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Q: How to give validation using JavaScript I have made a table where I have to enter some values for calculating the average of certain values. I have a submit button for submitting the form and store the values into a database.
When I don't enter any data and submit the form, 0 values are stored into the database. So I have to validate each of the text boxes. How can I do that with JavaScript?
function calcAvg(input_id, output_id, att_id) {
//Get all elements with 'class="select"'
var selects = document.getElementsByClassName(input_id);
//Initialize vars
var avg = 0;
var count = 0;
var grade = 0;
//Calculate average
for (var i = 0; i < selects.length; i++) {
if (selects[i].value != "") {
count++;
avg += Number(selects[i].value);
//Alert for debugging purposes
//alert(selects[i].value+" "+avg);
}
}
avg = avg / count;
//Output average
document.getElementById(output_id).value = avg;
if (avg >= 80)
grade = 'HIGH';
else if (avg >= 60 && avg < 80)
grade = 'MEDIUM';
else if (avg >= 40 && avg < 60)
grade = 'LOW'
else
grade = 'N/A'
//Output average
document.getElementById(att_id).value = grade;
}
<form method='POST'>
<table>
<tr>
<td height="41" colspan="12" align="center">ATTAINMENT OF PO( PO-CO MAPPING)
</td>
</tr>
<tr>
<td width="17%" rowspan="2" align="center">NAME OF THE MODULES</td>
<td height="34" colspan="11" align="center">PROGRAME OUTCOMES</td>
</tr>
<tr>
<td width="7%" align="center">PO1</td>
<td width="7%" align="center">PO2</td>
</tr>
<tr>
<td height="71" align="center">MATHEMATICS</td>
<td align="center"><input type="number" class="select1" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select1','calculation1','calatt1');" style="width:60px"> </td>
<td align="center"><input type="number" class="select2" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select2','calculation2','calatt2');" style="width:60px"> </td>
</tr>
<tr>
<td height="71" align="center">SCIENCE</td>
<td align="center"><input type="number" class="select1" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select1','calculation1','calatt1');" style="width:60px"> </td>
<td align="center"><input type="number" class="select2" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select2','calculation2','calatt2');" style="width:60px">
</td>
</tr>
<tr bgcolor="#9999CC">
<td height="71" align="center">PO AVERAGES</td>
<td align="center"><input type="text" name="Avg" id="calculation1" readonly style="width:60px"> </td>
<td align="center"><input type="text" name="Avg1" id="calculation2" readonly style="width:60px"> </td>
</tr>
<tr>
<td height="71" align="center">PO ATTAINMENT</td>
<td align="center"><input type="text" name="Att1" id="calatt1" readonly style="width:60px"> </td>
<td align="center"><input type="text" name="Att2" id="calatt2" readonly style="width:60px"> </td>
</tr>
</table>
<p align="center"><input type="submit" name="submit" value="submit"></p>
</form>
<script type="text/javascript">
function calcAvg(input_id, output_id, att_id) {
//Get all elements with 'class="select"'
var selects = document.getElementsByClassName(input_id);
//Initialize vars
var avg = 0;
var count = 0;
var grade = 0;
//Calculate average
for (var i = 0; i < selects.length; i++) {
if (selects[i].value != "") {
count++;
avg += Number(selects[i].value);
//Alert for debugging purposes
//alert(selects[i].value+" "+avg);
}
}
avg = avg / count;
//Output average
document.getElementById(output_id).value = avg;
if (avg >= 80)
grade = 'HIGH';
else if (avg >= 60 && avg < 80)
grade = 'MEDIUM';
else if (avg >= 40 && avg < 60)
grade = 'LOW'
else
grade = 'N/A'
//Output average
document.getElementById(att_id).value = grade;
}
</script>
<form method='POST'>
<table>
<tr>
<td height="41" colspan="12" align="center">ATTAINMENT OF PO( PO-CO MAPPING)
</td>
</tr>
<tr>
<td width="17%" rowspan="2" align="center">NAME OF THE MODULES</td>
<td height="34" colspan="11" align="center">PROGRAME OUTCOMES</td>
</tr>
<tr>
<td width="7%" align="center">PO1</td>
<td width="7%" align="center">PO2</td>
</tr>
<tr>
<td height="71" align="center">MATHEMATICS</td>
<td align="center"><input type="number" class="select1" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select1','calculation1','calatt1');"
style="width:60px"> </td>
<td align="center"><input type="number" class="select2" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select2','calculation2','calatt2');"
style="width:60px"> </td>
</tr>
<tr>
<td height="71" align="center">SCIENCE</td>
<td align="center"><input type="number" class="select1" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select1','calculation1','calatt1');"
style="width:60px"> </td>
<td align="center"><input type="number" class="select2" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select2','calculation2','calatt2');"
style="width:60px">
</td>
</tr>
<tr bgcolor="#9999CC">
<td height="71" align="center">PO AVERAGES</td>
<td align="center"><input type="text" name="Avg" id="calculation1" readonly style="width:60px"> </td>
<td align="center"><input type="text" name="Avg1" id="calculation2" readonly style="width:60px"> </td>
</tr>
<tr>
<td height="71" align="center">PO ATTAINMENT</td>
<td align="center"><input type="text" name="Att1" id="calatt1" readonly style="width:60px"> </td>
<td align="center"><input type="text" name="Att2" id="calatt2" readonly style="width:60px"> </td>
</tr>
</table>
<p align="center"><input type="submit" name="submit" value="submit"></p>
</form>
A: You could set a common class to the required fields and then, create a new function to check if they are not empty or = 0, calling it on submit button onclick event.
