutually orthogonal pilot sequences, $c_{mk}$ and $\gamma_{mk}$ simplify to $c_{mk}^o$ and $\gamma_{mk}^o$, respectively, as follows
\begin{equation}
c_{mk}^o = \frac{\sqrt{p\tau_p} \big(\beta_{mk} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk}\big) \big)}{p\tau_p \big( \beta_{mk} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk}\big) \big)+1}, \, \gamma_{mk}^o =\sqrt{p\tau_p} \big( \beta_{mk} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) \big) c_{mk}^o.
\end{equation}
This implies that, in the absence of pilot contamination, we have $\gamma_{mk}^o \rightarrow \beta_{mk} + \mathrm{tr}\big(\pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)$ as $\tau_p \rightarrow \infty$, i.e., the variance of the channel estimation error in \eqref{eq:gammamk} is equal to zero. The channel estimates given in Lemma~\ref{lemma:ChannelEst} can be applied to an arbitrary set of phase shifts and covariance matrices. In the following, the analytical expression of the channel estimates in Lemma~\ref{lemma:ChannelEst} are employed for signal detection in the uplink and for beamforming in the downlink. They are used also to optimize the phase shifts of the RIS in order to minimize the channel estimation error and to evaluate the corresponding ergodic net throughput.
\vspace{-0.3cm}
\subsection{RIS Phase Shift Control and Optimization}
\vspace{-0.15cm}
Channel estimation is a critical aspect in Cell-Free Massive MIMO. As discussed in previous text, in many scenarios, non-orthogonal pilots have to be used. This causes pilot contamination, which may reduce the system performance significantly. In this section, we design an RIS-assisted phase shift control scheme that is aimed to improve the quality of channel estimation. To this end, we introduce the normalized mean square error (NMSE) of the channel estimate of the user~$k$ at the AP~$m$ as follows
\begin{equation} \label{eq:NMSEmk}
\mathrm{NMSE}_{mk} = \frac{\mathbb{E}\{|e_{mk}|^2\}}{\mathbb{E}\{|u_{mk}|^2\}} =1 - \frac{p \tau_p \left(\beta_{mk} + \mathrm{tr}(\pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} ) \right)}{p \tau_p \sum_{k' \in \mathcal{P}_{k}}\left( \beta_{mk'} + \mathrm{tr}(\pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} ) \right) + 1},
\end{equation}
where the last equality is obtained from \eqref{eq:2Order} and \eqref{eq:EstError}. We propose to optimize the phase shift matrix $\pmb{\Phi}$ of the RIS so as to minimize the total NMSE obtained from all the users and all the APs as follows
\begin{equation} \label{Prob:NMSEk}
\begin{aligned}
& \underset{\{ \theta_n \} }{\mathrm{minimize}}
&& \sum_{m=1}^M \sum_{k=1}^K \mathrm{NMSE}_{mk} \\
& \,\,\mathrm{subject \,to}
& & -\pi \leq \theta_n \leq \pi, \forall n.
\end{aligned}
\end{equation}
We emphasize that the optimal phase shifts solution to problem~\eqref{Prob:NMSEk} is obtained by exploiting the statistical CSI that include the large-scale fading coefficients and the covariance matrices. Problem~\eqref{Prob:NMSEk} is a fractional program, whose globally-optimal solution is not simple to be obtained for an RIS with a large number of independently tunable elements. Nonetheless, in the special network setup where the direct links from the APs to the users
are weak enough to be negligible with respect to the RIS-assisted links, the optimal solution to problem~\eqref{Prob:NMSEk} is available in a closed-form expression as summarized in Corollary~\ref{corollary:EqualPhase}.
\begin{corollary} \label{corollary:EqualPhase}
If the direct links are weak enough to be negligible and the RIS-assisted channels are spatially correlated as formulated in \eqref{eq:CovarMa}, the optimal maximizer of the optimization problem in \eqref{Prob:NMSEk} is $\theta_1 = \ldots = \theta_N$, i.e., the equal phase shift design is optimal.
\end{corollary}
\begin{proof}
See Appendix~\ref{appendix:CorMSEk}.
\end{proof}
Corollary~\ref{corollary:EqualPhase} provides a simple but effective option to design the phase shifts of the RIS while ensuring the optimal estimation of the aggregated channels according to the sum-NMSE minimization criterion, provided that the direct link are completely blocked and the spatial correlation model in \eqref{eq:CovarMa} holds true. Therefore, an efficient channel estimation protocol can be designed even in the presence of an RIS with a large number of scattering elements. The numerical results in Section~\ref{Sec:NumRes}
show that the phase shift design in Corollary~\ref{corollary:EqualPhase} offers good gains in terms of net throughput even if the direct links are not negligible.
\begin{remark}\label{RemarkProb}
The proposed optimization method of the phase shifts of the RIS is based on the minimization of the sum-NMSE, and it is, therefore, based on improving the channel estimation quality. This is a critical objective in Massive MIMO systems, since improving the accuracy of channel estimation results in a noticeable enhancement of the uplink and downlink net throughput \cite{mai2018pilot,van2018joint}. Another option would be to optimize the phase shifts of the RIS based on the maximization of the uplink or downlink ergodic net throughput. The solution of the corresponding optimization problem is, however, challenging and depends on whether the uplink or the downlink transmission phases are considered. Due to space limitations, therefore, we postpone this latter criterion for optimizing the phase shifts of the RIS to a future research work.
\end{remark}
\vspace*{-0.3cm}
\section{Uplink Data Transmission and Performance Analysis With MR Combining}\label{Sec:UL}
\vspace*{-0.15cm}
In this section, we first introduce a procedure to detect the uplink transmitted signals by capitalizing on the channel estimation method introduced in the previous section. Then, we derive an asymptotic closed-form expression of the ergodic net throughput.
\vspace*{-0.5cm}
\subsection{Uplink Data Transmission Phase}
\vspace*{-0.15cm}
In the uplink, all the $K$ users transmit their data to the $M$ APs simultaneously. Specifically, the user~$k$ transmits a modulated symbol $s_k$ with $\mathbb{E}\{|s_k|^2\} =1$. This symbol is weighted by a power control factor $\sqrt{\eta_k}$, $0 \leq \eta_k \leq 1$. Then, the received baseband signal, $y_{um} \in \mathbb{C},$ at the AP~$m$ is
\begin{equation} \label{eq:yum}
\begin{split}
y_{um} & = \sqrt{\rho_u} \sum_{k=1}^K \sqrt{\eta_k} \big(g_{mk} + \mathbf{h}_{m}^H \pmb{\Phi} \mathbf{z}_{mk} \big)s_k + w_{um} = \sqrt{\rho_u} \sum_{k=1}^K \sqrt{\eta_{k}} u_{mk} s_k + w_{um},
\end{split}
\end{equation}
where $\rho_u$ is the normalized uplink SNR of each data symbol and $w_{um}$ is the normalized additive noise with $w_{um} \sim \mathcal{CN}(0,1)$. For data detection, the MR combining method is used at the CPU, i.e., $\hat{u}_{mk}, \forall m,k,$ in \eqref{eq:ChannelEst} is employed to detect the data transmitted by the user~$k$. In mathematical terms, the corresponding decision statistic is
\begin{equation} \label{eq:ruk}
r_{uk} = \sum_{m=1}^M \hat{u}_{mk}^\ast y_{um}= \sqrt{\rho_u} \sum_{m=1}^M \sum_{k'=1}^K \sqrt{\eta_k} \hat{u}_{mk}^\ast u_{mk'} s_{k'} + \sum_{m=1}^M \hat{u}_{mk}^\ast w_{um}.
\end{equation}
Based on the observation $r_{uk}$, the uplink ergodic net throughput of the user~$k$ is analyzed in the next subsection.
\vspace*{-0.5cm}
\subsection{Asymptotic Analysis of the Uplink Received Signal} \label{subsec:Asymul}
\vspace*{-0.15cm}
In the considered system model, the number of APs, $M$, and the number of tunable elements of the RIS, $N$, can be large. Therefore, we analyze the performance of two case studies: $(i)$ $N$ is fixed and $M$ is large; and $(ii)$ both $N$ and $M$ are large but their ratio is fixed. The asymptotic analysis is conditioned upon a given setup of the CSI, which includes the large-scale fading coefficients, the covariance matrices, and the power utilized for pilot and data transmissions. To this end, the uplink weighted signal in \eqref{eq:ruk} is split into three terms based on the pilot reuse set $\mathcal{P}_k$, as follows
\begin{equation} \label{eq:rukv1}
r_{uk} = \underbrace{\sqrt{\rho_u} \sum_{k' \in \mathcal{P}_k } \sum_{m=1}^M \sqrt{\eta_{k'}} \hat{u}_{mk}^\ast u_{mk'} s_{k'}}_{\mathcal{T}_{k1}} + \underbrace{\sqrt{\rho_u} \sum_{k' \notin \mathcal{P}_k} \sum_{m=1}^M \sqrt{\eta_{k'}} \hat{u}_{mk}^\ast u_{mk'} s_{k'}}_{\mathcal{T}_{k2}} + \underbrace{\sum_{m=1}^M \hat{u}_{mk}^\ast w_{um}}_{\mathcal{T}_{k3}},
\end{equation}
where $\mathcal{T}_{k1}$ accounts for the signals received from all the users in $\mathcal{P}_k$, and $\mathcal{T}_{k2}$ accounts for the mutual interference from the users that are assigned orthogonal pilot sequences. The impact of the additive noise obtained after applying MR combining is given by $\mathcal{T}_{k3}$. From \eqref{eq:ReceivedPilotv1}, \eqref{eq:umk}, and \eqref{eq:ChannelEst}, we obtain the following identity
\begin{equation} \label{eq:Termv1}
\begin{split}
& \sum_{m=1}^M \sqrt{\eta_{k'}} \hat{u}_{mk}^\ast u_{mk'} = \sum_{m=1}^M \sqrt{\eta_{k'}} c_{mk} u_{mk'} \left( \sum_{k'' \in \mathcal{P}_k} \sqrt{p} \tau_p u_{mk''}^\ast + w_{pmk}^\ast \right) \\
& = \sum_{m=1}^M \sqrt{\eta_{k'} p \tau_p} c_{mk} |u_{mk'}|^2 + \sum_{k'' \in \mathcal{P}_k \setminus \{k' \} } \sum_{m=1}^M \sqrt{\eta_{k'}p\tau_p} c_{mk} u_{mk'} u_{mk''}^{\ast} + \sum_{m=1}^M \sqrt{\eta_{k'}} c_{mk} u_{mk'} w_{pmk}^\ast.
\end{split}
\end{equation}
\subsubsection{Case I} $N$ is fixed and $M$ is large, i.e., $M \rightarrow \infty$. In this case, we divide both sides of \eqref{eq:Termv1} by $M$ and exploits Tchebyshev's theorem \cite{cramer2004random}\footnote{Let $X_1, \ldots, X_n$ be independent random variables such that $\mathbb{E}\{ X_i \} = \bar{x}_i$ and $\mathsf{Var}\{ X_i\} \leq c < \infty$. Then, Tchebyshev's theorem states $\frac{1}{n}\sum_{n'=1}^n X_{n'} \xrightarrow[n \rightarrow \infty]{P} \frac{1}{n} \sum_{n'} \bar{x}_{n'}.$} and \eqref{eq:2Order} to obtain
\begin{equation} \label{eq:Tchev1}
\frac{1}{M} \sum_{m=1}^M \sqrt{\eta_{k'}} \hat{u}_{mk}^\ast u_{mk'} \xrightarrow[M \rightarrow \infty ]{P} \frac{1}{M} \sum_{m=1}^M \sqrt{\eta_{k'} p \tau_p} c_{mk} \big(\beta_{mk'} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big) \big),
\end{equation}
where $\xrightarrow{P}$ denotes the convergence in probability.\footnote{A sequence $\{ X_n \}$ of random variables converges in probability to the random variable $X$ if, for all $\epsilon > 0$, it holds that $\lim_{n \rightarrow \infty} \mathrm{Pr}(|X_n - X| > \epsilon ) = 0$, where $\mathrm{Pr}(\cdot)$ denotes the probability of an event.} Note that the second and third terms in \eqref{eq:Termv1} converge to zero. By inserting \eqref{eq:Tchev1} into the decision variable in \eqref{eq:rukv1}, we obtain the following deterministic value
\begin{equation} \label{eq:Asympt1}
\frac{1}{M}r_{uk} \xrightarrow[M\rightarrow \infty]{P} \frac{1}{M} \sum_{k' \in \mathcal{P}_k} \sum_{m=1}^M \sqrt{\eta_{k'} p \tau_p} c_{mk} \big(\beta_{mk'} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big) \big) s_{k'},
\end{equation}
because $\mathcal{T}_{k2}/M \rightarrow 0$ and $\mathcal{T}_{k3}/M \rightarrow 0 $ as $M \rightarrow \infty$. The result in \eqref{eq:Asympt1} unveils that, for a fixed $N$, the channels become asymptotically orthogonal. In particular, the small-scale fading, the non-coherent interference, and the additive noise vanish. The only residual impairment is the pilot contamination caused by the users that employ the same pilot sequence. This result is the evidence that, due to pilot contamination, the system performance cannot be improved by adding more APs if MR combining is used. The contributions of both the direct and RIS-assisted indirect channels appear explicitly in \eqref{eq:Asympt1} through the terms $\beta_{mk'}$ and $\mathrm{tr}(\pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'})$, respectively.
\subsubsection{Case II}
Both $N$ and $M$ are large, i.e., $N\rightarrow \infty$ and $M \rightarrow \infty$. To analyze this case study, we need some assumptions on the covariance matrices $\mathbf{R}_m$ and $\widetilde{\mathbf{R}}_{mk}$, as summarized as follows.
\begin{assumption} \label{Assumption1}
For $m= 1,\ldots,M$ and $k=1,\ldots,K,$ the covariance matrices $\mathbf{R}_m$ and $\widetilde{\mathbf{R}}_{mk}$ are assumed to fulfill the following properties
\begin{align}
&\underset{N}{\limsup} \, \| \mathbf{R}_m\|_2 < \infty, \underset{N}{\liminf} \, \frac{1}{N} \mathrm{tr} ( \mathbf{R}_m) > 0, \label{eq:Asymp1}\\
&\underset{N}{\limsup} \, \| \widetilde{\mathbf{R}}_{mk} \|_2 < \infty, \underset{N}{\liminf} \, \frac{1}{N} \mathrm{tr} ( \widetilde{\mathbf{R}}_{mk}) > 0. \label{eq:Asymp2v1}
\end{align}
\end{assumption}
The assumptions in \eqref{eq:Asymp1} and \eqref{eq:Asymp2v1} imply that the largest singular value and the sum of the eigenvalues (counted with their mutiplicity) of the $ N \times N $ covariance matrices that characterize the spatial correlation among the channels of the RIS elements are finite and positive. Dividing both sides of \eqref{eq:Termv1} by $MN$ and then applying Tchebyshev's theorem and \eqref{eq:2Order}, we obtain
\begin{equation}\label{eq:AsymMNUL}
\frac{1}{MN} \sum_{m=1}^M \sqrt{\eta_{k'}} \hat{u}_{mk}^\ast u_{mk'} \xrightarrow[\substack{M \rightarrow \infty\\ N \rightarrow \infty} ]{P} \frac{1}{MN} \sum_{m=1}^M \sqrt{\eta_{k'} p\tau_p} c_{mk} \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big).