See my example below:
function calcAvg(input_id, output_id, att_id) {
//Get all elements with 'class="select"'
var selects = document.getElementsByClassName(input_id);
//Initialize vars
var avg = 0;
var count = 0;
var grade = 0;
//Calculate average
for (var i = 0; i < selects.length; i++) {
if (selects[i].value != "") {
count++;
avg += Number(selects[i].value);
//Alert for debugging purposes
//alert(selects[i].value+" "+avg);
}
}
avg = avg / count;
//Output average
document.getElementById(output_id).value = avg;
if (avg >= 80)
grade = 'HIGH';
else if (avg >= 60 && avg < 80)
grade = 'MEDIUM';
else if (avg >= 40 && avg < 60)
grade = 'LOW'
else
grade = 'N/A'
//Output average
document.getElementById(att_id).value = grade;
}
function validateForm() {
var required = document.getElementsByClassName('required');
for (var i = 0; i < required.length; i++) {
var value = required[i].value;
if (Number(value) === 0 || value == '') {
console.log('Please, fill all the required fields before submitting');
return false;
}
}
}
<form method='POST'>
<table>
<tr>
<td height="41" colspan="12" align="center">ATTAINMENT OF PO( PO-CO MAPPING)
</td>
</tr>
<tr>
<td width="17%" rowspan="2" align="center">NAME OF THE MODULES</td>
<td height="34" colspan="11" align="center">PROGRAME OUTCOMES</td>
</tr>
<tr>
<td width="7%" align="center">PO1</td>
<td width="7%" align="center">PO2</td>
</tr>
<tr>
<td height="71" align="center">MATHEMATICS</td>
<td align="center">
<input type="number" class="select1 required" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select1','calculation1','calatt1');" style="width:60px"> </td>
<td align="center">
<input type="number" class="select2 required" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select2','calculation2','calatt2');" style="width:60px"> </td>
</tr>
<tr>
<td height="71" align="center">SCIENCE</td>
<td align="center">
<input type="number" class="select1 required" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select1','calculation1','calatt1');" style="width:60px"> </td>
<td align="center">
<input type="number" class="select2 required" name="value[]" onKeyPress="if(this.value.length==2) return false;" onChange="calcAvg('select2','calculation2','calatt2');" style="width:60px">
</td>
</tr>
<tr bgcolor="#9999CC">
<td height="71" align="center">PO AVERAGES</td>
<td align="center">
<input type="text" name="Avg" id="calculation1" readonly style="width:60px"> </td>
<td align="center">
<input type="text" name="Avg1" id="calculation2" readonly style="width:60px"> </td>
</tr>
<tr>
<td height="71" align="center">PO ATTAINMENT</td>
<td align="center">
<input type="text" name="Att1" id="calatt1" readonly style="width:60px"> </td>
<td align="center">
<input type="text" name="Att2" id="calatt2" readonly style="width:60px"> </td>
</tr>
</table>
<p align="center">
<input type="button" name="submit" value="submit" onclick='validateForm()'>
</p>
</form>
A: The issue here may help you for parsing an input and checking if it is a number and positive. Or with some modification on your code:
//Calculate average
for (var i = 0; i < selects.length; i++) {
if (selects[i].value == "") {
// Alert if one of inputs is null
alert("Cannot be empty");
}
else {
count++;
avg += Number(selects[i].value);
//Alert for debugging purposes
//alert(selects[i].value+" "+avg);
}
}
Or with HTML5 you can use reuired attribute on the inputs themselves like:
<input type="text" name="foo" required>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,390 |
{"url":"https:\/\/vsoch.github.io\/2013\/mutual-information-for-image-registration\/","text":"Mutual information is an important concept for medical imaging, because it is so prominently used in image registration. \u00a0Let\u2019s set up a typical registration problem. \u00a0Here we have two pictures of something that you see everyday, a dancing pickle!\n\nI\u2019ve done two things in creating these images. \u00a0First I\u2019ve applied a simple translation to our pickle, meaning that I\u2019ve moved him down and right (a linear transformation), and I haven\u2019t warped or stretched him in any way (a non-linear transformation). \u00a0I\u2019ve also changed his intensity values so that a bright pixel in one image does not correspond to a bright pixel in another image. \u00a0Our goal, is to register these two images. Abstractly, registration is pretty simple. We need to find some function that measures the difference between two images, and then minimize it.