\end{equation}
We first observe that $\pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'}$ is similar to $ \widetilde{\mathbf{R}}_{mk'}^{1/2} \pmb{\Phi} \mathbf{R}_{m} \pmb{\Phi}^H \widetilde{\mathbf{R}}_{mk'}^{1/2}$, which is a positive semi-definite matrix.\footnote{Two matrices $\mathbf{A}$ and $\mathbf{B}$ of size $N \times N$ are similar if there exists an invertible $N \times N$ matrix $\mathbf{U}$ such that $\mathbf{B} = \mathbf{U}^{-1} \mathbf{A} \mathbf{U}$.} Because similar matrices have the same eigenvalues, it follows that $\mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big) >0$. Based on Assumption~\ref{Assumption1}, we obtain the following inequalities
\begin{equation} \label{eq:traceConv}
\frac{1}{N}\mathrm{tr}\big(\pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big) \stackrel{(a)}{\leq} \frac{1}{N} \| \pmb{\Phi} \|_2 \mathrm{tr}\big( \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big) \stackrel{(b)}{=} \frac{1}{N}\mathrm{tr}\big( \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \mathbf{R}_m \big) \stackrel{(c)}{\leq} \frac{1}{N} \| \widetilde{\mathbf{R}}_{mk'} \|_2 \mathrm{tr}(\mathbf{R}_{m}),
\end{equation}
where $(a)$ is obtained from Lemma~\ref{lemma:trace} in Appendix~\ref{Appendix:UsefulLemmas}; $(b)$ follows because $\|\pmb{\Phi}\|_2 = 1$; and $(c)$ is obtained by applying again Lemma~\ref{lemma:trace}. Based on Assumption~\ref{Assumption1}, the last inequality in \eqref{eq:traceConv} is bounded by a positive constant. From \eqref{eq:AsymMNUL} and \eqref{eq:traceConv}, therefore, the decision variable in \eqref{eq:rukv1} can be formulated as
\begin{equation} \label{eq:Asymp2}
\frac{1}{MN}r_{uk} \xrightarrow[\substack{M\rightarrow \infty\\ N \rightarrow \infty}]{P} \frac{1}{MN} \sum_{k' \in \mathcal{P}_k} \sum_{m=1}^M \sqrt{\eta_{k'} p\tau_p} c_{mk} \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big) s_{k'}.
\end{equation}
The expression obtained in \eqref{eq:Asymp2} reveals that, as $M,N \rightarrow \infty$, the post-processed signal at the CPU consists of the desired signal of the intended user $k$ and the interference from the other users in $\mathcal{P}_k$. Compared with \eqref{eq:Asympt1}, we observe that \eqref{eq:Asymp2} is independent of the direct links and depends only on the RIS-assisted indirect links. This highlights the potentially promising contribution of the RIS, in the limiting regime $M,N \to \infty$, for enhancing the system performance.
\vspace*{-0.5cm}
\subsection{Uplink Ergodic Net Throughput with a Finite Number of APs and RIS Elements}
\vspace*{-0.15cm}
In this section, we focus our attention on the practical setup in which $M$ and $N$ are both finite. By utilizing the user-and-then forget channel capacity bounding method \cite{Marzetta2016a}, the uplink ergodic net throughput of the user $k$ can be written as follows
\begin{equation} \label{eq:ULRate}
R_{uk} = B \nu_u \left( 1 - \tau_u/\tau_p \right) \log_2 \left( 1 + \mathrm{SINR}_{uk} \right), \mbox{[Mbps]},
\end{equation}
where $B$ is the system bandwidth measured in MHz and $0\leq \nu_u \leq 1$ is the portion of each coherence interval that is dedicated to the uplink data transmission. The effective uplink signal-to-noise-plus-interference ratio (SINR), which is denoted by $\mathrm{SINR}_{uk}$, is defined as follows
\begin{equation} \label{eq:ULSINR}
\mathrm{SINR}_{uk} = \frac{|\mathsf{DS}_{uk}|^2}{\mathbb{E} \{|\mathsf{BU}_{uk}|^2 \} + \sum_{k'=1, k' \neq k}^K \mathbb{E} \{|\mathsf{UI}_{uk'k}|^2 \} + \mathbb{E}\{ |\mathsf{NO}_{uk}|^2 \}},
\end{equation}
where the following definitions hold
\begin{equation}
\begin{split}
&\mathsf{DS}_{uk} = \sqrt{\rho_u\eta_k} \mathbb{E} \left\{ \sum_{m=1}^M \hat{u}_{mk}^\ast u_{mk} \right\}, \mathsf{BU}_{uk} = \sqrt{\rho_u\eta_k} \left( \sum_{m=1}^M \hat{u}_{mk}^\ast u_{mk} - \mathbb{E} \left\{ \sum_{m=1}^M \hat{u}_{mk}^\ast u_{mk} \right\} \right), \\
&\mathsf{UI}_{uk'k} = \sqrt{\rho_u \eta_{k'}} \sum_{m=1}^M \hat{u}_{mk}^\ast u_{mk'}, \mathsf{NO}_{uk} = \sum_{m=1}^M \hat{u}_{mk}^\ast w_{um}.
\end{split}
\end{equation}
In particular, $\mathsf{DS}_{uk}$ denotes the (average) strength of desired signal, $\mathsf{BU}_{uk}$ denotes the beamforming uncertainty, $\mathsf{UI}_{uk'k}$ denotes the interference caused by the user~$k'$ to the user~$k$, and $\mathsf{NO}_{uk}$ denotes the additive noise. We emphasize that the net throughput in \eqref{eq:ULRate} is achievable since it is a lower bound of the channel capacity. A closed-form expression for \eqref{eq:ULRate} is given in Theorem~\ref{theorem:ULMR}.
\begin{theorem} \label{theorem:ULMR}
If the CPU utilizes the MR combining method, a lower-bound closed-form expression for the uplink net throughput of the user~$k$ is given by \eqref{eq:ULRate}, where the SINR is
\begin{equation} \label{eq:ClosedFormSINR}
\mathrm{SINR}_{uk} = \frac{\rho_u \eta_{k} \left( \sum_{m=1}^M \gamma_{mk} \right)^2}{\mathsf{CI}_{uk} + \mathsf{NI}_{uk} + \mathsf{NO}_{uk}},
\end{equation}
where $\mathsf{CI}_{uk}$ is the coherent interference, $\mathsf{NI}_{uk}$ is the non-coherent interference, and $\mathsf{NO}_{uk}$ is the noise, which are formulated as follows
\begin{equation}
\begin{split}
& \mathsf{CI}_{uk} = 2 \rho_u p \tau_p \sum\limits_{k'\in \mathcal{P}_k \setminus \{ k\}} \sum\limits_{m=1}^M\eta_{k'}c_{mk}^2\xi_{mk'}+p \tau_p \rho_u \sum\limits_{k' \in \mathcal{P}_k \setminus \{k\} }\eta_{k'} \left(\sum\limits_{m=1}^M c_{mk} \delta_{mk'} \right)^2,\\
&\mathsf{NI}_{uk} = 2 \rho_u p \tau_p \sum\limits_{m=1}^M\eta_{k}c_{mk}^2\xi_{mk} + \rho_u \sum\limits_{k'=1}^{K} \sum\limits_{m=1}^M \eta_{k'} \gamma_{mk} \delta_{mk'},\mathsf{NO}_{uk} = \sum_{m=1}^M \gamma_{mk},
\end{split}
\end{equation}
with $\xi_{mk'} = \mathrm{tr}\big( ( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'})^2 \big)$, $\delta_{mk'} =\beta_{mk'} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big)$, and $\gamma_{mk}$ given in \eqref{eq:gammamk}.
\end{theorem}
\begin{proof}
See Appendix~\ref{appendix:ULMR}.
\end{proof}
By direct inspection of the SINR in \eqref{eq:ClosedFormSINR}, the numerator increases with the square of the sum of the variances of the channel estimates, $\gamma_{mk},\forall m$ thanks to the joint coherent transmission, i.e., all the $M$ APs transmit the same data to the user. On the other hand, the first term in the denominator represents the power of the coherent interference. Due to the limited and finite number of orthogonal pilot sequences being used, the second term in the denominator represents the impact of pilot contamination. The last term is the additive noise. If the coherence time is sufficiently large that every user can utilize its own orthogonal pilot sequence, the uplink net throughput of the user~$k$ can still be obtained from \eqref{eq:ULRate}, but the effective SINR simplifies to
\begin{equation} \label{eq:SINRULv1}
\mathrm{SINR}_{uk} = \frac{\rho_u \eta_k \left( \sum_{m=1}^M \gamma_{mk}^o \right)^2}{2 \rho_u p \tau_p \sum_{m=1}^M\eta_{k'}(c_{mk}^o)^2\xi_{mk'} + \rho_u \sum_{k'=1}^{K} \sum_{m=1}^M \eta_{k'} \gamma_{mk}^o + \sum_{m=1}^M \gamma_{mk}^o}.