\n\n### Registration with minimizing the sum of squared distances\n\nThe most basic approach would be to say \u201cok, I want to find the alignment that minimizes the sum of squared distances,\u201d because when pickle one is lined up with pickle two, the pixels should be the same, and so our error is small, right? \u00a0Wrong! \u00a0This is not a good approach, because a high intensity in one image does not mean the same thing as a high intensity in the other. \u00a0An example of this from medical imaging is with T1 and T2 images. \u00a0A T1 image of a brain has shows cerebral spinal fluid as black, while a T2 image shows it as bright white. \u00a0If we use an algorithm that finds a registration based on minimizing the summed squared distance between pixels, it wouldn\u2019t work. \u00a0I\u2019ll try it anyway to prove it to you:\n\n\n% Mutual Information\n% This shows why mutual information is important for registration, more-so\n% than another metric like the least sum of squares.\n\n% First, let's read in two pickle images that we want to register\n\n% Let's look at our lovely pickles, pre registration\nfigure(1); subplot(1,2,1); imshow(pickle1); title('Pre-registration')\nsubplot(1,2,2); imshow(pickle2);\n\n% Matlab has a nice function, imregister, that we can configure to do the heavy lifting\ntemplate = pickle1; % fixed image\ninput = pickle2; % We will register pickle2 to pickle1\ntransformType = 'affine'; % Move it anyhow you please, Matlab.\n\n% Matlab has a function to help us specify how we will optimize the\n% registration. We actually are going to tell it to do 'monomodal' because\n% this should give us the least sum of squared distances...\n[optimizer, metric] = imregconfig('monomodal');\n\n% Now let's look at what \"metric\" is:\nmetric =\n\nregistration.metric.MeanSquares\n\n% Now let's look at the documentation in the script itself:\n% A MeanSquares object describes a mean square error metric configuration\n% that can be passed to the function imregister to solve image\n% registration problems. The metric is an element-wise difference between\n% two input images. The ideal value of the metric is zero.\n\n% This is what we want! Let's do the registration:\nmoving_reg = imregister(input,template,transformType,optimizer,metric);\n\n% And take a look at our output, compared to the image that we wanted to\n% register to:\nfigure(2); subplot(1,2,1); imshow(template); title('Registration by Min. Mean Squared Error')\nsubplot(1,2,2); imshow(moving_reg);\n\n\n\nWhere did my pickle go??\n\n### Registration by Maximizing Mutual Information\n\nWe are going to make this work by not trying to minimize the total sum of squared error between pixel intensities, but rather by maximizing mutual information, or the extent to which knowing one image intensity tells us the other image intensity. This makes sense because in our pickle pictures, all of the black background pixels in the first correspond to white background pixels in the second. This mapping of the black background to the white background represents a consistent relationship across the image. If we know a pixel is black in one image, then we also know that the same pixel is white in the other.\n\nIn terms of probability, what we are talking about is the degree of dependence between our two images. Let\u2019s call pickle one variable \u201cA\u201d and pickle two variable \u201cB.\u201d\n\nIf A and B are completely independent, then the probability of A and B (the joint probability, P(AB)) is going to be the same thing as multiplying their individual probabilities P(A)*P(B). We learned that in our introductory statistics class.\n\nIf A and B are completely dependent, then when we know A, we also know B, and so the joint probability, P(AB), is equal to the separate probabilities, P(AB) = P(A) = P(B).\n\nIf they are somewhere between dependent and independent, we fall somewhere between there, and perhaps knowing A can allow us to make a good guess about B, even if we aren\u2019t perfect.\n\nSo perfect (high) mutual information means that the pixels in the image are highly dependent on one another, or there is some consistent relationship between the pixels so that we can very accurately predict one from the other. Since we have the raw images we know the individual probability distributions P(A) and P(B), and so we can measure how far away P(AB) is from P(A)P(B) to assess this level of dependence. If the distance between P(AB) and P(A)P(B) is very large, then we can infer that A and B are somewhat dependent. More on this computation later - let\u2019s take a look at mutual information in action.\n\n\n% Now let's select \"multimodal\" and retry the registration. This will use\n% mutual information.\n[optimizer, metric] = imregconfig('multimodal');\n\nMattesMutualInformation Mutual information metric configuration object\n\nA MutualInformation object describes a mutual information metric\nconfiguration that can be passed to the function imregister to solve\nimage registration problems.\n\n% Now the metric is mutual information! Matlab, you are so smart!