\end{equation}
The SINR in \eqref{eq:ClosedFormSINR} is a multivariate function of the matrix of phase shifts of the RIS and of the channel statistics, i.e., the channel covariance matrices. Table~\ref{Table:CompareUL} gives a comparison of the uplink SINR expression of the user $k$ with and without the presence of the RIS. By direct inspection of $|\mathsf{DS}_{uk}|^2$, we evince that the strength of the desired signal gets better thanks to the assistance of the RIS. However, the coherent and non-coherent interference become more severe as well, due to the need of estimating both the direct and indirect links in the presence of the RIS. By assigning the orthogonal pilot sequences to all the $K$ users, the coherent interference represented by $\mathsf{CI}_{uk}$ can be completely suppressed. In Section~\ref{Sec:NumRes}, the performance of Cell-Free Massive MIMO and RIS-assisted Cell-Free Massive MIMO are compared with the aid of numerical simulations.
\begin{table}[t]
\caption{Comparison of the uplink SINR between Cell-Free Massive MIMO and RIS-Assisted Cell-Free Massive MIMO} \label{Table:CompareUL}
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Uplink SINR} & Cell-Free Massive MIMO & RIS-Assisted Cell-Free Massive MIMO \\
\hline
\multirow{9}{*}{\eqref{eq:ClosedFormSINR}} & $c_{mk}$ & $\frac{\sqrt{p\tau_p} \beta_{mk}}{p\tau_p \sum_{k' \in \mathcal{P}_k} \beta_{mk'} + 1}$ & $\frac{\sqrt{p\tau_p} \big( \beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)\big) }{p\tau_p \sum_{k' \in \mathcal{P}_k} \big( \beta_{mk'} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk'} \big)\big) + 1}$ \\
\cline{2-4}
& $\gamma_{mk}$ & $ \sqrt{p\tau_p} \beta_{mk}c_{mk}$ & $ \sqrt{p\tau_p} \big(\beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) \big)c_{mk}$ \\
\cline{2-4}
& $\delta_{mk}$ & $\beta_{mk}$ & $\beta_{mk} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)$ \\
\cline{2-4}
&$\xi_{mk}$ & - & $\mathrm{tr}\big( ( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk})^2 \big)$ \\
\cline{2-4}
& $|\mathsf{DS}_{uk}|^2$ & $\rho_u \eta_{k} \left( \sum_{m=1}^M \gamma_{mk} \right)^2$ & $\rho_u \eta_{k} \left( \sum_{m=1}^M \gamma_{mk} \right)^2$\\
\cline{2-4}
& $\mathsf{CI}_{uk}$ & $p \tau_p \rho_u \sum\limits_{k' \in \mathcal{P}_k \setminus \{k\} }\eta_{k'} \left(\sum\limits_{m=1}^M c_{mk} \delta_{mk'} \right)^2$ & $2 \rho_u p \tau_p \sum\limits_{k'\in \mathcal{P}_k \setminus \{ k \} } \sum\limits_{m=1}^M\eta_{k'}c_{mk}^2\xi_{mk'}+p \tau_p \rho_u \sum\limits_{k' \in \mathcal{P}_k \setminus \{k\} }\eta_{k'} \left(\sum\limits_{m=1}^M c_{mk} \delta_{mk'} \right)^2$ \\
\cline{2-4}
& $\mathsf{NI}_{uk}$ & $\rho_u \sum\limits_{k'=1}^{K} \sum\limits_{m=1}^M \eta_{k'} \gamma_{mk} \delta_{mk'}$ & $2 \rho_u p \tau_p \sum\limits_{m=1}^M\eta_{k}c_{mk}^2\xi_{mk} + \rho_u \sum\limits_{k'=1}^{K} \sum\limits_{m=1}^M \eta_{k'} \gamma_{mk} \delta_{mk'}$ \\
\cline{2-4}
& $\mathsf{NO}_{uk}$ & $ \sum_{m=1}^M \gamma_{mk}$ & $ \sum_{m=1}^M \gamma_{mk}$\\
\cline{1-4}
\multirow{6}{*}{\eqref{eq:SINRULv1}} & $c_{mk}^o$ & $\frac{\sqrt{p\tau_p} \beta_{mk}}{p\tau_p \beta_{mk} + 1}$ & $\frac{\sqrt{p\tau_p} \big( \beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)\big) }{p\tau_p \big( \beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)\big) + 1}$ \\
\cline{2-4}
& $\gamma_{mk}^o$ & $ \sqrt{p\tau_p} \beta_{mk}c_{mk}^o$ & $ \sqrt{p\tau_p} \big(\beta_{mk} + \mathrm{tr} \big( \pmb{\Phi}^H \mathbf{R}_{m} \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) \big)c_{mk}^o$ \\
\cline{2-4}
&
$|\mathsf{DS}_{uk}|^2$ & $\rho_u \eta_{k} \left( \sum_{m=1}^M \gamma_{mk}^o \right)^2$ & $\rho_u \eta_{k} \left( \sum_{m=1}^M \gamma_{mk}^o \right)^2$ \\
\cline{2-4}
& $\mathsf{CI}_{uk}$ & - & - \\
\cline{2-4}
& $\mathsf{NI}_{uk}$ & $\rho_u \sum\limits_{k'=1}^{K} \sum\limits_{m=1}^M \eta_{k'} \gamma_{mk}^{o} \delta_{mk'}$ & $2 \rho_u p \tau_p \sum\limits_{m=1}^M\eta_{k}(c_{mk}^o)^2\xi_{mk} + \rho_u \sum\limits_{k'=1}^{K} \sum\limits_{m=1}^M \eta_{k'} \gamma_{mk}^o \delta_{mk'}$ \\
\cline{2-4}
& $\mathsf{NO}_{uk}$ & $ \sum_{m=1}^M \gamma_{mk}^o$ & $ \sum_{m=1}^M \gamma_{mk}^o$\\
\cline{1-4}
\end{tabular}}
\vspace*{-0.5cm}
\end{table}
\vspace*{-0.3cm}
\section{Downlink Data Transmission and Performance Analysis With MR Precoding} \label{Sec:Downlink}
\vspace*{-0.15cm}
In this section, we consider the downlink data transmission phase and analyze the received signal at the users when the number of APs is large. A closed-form expression of the downlink ergodic net throughput that is attainable with MR precoding and for an arbitrary phase shift matrix of the RIS elements is provided.
\vspace*{-0.5cm}
\subsection{Downlink Data Transmission Phase}
\vspace*{-0.15cm}
By exploiting channel reciprocity, the AP~$m$ treats the channel estimates obtained in the uplink as the true channels in order to construct the beamforming coefficients. Accordingly, the downlink signal transmitted from the AP~$m$ is
\begin{equation} \label{eq:TransSig}
x_m = \sqrt{\rho_{d}}\sum_{k=1}^K \sqrt{\eta_{mk}} \hat{u}_{mk}^\ast q_k,
\end{equation}
where $\rho_d$ is the normalized SNR in the downlink; $q_k$ is the complex data symbol that is to be sent (cooperatively) by the $M$ APs to the user~$k$, with $\mathbb{E}\{ |q_k|^2 \} = 1$; and $\eta_{mk}$ is the power control coefficient of the AP~$m$, which satisfies the limited power budget constraint as follows
\begin{equation} \label{eq:PowerConst}
\mathbb{E} \{|x_m|^2\} \leq \rho_d \Rightarrow
\sum_{k=1}^K \eta_{mk} \gamma_{mk} \leq 1.
\end{equation}
The cooperation among the $M$ APs for jointly transmitting the same data symbol to a particular user creates the major distinction between the downlink and uplink data transmission phases. Based on \eqref{eq:TransSig}, the received signal at the user~$k$ is the superposition of the signals transmitted by the $M$ APs as
\begin{equation} \label{eq:rdk}
\begin{split}
r_{dk} =& \sum_{m=1}^M u_{mk} x_m + w_{dk} = \sqrt{\rho_{d}} \sum_{m=1}^M \sum_{k'=1}^K \sqrt{\eta_{mk'}} u_{mk} \hat{u}_{mk'}^\ast q_{k'} + w_{dk}.
\end{split}
\end{equation}
where $w_{dk}$ is the additive noise at the user~$k$ with $w_{dk} \sim \mathcal{CN}(0,1)$. The user~$k$ decodes the desired data symbol based on the observation in \eqref{eq:rdk}.
\vspace*{-0.5cm}
\subsection{Asymptotic Analysis of the Downlink Received Signal}
\vspace*{-0.15cm}
In contrast with the uplink data processing where the CPU needs only the channel estimate $\hat{u}_{mk}$ for detecting the data of the user~$k$, as displayed in \eqref{eq:rukv1}, the received signal in \eqref{eq:rdk} depends on the channel estimates of the $K$ users in the network, since the channel estimates from the $K$ users are used for MR precoding. Therefore, the analysis of the uplink and downlink data transmission phases are different. First, we split \eqref{eq:rukv1} into three terms, by virtue of the pilot reuse pattern $\mathcal{P}_k$, as follows
\begin{equation} \label{eq:rdkv1}
r_{dk} = \sqrt{\rho_{d}} \sum_{k'\in \mathcal{P}_k} \sum_{m=1}^M \sqrt{\eta_{mk'}} u_{mk} \hat{u}_{mk'}^\ast q_{k'} + \sqrt{\rho_{d}} \sum_{k'\notin \mathcal{P}_k } \sum_{m=1}^M \sqrt{\eta_{mk'}} u_{mk} \hat{u}_{mk'}^\ast q_{k'} +w_{dk}.
\end{equation}
Then, we investigate the two asymptotic regimes for $M \to \infty$ and $M,N \to \infty$ while keeping the $M/N$ ratio fixed. In particular, the first term in \eqref{eq:rdkv1} can be rewritten as
\begin{equation} \label{eq:rdkAsymtotic}
\begin{split}
& \sum_{m=1}^M \sqrt{\eta_{mk'}} u_{mk} \hat{u}_{mk'}^\ast \stackrel{(a)}{=} \sum_{m=1}^M \sqrt{\eta_{mk'}} c_{mk'} u_{mk} \left( \sum_{k'' \in \mathcal{P}_k} \sqrt{p} \tau_p u_{mk''}^\ast + w_{pmk'}^\ast \right) \\
&\stackrel{(b)}{=} \sum_{m=1}^M \sqrt{\eta_{mk'}p} \tau_p c_{mk'} |u_{mk}|^2 + \sum_{k'' \in \mathcal{P}_k \setminus \{k\} } \sum_{m=1}^M \sqrt{\eta_{mk'}p} \tau_p c_{mk'} u_{mk} u_{mk''}^\ast + \sum_{m=1}^M \sqrt{\eta_{mk'}} c_{mk'} u_{mk} w_{pmk'}^\ast,
\end{split}
\end{equation}
where $(a)$ is obtained by utilizing the channel estimates in \eqref{eq:ChannelEst} and $(b)$ is obtained by extracting the aggregated channel of the user~$k$ from the summation. By letting $M$ and/or $N$ be large, similar to the uplink transmission phase, we obtain the following asymptotic results
\begin{align}
&\frac{1}{M} \sum_{m=1}^M \sqrt{\eta_{mk'}} u_{mk} \hat{u}_{mk'}^\ast \xrightarrow[M \rightarrow \infty]{P} \frac{1}{M} \sum_{m=1}^M \sqrt{\eta_{mk'} p} \tau_p c_{mk'} \big( \beta_{mk} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) \big),\\
& \frac{1}{MN} \sum_{m=1}^M \sqrt{\eta_{mk'}} u_{mk} \hat{u}_{mk'}^\ast \xrightarrow[\substack{M \rightarrow \infty\\
N \rightarrow \infty}]{P} \frac{1}{MN} \sum_{m=1}^M \sqrt{\eta_{mk'} p} \tau_p c_{mk'} \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big),
\end{align}
which are bounded based on Assumption~\ref{Assumption1}. Consequently, the received signal at the user~$k$ converges to a deterministic equivalent as the number of APs is large, i.e., $M \rightarrow \infty$, and as the number of APs and RIS elements are large, i.e., $M,N \rightarrow \infty$, but their ratio is kept fixed. More precisely, the received signal converges (asymptotically) to
\begin{align}
&\frac{1}{M} r_{dk} \xrightarrow[M \rightarrow \infty]{P} \frac{1}{M} \sum_{k' \in \mathcal{P}_k} \sum_{m=1}^M \sqrt{\eta_{mk'} p} \tau_p c_{mk'} \big(\beta_{mk} + \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big) \big)s_{k'},\label{eq:DeEquiv1}\\
&\frac{1}{MN} r_{dk} \xrightarrow[M \rightarrow \infty]{P} \frac{1}{MN} \sum_{k' \in \mathcal{P}_k} \sum_{m=1}^M \sqrt{\eta_{mk'} p} \tau_p c_{mk'} \mathrm{tr}\big( \pmb{\Phi}^H \mathbf{R}_m \pmb{\Phi} \widetilde{\mathbf{R}}_{mk} \big)s_{k'},\label{eq:DeEquiv2}
\end{align}
which indicates an inherent coexistence of the users in $\mathcal{P}_k$. The deterministic equivalents in \eqref{eq:DeEquiv1} and \eqref{eq:DeEquiv2} unveil that the impact of the channel estimation accuracy and the channel statistics is different between the uplink and the downlink. In particular, the asymptotic received signal in the uplink only depends on the channel estimation quality of each individual user, which is expressed by the coefficient $c_{mk}$. The asymptotic received signal in the downlink depends, on the other hand, on the channel estimation quality of all the users that share the same orthogonal pilot sequences, i.e., $c_{m,k'}, \forall k' \in \mathcal{P}_{k}$.
\vspace*{-0.5cm}
\subsection{Downlink Ergodic Net Throughput with a Finite Number of APs and RIS Elements}
\vspace*{-0.15cm}
By utilizing the channel capacity bounding technique \cite{Marzetta2016a}, similar to the analysis of the uplink data transmission phase, the downlink ergodic net throughput of the user $k$ can be written as follows
\begin{equation} \label{eq:DLRate}
R_{dk} = B\nu_d \left( 1- \tau_p/\tau_c \right) \log_2 \left( 1 + \mathrm{SINR}_{dk} \right),\mbox{[Mbps]},
\end{equation}
where $0\leq \nu_d \leq 1$ is the portion of each coherence interval dedicated to the downlink data transmission, with $\nu_u + \nu_d = 1$, and the effective downlink SINR is defined as
\begin{equation} \label{eq:DLSINR}
\mathrm{SINR}_{dk} = \frac{|\mathsf{DS}_{dk}|^2}{\mathbb{E} \{|\mathsf{BU}_{dk}|^2 \} + \sum_{k'=1, k' \neq k}^K \mathbb{E} \{|\mathsf{UI}_{dk'k}|^2 \} + 1},
\end{equation}
where the following definitions hold
\begin{equation}
\begin{split}
\mathsf{DS}_{dk} &= \sqrt{\rho_{d}} \mathbb{E}\left\{ \sum_{m=1}^M \sqrt{\eta_{mk}} u_{mk}\hat{u}_{mk}^\ast \right\}, ~ \mathsf{UI}_{dk'k} = \sqrt{\rho_{d}} \sum_{m=1}^M \sqrt{\eta_{mk'}} u_{mk}\hat{u}_{mk'}^\ast,\\
\mathsf{BU}_{dk} & = \sqrt{\rho_{d}} \left( \sum_{m=1}^M \sqrt{\eta_{mk}} u_{mk}\hat{u}_{mk}^\ast - \mathbb{E}\left\{ \sum_{m=1}^M \sqrt{\eta_{mk}} u_{mk}\hat{u}_{mk}^\ast \right\} \right),
\end{split}
\end{equation}
In particular, $\mathsf{DS}_{dk}$ denotes the (average) strength of the desired signal received by the user~$k$, $\mathsf{BU}_{dk}$ denotes the beamforming uncertainty, $\mathsf{UI}_{dk'k}$ denotes the interference caused to the user~$k$ by the signal intended to the user~$k'$. The downlink ergodic net throughput in \eqref{eq:DLRate} is achievable since it is a lower bound of the channel capacity, similar to the uplink data transmission. In contrast to the uplink ergodic net throughput, which only depends on the combining coefficients of each individual user, the downlink net throughput of the user~$k$ depends on the precoding coefficients of all the $K$ users. A closed-form expression for \eqref{eq:DLRate} is given in Theorem~\ref{theorem:DLMR}.