\nmoving_reg = imregister(input,template,transformType,optimizer,metric);\n\n% Let's take one last look...\nfigure(3); subplot(1,2,1); imshow(template); title('Registration by Max. Mutual Information')\nsubplot(1,2,2); imshow(moving_reg);\n\n\n\nHooray! Success!\n\n### Mutual Information - the Math Behind It!\n\nWe\u2019ve shown that mutual information is the way to go to register two images with different intensity values (i.e., different modalities). \u00a0Here is the actual equation to measure mutual information (this is the value that we would iteratively try to maximize)\n\nThis is for discrete distributions, and if we had continuous distributions the summations would turn to integrals, and we would add dxdy to the end. \u00a0Given that we are summing probabilities, this value always has to be positive. \u00a0Can you see what happens when our variables are completely independent, when p(x)*p(y) = p(x,y)? \u00a0We take a log of 1, and since that is 0, our mutual information \u00a0is also 0. \u00a0This tells us that knowing A says nothing about B.\n\n### Kullback-Leibler Divergence - Another measure for comparison\n\n KL Divergence is another way to assess the difference between two probability distributions. \u00a0It\u2019s even special enough to have it\u2019s own format, specifically if you \u00a0see\u00a0DKL(A B) those symbols mean \u201cthe kullback leibler divergence between variables A and B.\u201d \u00a0In real applications of KL, the value A usually represents some true distribution of something, and B is how we model it, and so DKL(A B) \u00a0is getting at\u00a0the\u00a0amount of information lost when we use our model, B to guess the reality, A. \u00a0Here comes the math! (thanks wikipedia!)\n\nThe above reads, \u201cthe KL divergence between some real distribution, P, and our model of it, Q, is\u00a0the expectation of the logarithmic difference (remember that a fraction in a log can be rewritten as log(top) - log(bottom)) between the probabilities\u00a0P\u00a0and\u00a0Q, where the expectation is taken using the probabilities\u00a0P.\u201d \u00a0 Again, when our model (Q) fits the reality well, (P) ad the two probabilities are equal, we take the ln(1) = 0, and so we would say that \u201czero information is lost when we use Q to model P.\u201d\n\nThe above code is also available as a gist here.","date":"2022-01-23 14:28:59","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\": 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\": 0.7730920910835266, \"perplexity\": 1199.5838043347312}, \"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-2022-05\/segments\/1642320304287.0\/warc\/CC-MAIN-20220123141754-20220123171754-00176.warc.gz\"}"} | null | null |
\section{Introduction}
Understanding why and how neurons behave is now foundational for both machine learning and neuroscience. Such understanding can lead to better, more interpretable artificial neural networks, as well as provide insights into how biological networks mediate cognition. A key to both these pursuits lies in understanding how neurons can best structure their firing patterns to solve tasks.
Neuroscientists have some understanding of how task demands affect both early single neuron responses \citep{olshausen_emergence_1996, yamins_performance-optimized_2014,ocko_emergence_2018, mcintosh_deep_2016} and population level measures such as dimensionality \citep{gao_theory_2017, stringer_high-dimensional_2019}. However, there is little understanding of neural population structure in higher brain areas. As an example, we do not even understand why many different bespoke cellular responses exist for physical space, such as grid cells \citep{hafting_microstructure_2005}, object-vector cells \citep{hoydal_object-vector_2019}, border vector cells \citep{solstad_representation_2008, lever_boundary_2009}, band cells \citep{krupic_neural_2012}, or many other cells \citep{okeefe_hippocampus_1971, gauthier_dedicated_2018, sarel_vectorial_2017, deshmukh_influence_2013}. Each cell has a well defined, specific cellular response pattern to space, objects, \textit{or} borders, as opposed to a mixed response to space, objects, \textit{and} borders. Similarly, we don't understand why neurons in inferior temporal cortex are aligned to axes of data generative factors \citep{chang_code_2017, bao_map_2020, higgins_unsupervised_2021}, why visual cortical neurons are de-correlated \citep{ecker_decorrelated_2010}, why neurons in parietal cortex are selective only for specific tasks \citep{lee_task_2022}, why prefrontal neurons are apparently mixed-selective \citep{rigotti_importance_2013}, and why grid cells sometimes warp towards rewarded locations \citep{boccara_entorhinal_2019} and sometimes don't \citep{butler_remembered_2019}. In essence, why are some neural representations entangled and others not?