\begin{theorem} \label{theorem:DLMR}
If the CPU utilizes the MR precoding method, a lower-bound closed-form expression for the downlink net throughput of the user~$k$ is given by \eqref{eq:DLRate}, where the SINR is
\begin{equation} \label{eq:DLSINRMRT}
\mathrm{SINR}_{dk} = \frac{\rho_{d} \big( \sum_{m=1}^M \sqrt{\eta_{mk}} \gamma_{mk} \big)^2}{\mathsf{CI}_{dk} + \mathsf{NI}_{dk} + 1},
\end{equation}
where $\mathsf{CI}_{dk}$ is the coherent interference and $\mathsf{NI}_{dk}$ is the non-coherent interference, which are defined as follows
\begin{align}
\mathsf{CI}_{dk} &= 2\rho_d p \tau_p \sum\limits_{k' \in \mathcal{P}_k \setminus \{k \}} \sum\limits_{m=1}^M \eta_{mk'} c_{mk'}^2 \xi_{mk} + \rho_d p \tau_p \sum\limits_{k' \in \mathcal{P}_k \setminus \{ k\}} \left(\sum\limits_{m=1}^M \sqrt{\eta_{mk'} } c_{mk'} \delta_{mk} \right)^2,\\
\mathsf{NI}_{dk} &= 2\rho_d p \tau_p \sum\limits_{m=1}^M \eta_{mk} c_{mk}^2 \xi_{mk} + \rho_d \sum\limits_{k' =1}^K \sum\limits_{m=1}^M \eta_{mk'} \gamma_{mk'} \delta_{mk}.
\end{align}
\end{theorem}
\begin{proof}
The main steps of the proof are similar to those of the proof of Theorem~\ref{theorem:ULMR}. However, there are also major differences that are due to the coherent joint data transmission among the APs. The details of the proof are available in Appendix~\ref{appendix:DLMR}.
\end{proof}
From Theorem~\ref{theorem:DLMR}, we observe that the effective downlink SINR has some similarities and differences as compared with its uplink counterpart in Theorem~\ref{theorem:ULMR}. Similar to the uplink, the numerator of \eqref{eq:DLSINRMRT} is a quadratic function that depends on the channel estimation quality and the coherent joint transmission processing. Differently from the uplink, the transmit power coefficients appear explicitly in the numerator of \eqref{eq:DLSINRMRT} as a result of the cooperation among the APs. The first term in the denominator of \eqref{eq:DLSINRMRT} unveils the impact of pilot contamination. In contrast to the uplink, we observe that it depends on all the transmit power coefficients. In addition, we observe that the impact of pilot contamination scales up with the number of APs and with the number of elements of the RIS. Finally, the second term accounts for the non-coherent interference from the other users in the network. When the $K$ users employ orthogonal pilot sequences, the downlink SINR expression of the user~$k$ simplifies to
\begin{equation} \label{eq:DLOrt}
\mathrm{SINR}_{dk} = \frac{\rho_{d} \big( \sum_{m=1}^M \sqrt{\eta_{mk}} \gamma_{mk} \big)^2}{2\rho_d p \tau_p \sum_{m=1}^M \eta_{mk} (c_{mk}^o)^2 \xi_{mk}+ \rho_d \sum_{k' =1}^K \sum_{m=1}^M \eta_{mk'} \gamma_{mk'}^o \delta_{mk} +1}.
\end{equation}
\begin{table}[t]
\caption{Comparison of the Downlink SINR between Cell-Free Massive MIMO and RIS-Assisted Cell-Free Massive MIMO (some parameters are defined in Table~\ref{Table:CompareUL})} \label{Table:CompareDL}
\centering
\resizebox{\textwidth}{!}{\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\multicolumn{2}{|c|}{Downlink SINR} & Cell-Free Massive MIMO & RIS-Assisted Cell-Free Massive MIMO \\
\hline
\multirow{5}{*}{\eqref{eq:DLSINRMRT}}
& $|\mathsf{DS}_{dk} |^2$ & $\rho_{d} \big( \sum_{m=1}^M \sqrt{\eta_{mk}} \gamma_{mk} \big)^2$ & $\rho_{d} \big( \sum_{m=1}^M \sqrt{\eta_{mk}} \gamma_{mk} \big)^2$ \\
\cline{2-4}
& $\mathsf{CI}_{dk}$ & $\rho_d p \tau_p \sum\limits_{k' \in \mathcal{P}_k \setminus \{ k\}} \left(\sum\limits_{m=1}^M \sqrt{\eta_{mk'} } c_{mk'} \delta_{mk} \right)^2$ & $2\rho_d p \tau_p \sum\limits_{k' \in \mathcal{P}_k \setminus \{ k\}} \sum\limits_{m=1}^M \eta_{mk'} c_{mk'}^2 \xi_{mk} + \rho_d p \tau_p \sum\limits_{k' \in \mathcal{P}_k \setminus \{ k\}} \left(\sum\limits_{m=1}^M \sqrt{\eta_{mk'} } c_{mk'} \delta_{mk} \right)^2$ \\
\cline{2-4}
& $\mathsf{NI}_{dk}$ & $\rho_d \sum\limits_{k' =1}^K \sum\limits_{m=1}^M \eta_{mk'} \gamma_{mk'} \delta_{mk}$ & $2\rho_d p \tau_p \sum\limits_{m=1}^M \eta_{mk} c_{mk}^2 \xi_{mk} + \rho_d \sum\limits_{k' =1}^K \sum\limits_{m=1}^M \eta_{mk'} \gamma_{mk'} \delta_{mk}$\\
\cline{2-4}
& $\mathsf{NO}_{dk}$ & 1 & 1 \\
\cline{1-4}
\multirow{5}{*}{\eqref{eq:DLOrt}}
& $|\mathsf{DS}_{dk} |^2$ & $\rho_{d} \big( \sum_{m=1}^M \sqrt{\eta_{mk}} \gamma_{mk}^o \big)^2$ & $\rho_{d} \big( \sum_{m=1}^M \sqrt{\eta_{mk}} \gamma_{mk}^o \big)^2$ \\
\cline{2-4}
& $\mathsf{CI}_{dk}$ & - & - \\
\cline{2-4}
& $\mathsf{NI}_{dk}$ & $\rho_d \sum_{k' =1}^K \sum_{m=1}^M \eta_{mk'} \gamma_{mk'}^o \delta_{mk}$ & $2\rho_d p \tau_p \sum_{m=1}^M \eta_{mk} (c_{mk}^o)^2 \xi_{mk} + \rho_d \sum_{k' =1}^K \sum_{m=1}^M \eta_{mk'} \gamma_{mk'}^o \delta_{mk}$ \\
\cline{2-4}
& $\mathsf{NO}_{dk}$ & $1$ & $1$\\
\cline{1-4}
\end{tabular}}
\vspace*{-0.5cm}
\end{table}
A comparison of the downlink SINR for Cell-Free Massive MIMO and RIS-assisted Cell-Free Massive MIMO systems is given in Table~\ref{Table:CompareDL}. By comparing Table~\ref{Table:CompareUL} and Table~\ref{Table:CompareDL}, the difference and similarities between the uplink and downlink transmission phases can be identified as well. With the aid of numerical results, in Section~\ref{Sec:NumRes}, we will illustrate the advantages of RIS-assisted Cell-Free Massive MIMO especially if the direct links are not sufficiently reliable (e.g., they are blocked) with high probability.
\begin{remark} \label{RemarkPhase}
We observe that the ergodic net throughput in the uplink (Theorem~\ref{theorem:ULMR}) and downlink (Theorem~\ref{theorem:DLMR}) data transmission phases depend only on the large-scale fading statistics and on the channel covariance matrices, while they are independent of the instantaneous CSI. This simplifies the deployment and optimization of RIS-assisted Cell Free Massive MIMO systems. As anticipated in Remark~\ref{RemarkProb}, in fact, the phase shifts of the RIS can be optimized based on the (long-term) analytical expressions of the ergodic net throughputs in Theorem~\ref{theorem:ULMR} and Theorem~\ref{theorem:DLMR}, which are independent of the instantaneous CSI. In this paper, we have opted for optimizing the phase shifts of the RIS in order to minimize the channel estimation error, which determines the performance of both the uplink and downlink transmission phases. The optimization of the phase shifts of the RISs based on the closed-form expressions in Theorem~\ref{theorem:ULMR} and Theorem~\ref{theorem:DLMR} is postponed to a future research work.
\end{remark}
\vspace*{-0.3cm}
\section{Numerical Results} \label{Sec:NumRes}
\vspace*{-0.15cm}
In this section, we report some numerical results in order to illustrate the performance of the RIS-assisted Cell-Free Massive MIMO system introduced in the previous sections. We consider a geographic area of size $1$~km$^2$, where the locations of the APs and users are given in terms of $(x,y)$ coordinates. The four vertices of the considered region are $[-0.5, -0.5]$ km, $[-0.5, 0.5]$ km, $[0.5, 0.5]$ km, $[0.5, -0.5]$ km. To minimize the border effects, the considered region is wrapped around at the edges. To simulate a harsh communication environment, the $M$ APs are uniformly distributed in the sub-region $x,y \in [-0.5, -0.25]$~km, while the $K$ users are uniformly distributed in the sub-region $x,y \in [0.25, 0.25]$~km. The RIS is located at the origin, i.e., $(x,y) = (0,0)$. The carrier frequency is $1.9$~GHz and the system bandwidth is $20$~MHz. Each coherence interval comprises $\tau_c = 200$ symbols, which may correspond to a coherence bandwidth $B_c = 200$~KHz and a coherence time $T_c = 1$~ms. We assume $\tau_p =5$ orthonormal pilot sequences that are shared by all the users. The large-scale fading coefficients in dB are generated according to the three-slope propagation model in \cite{ngo2017cell}, where the path loss exponent depends on the distance between the transmitter and the receiver. The shadow fading has a log-normal distribution with standard deviation equal to $8$~dB. The distance thresholds for the three slopes are $10$~m and $50$~m. The height of the APs, RIS, and users is $15$~m, $30$~m, and $1.65$~m, respectively. The direct links, $g_{mk}, \forall m,k$, are assumed to be unblocked with a given probability. More specifically, the large-scale fading coefficient $\beta_{mk}$ is formulated as follows
\begin{equation}
\beta_{mk} = \bar{\beta}_{mk}a_{mk},
\end{equation}
where $\bar{\beta}_{mk}$ accounts for the path loss due to the transmission distance and the shadow fading according to the three-slope propagation model in \cite{ngo2017cell}. The binary variables $a_{mk}$ accounts for the probability that the direct links are unblocked, and it is defined as
\begin{equation} \label{eq:amk}
a_{mk} = \begin{cases}
1, & \mbox{ with a probability } p,\\
0, & \mbox{ with a probability } 1-p,
\end{cases}
\end{equation}
where $p \in [0,1]$ is the probability that the direct link is not blocked. The noise variance is $-92$~dBm, which corresponds to a noise figure of $9$~dB. The covariance matrices are generated according to the spatial correlation model in \eqref{eq:CovarMa}. The power of the pilot sequences is $100$~mW and the power budget of each AP is $200$~mW. The time intervals of data transmission, in each coherence interval, that is allocated for the uplink and downink transmissions are $\nu_u = \nu_d = 0.5$. The uplink and downlink power control coefficients are $\eta_k = 1, \forall k,$ and $\eta_{mk} = ( \sum_{k'=1}^K \gamma_{mk'} )^{-1},\forall m,k$.\footnote{In the worst case, if all the direct links are blocked, we introduce a damping constant when Cell-Free Massive MIMO systems in the absence of RIS are considered, since in those cases we have $\sum_{m=1}^M \gamma_{mk'} = 0$.} As far as the optimization of the phase shifts of the RIS elements are concerned, we assume that they are optimized according to the sum-NMSE minimization criterion in the absence of direct links, according to Corollary~\ref{corollary:EqualPhase} and Remark~\ref{RemarkProb}. Without loss of generality, in particular, the $N$ phase shifts in $\pmb{\Phi}$ are all set equal to $\pi/4$, except in Figs.~\ref{FigRandomEqualUL} and \ref{FigRandomEqualDL} where different phase shifts are considered for comparison. In order to evaluate the advantages and limitations of RIS-assisted Cell-Free Massive MIMO systems, three system configurations are considered for comparison:
\begin{itemize}
\item[$i)$] \textit{RIS-Assisted Cell-Free Massive MIMO}: This is the proposed system model in which the direct links are blocked according to \eqref{eq:amk}. This setup is denoted by ``RIS-CellFree''.
\item[$ii)$] \textit{(Conventional) Cell-Free Massive MIMO}: This setup is the same as the previous one with the only exception that the RIS is not deployed in the network. This setup is denoted by ``CellFree''.
\item[$iii)$] \textit{RIS-Assisted Cell-Free Massive MIMO with blocked direct links}: This is the worst case study in which the direct links are blocked with unit probability and the uplink and downlink transmissions are ensured only through the RIS. This setup is denoted by ``RIS-CellFree-NoLOS''.
\end{itemize}
\begin{figure*}[t]
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigMonteCarloApr5} \vspace*{-0.6cm}
\caption{Monte Carlo simulations versus the analytical frameworks with $M= 20$, $K=5$, $N= 64$, $\tau_p = 2,$ and $d_H= d_V = \lambda/4$. The unblocked probability of the direct links is $p = 1.0$.}
\label{FigMonteCarloClosedForm}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.2cm}
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigActiveProbSumUser} \vspace*{-0.6cm}
\caption{The sum net throughput [Mbps] versus the unblocked probability of the direct links $p$ with $M= 100$, $K=10$, $N= 900, \tau_p = 5,$ and $d_H = d_V = \lambda/4$.}
\label{FigActiveProbSumUser}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.2cm}
\end{figure*}
In Fig.~\ref{FigMonteCarloClosedForm}, we illustrate the cumulative distribution function (CDF) obtained by using Monte Carlo methods from \eqref{eq:ULRate} and \eqref{eq:DLRate}, and the analytical expressions of the net throughput obtained in Theorem~\ref{theorem:ULMR} and Theorem~\ref{theorem:DLMR}. We observe a very good overlap between the numerical simulations and the obtained analytical expressions. From Fig.~\ref{FigMonteCarloClosedForm}, we evince that the downlink net throughput per user is about $2.1\times$ better than the uplink net throughput. This is due to the higher transmission power of the APs and the gain of the joint processing gain of the APs. Since the Monte Carlo simulations are not simple to obtain for larger values of the simulation parameters, the rest of the figures are obtained only by using the closed-form expressions derived in Theorem~\ref{theorem:ULMR} and Theorem~\ref{theorem:DLMR}.