Machine learning has long endeavoured to build models that disentangle factors of variation \citep{hinton_transforming_2011, higgins_beta-vae_2017, locatello_challenging_2019}. Such disentangled factors can facilitate compositional generalisation and reasoning \citep{higgins_scan_2018,higgins_darla_2017, whittington_constellation_2021}, as well as lead to more interpretable outcomes in which individual neurons represent meaningful quantities. Unfortunately, building models that disentangle is challenging \citep{locatello_challenging_2019}, with disentangling often coming at the cost of accurately modelling the data itself.
In this work we 1) prove simple biological constraints of \textbf{nonnegativity} and \textbf{minimising activity energy} lead to factorised representations in linear networks; 2) empirically show these constraints lead to disentangled representations in both linear and non-linear networks; 3) obtain competitive disentangling scores on a standard disentangling benchmark; 4) provide an understanding why neurons in the brain are characterised into specific cell types due to these same biological constraints; 5) empirically show these constraints lead to specific cell types; 6) suggest when and why neurons in the brain exhibit disentangling versus mixed-selectivity.
\subsection{Related work}
\textbf{Disentangling in machines.} Algorithms like PCA or ICA can extract linear factors of variation from data. However, in neural models, while one can learn the principle subspace \citep{pehlevan_hebbiananti-hebbian_2015} or learn principle components sequentially (e.g. with Oja's rule), learning single independent factors in individual neurons (i.e. disentangling) has only recently been achieved via specific weight regularization in linear autoencoders \citep{kunin_loss_2019}. For disentangling in \textit{non-linear} models, most modern methods use variational autoencoders (VAE; \citet{kingma_auto-encoding_2013}), with various choices that promote explicit factorisation of the learned latent space. For example \(\beta\)-VAEs \citep{higgins_beta-vae_2017} up-weight (by \(\beta\)) the term in the VAE loss that encourages the posterior to be a factorised distribution. Other variations of disentangling VAEs \citep{burgess_understanding_2018, kim_disentangling_2018, ridgeway_learning_2018, kumar_variational_2018, chen_isolating_2018} similarly try to explicitly enforce a factorised aggregate posterior. Importantly, while it has been shown disentangling is not possible without inductive bias in the data or model \citep{locatello_challenging_2019}, numerous inductive biases have been shown to be sufficient for disentangling (i.e. conditioning the prior on a third variable \citet{khemakhem_variational_2020}; or local isometry and non-Gaussianity \citet{horan_when_2021}). Our work also relates to modern self-supervised learning \citep{bardes_vicreg_2022} which seeks decorrelated representations with non-zero variance. Again, most algorithms have an explicit term to force decorrelation. Here instead, we show that simple biological constraints of nonnegativity and energy efficiency lead to emergent disentangling without an explicit decorrelation term in the objective.
\textbf{Disentangling in brains.} There is no formal understanding of when and why neurons in the brain factorise across task parameters. Nevertheless, single latent dimensions from disentangling models do predict single neuron activity, implying biological neurons are disentangled \citep{higgins_unsupervised_2021}. One conjecture in neuroscience is that while representing task factors is important for generalisation, mixed-selectivity is important for efficient readout \citep{behrens_what_2018, rigotti_importance_2013, bernardi_geometry_2020}. Here we show disentangled brain representations are preferred if the task is factorised into independent factors.
\textbf{Non-negativity.} Recent work on multitask learning in recurrent neural networks (RNNs) \citep{yang_task_2019, driscoll_flexible_2022} demonstrated that neural populations, with a nonnegative activation function, partition themselves into task specific modules \citep{driscoll_flexible_2022}. Non-negativity is also important in obtaining hexagonal, not square, grid cells \citep{dordek_extracting_2016, whittington_relating_2021,sorscher_unified_2019, sorscher_unified_2020}. Also, nonnegative matrix factorisation empirically yields spatially localised factors for images \citep{lee_algorithms_2000}. Our work demonstrates theoretically and empirically {\it why} nonnegativity leads to single neurons becoming selective for single factors.