\begin{figure*}[t]
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigCDFSumUser} \vspace*{-0.7cm}
\caption{CDF of the sum net throughput [Mbps] with $M= 100$, $K=10$, $N=900$, $\tau_p = 5,$ and $d_H = d_V = \lambda/4$. The unblocked probability of the direct links is $p=0.2$. }
\label{FigCDFSumUser}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.3cm}
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigRandomEqualPhaseUplink} \vspace*{-0.6cm}
\caption{CDF of the uplink sum net throughput [Mbps] with $M= 100$, $K= 10$, $N= 900$, $\tau_p = 5$, and $d_H = d_V = \lambda/4$. The unblocked probability of the direct links is $p=0.2$.}
\label{FigRandomEqualUL}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.3cm}
\end{figure*}
\begin{figure*}[t]
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigRandomEqualPhaseDownlink} \vspace*{-0.6cm}
\caption{CDF of the downlink sum net throughput [Mbps] with $M= 100$, $K=10$, $N= 900$, $\tau_p = 5$, and $d_H = d_V = \lambda/4$. The unblocked probability of the direct links is $p=0.2$. }
\label{FigRandomEqualDL}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.2cm}
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigDifferentdHdVUplink} \vspace*{-0.6cm}
\caption{CDF of the uplink sum net throughput [Mbps] with the different $\{d_H, d_V\}$, $M= 100 $, $K= 10$, $N= 900$, and $\tau_p = 5$. The unblocked probability of the direct links is $p=0.2$.}
\label{FigDifferentdHdVUplink}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.2cm}
\end{figure*}
In Fig.~\ref{FigActiveProbSumUser} we illustrate the sum net throughput as a function of the probability $p$ in \eqref{eq:amk}. In particular, the uplink sum net throughput is defined as $\sum_{k=1}^K R_{uk}$ and the downlink sum net throughput is defined as $\sum_{k=1}^K R_{dk}$. From the obtained results, we evince that Cell-Free Massive MIMO provides the worst performance if the blocking probability is large ($p$ is small). If the direct links are unreliable, as expected, the net throughput offered by Cell-Free Massive MIMO tends to zero if $p \to 0$. By assuming $p=0.1$, for example, Cell Free Massive MIMO is approximately $2.1\times$ and $1.6\times$ worse than the worst-case RIS-assisted Cell-Free Massive MIMO setup (RIS-CellFree-NoLOS) in the uplink and downlink, respectively. In the considered case study, in addition, we note that the proposed RIS-assisted Cell-Free Massive MIMO setup offers the best net throughput, since it can overcome the unreliability of the direct links thanks to the presence of the RIS. The presence of the RIS is particularly useful if $p$ is small, i.e., $p<0.2$ in Fig.~\ref{FigActiveProbSumUser}, since in this case the direct links are not able to support a high throughput. In this case, the combination of Cell-Free Massive MIMO and RISs is capable of providing a high throughput and signal reliability.
In Fig.~\ref{FigCDFSumUser}, we compare the three considered systems in terms of sum net throughput when $p=0.2$. We observe the net advantage of the proposed RIS-assisted Cell-Free Massive MIMO system, especially in the downlink. In the uplink, in addition, even the worst-case RIS-assisted Cell-Free Massive MIMO system setup (i.e., $p=0$) outperforms the Cell-Free Massive MIMO setup in the absence of an RIS. In the following figures, we focus our attention only on the RIS-asissted Cell-Free Massive MIMO setup, since it provides the best performance.
\begin{figure*}[t]
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigDifferentdHdVDownlink} \vspace*{-0.6cm}
\caption{The CDF of downlink sum net throughput [Mbps] with the different $\{d_H, d_V \}$, $M= 100$, $K=10$, $N= 900$, and $\tau_p = 5$. The unblocked probability of the direct links is $p=0.2$. }
\label{FigDifferentdHdVDownlink}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.2cm}
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[trim=3.2cm 9.8cm 3.7cm 10.4cm, clip=true, width=2.9in]{FigFixedAreaEqualPhase} \vspace*{-0.6cm}
\caption{The CDF of sum net throughput [Mbps] with $M= 100 $, $K= 10$, and $\tau_p = 5$. Here, $d=d_H=d_V$ and the unblocked probability of the direct links is $p=0.2$. }
\label{FigFixedArea}
\vspace*{-0.2cm}
\end{minipage}
\vspace*{-0.2cm}
\end{figure*}
In Figs.~\ref{FigRandomEqualUL} and \ref{FigRandomEqualDL}, we compare the uplink and downlink sum net throughput as a function of the phase shifts of the RISs (random and uniform phase shifts according to Corollary~\ref{corollary:EqualPhase}) in the presence of spatially-correlated and spatially-independent fading channels according to \eqref{eq:CovarMa}. In the presence of spatial correlation, we consider $\mathbf{R}_m = \alpha_m d_H d_V \mathbf{I}_N$ and $\widetilde{\mathbf{R}}_{mk} = \tilde{\alpha}_{mk} d_H d_V \mathbf{I}_N, \forall m,k$. We note different performance trends in spatially-correlated fading channels. If the spatial correlation is not considered, we observe that there is no significant difference between the random and uniform phase shifts setup. In the presence of spatial correlation, on the other hand, the proposed uniform phase shift design that is obtained from Corollary~\ref{corollary:EqualPhase} provides a much better throughput. This result highlights the relevance of using even simple optimization designs for RIS-assisted communications in the presence of spatial correlation.
In Figs.~\ref{FigDifferentdHdVUplink} and \ref{FigDifferentdHdVDownlink}, we analyze the impact of the size of the scattering elements of the RISs on the uplink and downlink net throughputs, respectively, while keeping the number $N$ of RIS elements fixed. The size of the considered RIS, which is a compact surface, is $N d_H d_V$, which implies that it increases as the size $d_H d_V$ of each element of the RIS increases. In this setup, we observe that the net throughput increases as the physical size of each radiating element of the RIS increases. In Fig.~\ref{FigFixedArea}, on the other hand, we analyze a setup in which the total size of the RIS is kept constant to $N d_H d_V = 10 \lambda \times 10 \lambda$ and the triplet $(N, d=d_H=d_V)$ is changed accordingly. With the considered fading spatial correlation model and for a size of the RIS elements no smaller than $\lambda/3$, we do not observe a significant difference of the achievable net throughput. Further studies are, however, necessary for deep sub-wavelength RIS structures, for different optimization criteria of the phase shifts of the RISs, and in the presence of mutual coupling in addition to the fading spatial correlation \cite{gradoni2021end,qian2021mutual}.
\vspace*{-0.5cm}
\section{Conclusion}\label{Sec:Conclusion}
\vspace*{-0.15cm}
Cell-Free Massive MIMO and RIS are two disruptive technologies for boosting the system performance of future wireless networks. These two technologies are not competing with each other, but have complementary features that can be integrated and leveraged to enhance the system performance in harsh communication environments. Therefore, we have considered an RIS-assisted Cell-Free Massive MIMO system that operates according to the TDD mode. An efficient channel estimation scheme has been introduced to overcome the high overhead that may be associated with the estimation of the individual channels of the RIS elements. Based on the proposed channel estimation scheme, an optimal design for the phase shifts of the RIS that minimizes the channel estimation error has been devised and has been used for system analysis. Based on the proposed channel estimation method, closed-form expressions of the ergodic net throughput for the uplink and downlink data transmission phases have been proposed. Based on them, the performance of RIS-assisted Cell-Free Massive MIMO has been analyzed as a function of the fading spatial correlation and the blocking probability of the direct AP-user links. The numerical results have shown that the presence of an RIS is particularly useful if the AP-user links are unreliable with high probability.
Possible generalizations of the results illustrated in this paper include the optimization of the phase shifts of the RIS that maximize the uplink or downlink throughput, the analysis of the impact of the fading spatial correlation for non-compact and deep sub-wavelength RIS structures, and the analysis and optimization of RIS-assisted systems in the presence of mutual coupling.
\vspace*{-0.5cm}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,516 |
Q: nginx how to remove suffix to static file location I am using nginx to server static image files. Within my directory, all images are stored without an extension. For example: abcdefg.png will be stored as /opt/file/ab/cdefg. And the image url is http://example.com/file/ab/cdefg.png.
If the url is http://example.com/file/ab/cdefg, the following configuration works. But it doesn't work for url http://example.com/file/ab/cdefg.png
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}
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"redpajama_set_name": "RedPajamaStackExchange"
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