\section{Linear disentangling with biological constraints}
We first provide a theorem that suggests why the combined biological constraints of nonnegativity and energy efficiency lead to neural disentangling (proofs of all theorems are in App. \ref{Appendix:Proofs}):
\textbf{Theorem 1}. Let \({\bm{e}} \in \mathbb{R}^k\) be a random vector whose $k$ independent components denote $k$ task factors. We assume each independent task factor \( {e}_i \) is drawn from a distribution that has mean \(0\), variance \( \sigma^2\), and maximum and minimum values of \( \min({e}_i) = -a \) and \( \max({e}_i) = a \). Also let \( {\bm{z}} \in \mathbb{R}^n \) be a linear neural representation of the task factors given by
\[
{\bm{z}} = {\bm{M}} {\bm{e}} + {\bm{b}}_z,
\]
where \( {\bm{M}} \in \mathbb{R}^{n\times k}\) are mixing weights and \( {\bm{b}}_z \in \mathbb{R}^n\) is a bias. We further assume two constraints: (1) the neural representation is \textit{nonnegative} with \({z}_i \ge 0\) for all $i=1,\dots,n$, and (2) the neural population variance is a nonzero constant, \( \sum_j Var({z}_j) = C \), so that the neural representation retains some information about the task variables. Under these two constraints we show that in the space of all possible neural representations (parameterised by \({\bm{M}}\) and \({\bm{b}}_z\)), the representations that achieve minimal activity energy \( \mathbb{E} || {\bm{z}} ||^2 \) also exhibit disentangling, by which we mean every neuron \( {z}_j\) is selective for at most one task parameter: i.e. \( | {M}_{jk}| | {M}_{jl} | = 0\) for \( k \neq l \). (Proof in App. \ref{Appendix:proof_constant_variance}).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.96\linewidth]{figures/proof_intuition.pdf}
\end{center}
\caption{\textbf{Proof intuition.} Two uniformly distributed independent factors represented with two entangled neurons (left). The representation can be made nonnegative at the expense of activity energy (middle). Activity energy is minimised under a nonnegativity (and variance) constraint when the neurons are axis aligned to task factors (i.e. disentangled, right). Grey boxes denote uniform distributions over neural activity induced by uniform distributions over task factors. Note our proof does not require uniformity.}
\label{fig:proof_intuition}
\end{figure}
\textbf{Intuition.} The intuition underlying the proof of the theorem is shown in Fig.\ref{fig:proof_intuition}, where the key idea can be seen with two neurons encoding two factors. In particular, the bias must make every \({z}_i \) nonnegative for all values of \({e}_1 \) and \( {e}_2 \). But since \({e}_1 \) and \( {e}_2 \) are independent, the minimum firing of neuron $1$ for example obeys \(\min(z_1) = \min(0.8{e}_1 - 0.6 {e}_2) = \min(0.8{e}_1 ) + \min(-0.6 {e}_2) = -a(0.8+0.6)\). Thus for neurons that mix factors, a larger bias term must be used to ensure nonnegativity, which leads to increased expected energy. Minimising this energy (subject to a constant variance) requires the smallest possible bias for each neuron, which occurs when each neuron is selective for a single task factor.
The above theorem, while simple, is restricted in two ways: (1) the independent task factors \({e}_i \) are {\it directly} available to the network; (2) representational collapse (i.e. setting \({\bm{z}}=0\)) under energy minimisation is prevented solely by a variance constraint. We thus consider a more general setting where a neural circuit receives not the independent task factor vector \( {\bm{e}} \), but instead receives the mixed combination \( {\bm{x}} = {\bm{D}} {\bm{e}} \). We further model the neural representation \( {\bm{z}} \) as a linear generative model that can predict observed data \( {\bm{x}} \) via
\(
{\bm{x}} = {\bm{W}} {\bm{z}} + {\bm{b}}_x.
\)
Thus prediction, not variance constraint, now prevents collapsing neural representations (proof in Appendix \ref{Appendix:proof_Variance_Fixed}). Furthermore in Appendix \ref{Appendix:proof_orthog_data} we prove the following:
\textbf{Theorem 2.} Let \( {\bm{x}} = {\bm{D}} {\bm{e}} \) be observed entangled data, where the independent task factor vector ${\bm{e}}$ obeys the same distributional assumptions as in Theorem 1. Let a neural representation \( {\bm{z}} \) exactly predict observed data via \({\bm{x}} = {\bm{W}} {\bm{z}} + {\bm{b}}_x \) with zero error. Then for all such data generation models (with parameters \( {\bm{D}} \)) and all such neural representations (with parameters \( {\bm{W}} \) and \( {\bm{b}}_x \)), as long as: (1) the columns of \({\bm{D}} \) are (scaled) orthonormal; (2) the norm of the read-out weights \( || {\bm{W}} ||^2_F \) is finite; (3) the neural representation is nonnegative (i.e. \({\bm{z}} > 0 \)), then out of all such neural representations, the minimum energy representations are also disentangled ones. By this we mean that each neuron \({z}_i \) will be selective for at most one hidden task factor \({e}_j \).
We note that \({\bm{D}}\) having (scaled) orthonormal columns may seem like a strong constraint, but it holds approximately for any random matrix \({\bm{D}}\) with many observations (dimensionality of \( {\bm{x}} \)) and few independent task factors (dimensionality of \( {\bm{e}} \)) (proof in Appendix \ref{Appendix:proof_random_matrix}). Appendix \ref{Appendix:no_orthonormal} discusses and provides intuition for when disentangling occurs as \({\bm{D}}\) takes more general forms.
Strikingly, the essential content of Theorem 2 is that any linear, nonnegative, optimally energetically efficient, generative neural representation that accurately predicts entangled observations that are linear mixtures of hidden task factors, will possess single neurons that are selective for individual task factors, despite {\it never} having {\it direct} access to them. In terms of applications, this theorem could apply in supervised or self-supervised machine learning settings in any neural network layer \( {\bm{z}} \) that is linearly read out from, or in neuroscience settings where \( {\bm{z}} \) reflects a neural population from which one attempts to linearly decode task factors. While Theorem 2 holds in a simple setting, we will show through simulations that its essential content, namely that nonnegativity and energy efficiency together promote disentangling, holds in practice in much more complex multilayer neural networks.
\section{Disentangling in machines}
We now present simulation results demonstrating that nonnegativity and energy efficiency (minimising either activity or weight energy) lead to single neuron selectivity for single task factors. We show this for supervised and unsupervised learning, both for linear and non-linear tasks and networks (simulation details in Appendix \ref{Appendix:network_architecure_sim_details}).
\textbf{A measure for disentangled subspaces.}
While our theory describes when single neurons become selective for single independent task factors, it does not limit the number of neurons selective for any given factor. For example in Theorem 2, four copies of the same neuron in \( {\bm{z}} \), each with half the activity, along with four copies of projecting weights each with half the values, predicts \( {\bm{x}} \) just as well and has exactly the same energy in both \( {\bm{z}} \) and \( {\bm{W}} \) \footnotemark. More interestingly, an underlying task factor may not be one-dimensional, e.g. spatial location, in which case the subspace that codes for this factor have at least the same dimension. This phenomena cannot be captured by many metrics of disentangling (e.g. the popular mutual information gap; MIG; \cite{chen_isolating_2018}) since they score highly if each factor is represented in just \textit{one} neuron. Thus we define a new metric (mutual information ratio; MIR) that instead scores highly if each neuron only cares about one factor (see Appendix \ref{Appendix:definitions} for details).
\footnotetext{Learning dynamics may favour fewer neurons per factor as there are fewer weights to align.}
\textbf{Regularizers as constraints.} We impose nonnegativity via a ReLU activation function, or softly via explicit regularization \( \mathcal{L}_{\text{nonneg}} = \beta_{\text{nonneg}} \sum_i \max ( - a_i, 0 ) \) where \( i \) indexes a neuron in the network, and \( \beta_{nonneg} \) determines the regularization strength. Similarly, we apply regularization to the activity energy and weight energy; \( \mathcal{L}_{\text{activity}} = \beta_{\text{activity}} \sum_l || {\bm{a}}_l ||^2 \) and \( \mathcal{L}_{\text{weight}} = \beta_{\text{weight}} \sum_{l} || {\bm{W}}_l ||^2 \). The role of \( \mathcal{L}_{\text{weight}} \) is to promote activity (variance) in the network, otherwise activity could be reduced via \( \mathcal{L}_{\text{activity}} \), and such reduced activity could be compensated for with arbitrarily large weights. The total loss we optimise is
\[
\mathcal{L} = \\
\underbrace{\mathcal{L}_{\text{nonneg}} + \mathcal{L}_{\text{activity}} + \mathcal{L}_{\text{weight}}}_{\text{Biological constraints}} + \\
\underbrace{\mathcal{L}_{\text{prediction}}}_{\text{Functional constraints}}.
\]
Here `functional constraints', are any prediction losses the network has i.e. error in predicting target labels in supervised learning, or reconstruction error in autoencoders.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.96\linewidth]{figures/SubspaceNet_Shallow.pdf}
\end{center}
\caption{\textbf{Shallow linear networks disentangle.} We train 1-hidden layer linear networks on linear data.
\textbf{A)} Schematic showing both input and output are entangled linear mixtures of factors (colours). Neurons colours schematically denote which of the factors it codes for. \( {\bm{W}}_{true} \) is the true mapping between \( {\bm{x}} \) and \( {\bm{y}} \).
\textbf{B)} A model without biological constraints learns entangled internal representations. Mutual information matrix shown on left, schematic on right. \( {\bm{W}}_{1} \) and \( {\bm{W}}_{2} \) are the learnable weights projecting to and from the hidden layer.
\textbf{C)} A model with our constraints learns disentangled representations.
\textbf{D)} Several model variants, in which only those with all our constraints learn disentangled representations. Average and standard error shown for 5 random seeds.}
\label{fig:subspace_net_shallow}
\end{figure}
\textbf{Disentangling in supervised shallow neural networks}. First we consider a dataset \( \mathcal{D} = \{ {\bm{x}}, {\bm{y}} \} \), where \( {\bm{x}} \) is a orthogonal mixture of six i.i.d. random variables (hidden independent task factors; uniform distribution), and \( {\bm{y}} \) is a linear transform of \( {\bm{x}} \) (dimension 6; Fig. \ref{fig:subspace_net_shallow}A). First we train shallow linear networks to read-out \({\bm{y}} \) from \({\bm{x}} \). Networks without biological constraints exhibit mixed internal representations (Fig. \ref{fig:subspace_net_shallow}B). However, with our constraints, networks learn distinct sub-networks for each task factor (Fig. \ref{fig:subspace_net_shallow}C). Removing any one of our constraints leads to entangled representations (Fig. \ref{fig:subspace_net_shallow}D). Lastly we note sparsity constraints do not induce disentangling (Appendix \ref{Appendix:sparsity_simulations}). Thus the disentangling effect of ReLUs is not from sparsity, but instead from nonnegativity.
\textbf{Disentangling in supervised deep neural networks}. Training deep non-linear (ReLU) networks on this data also leads to distinct sub-networks, with all layers learning disentangled representations (Fig. \ref{fig:subspace_net_deep}A). However with non-linear data (\( {\bm{x}} \leftarrow {\bm{x}}^3\), \( {\bm{y}} \) remaining the same), the early layers are mixed-selective, whereas the later layers are disentangled (Fig. \ref{fig:subspace_net_deep}B-C).
Understanding why the final hidden layer disentangles is easy, since it linearly projects to the target and so our theory directly applies. By extrapolating our theory, we conjecture that our biological constraints encourage any layer to be as linearly related to task factors and as disentangled as possible. However, early layers cannot be linear in hidden task factors since they are required to perform non-linear computations on the non-linear data, and thus only once activity becomes linearly related to independent task factors in later layers does disentangling set in (as predicted by our linear theory).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.72\linewidth]{figures/SubspaceNet_Deep.pdf}
\end{center}
\caption{\textbf{Deep non-linear networks disentangle.} We train 5-hidden layer nonlinear networks with our constraints on linear and non-linear data.
\textbf{A)} For \textit{linear} data, all layers in the network learn a disentangled representation.
\textbf{B)} For \textit{non-linear} data, only later layers learn a disentangled representations.
\textbf{C)} Example mutual information matrix from the penultimate hidden layer.}
\label{fig:subspace_net_deep}
\end{figure}
\textbf{Disentangling in unsupervised neural networks}. We now consider unsupervised learning, i.e. \( \mathcal{D} = \{ {\bm{x}} \} \), where \( {\bm{x}} \) is a linear mixture of multiple independent task factors as in Theorem 2. Training 0-hidden layer autoencoders on this data, with our biological constraints, recovers the independent task factors in individual neural subspaces (Fig. \ref{fig:subspace_ae}A/B). Moreover, this only occurs when all constraints are present (Fig. \ref{fig:subspace_ae}A). Again, even though our theory applies to the linear setting, the same phenomena occur when training deep non-linear autoencoders on non-linear data; i.e. when \( {\bm{x}} \) is a \textit{non-linear} mixture of multiple i.i.d. random variables, i.e. \( \mathcal{D} = \{ f({\bm{x}}) \} \) (Fig. \ref{fig:subspace_ae}C-D).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\linewidth]{figures/AE.pdf}
\end{center}
\caption{\textbf{Learning data generative factors with autoencoders.}
\textbf{A)} Training linear autoencoders on linear data. Only models with our constraints learn disentangled representations.
\textbf{B)} Example mutual information matrix from a high MIR model.
\textbf{C)} Non-linear autoencoders trained on non-linear data. Only models with our constraints learn disentangled representations.
\textbf{D)} Example mutual information matrix from a high MIR model. All learning curves show mean and standard error from 4 mean from 5 random seeds.}
\label{fig:subspace_ae}
\end{figure}
\textbf{Disentangling on a standard benchmark with VAEs}. We now consider a standard disentangling dataset (Fig. \ref{fig:subspace_vae}A; \citet{kim_disentangling_2018}. To be consistent with, and to compare to, the disentangling literature we use a VAE and measure disentangling with the familiar mutual-information gap (MIG) metric \citep{chen_isolating_2018}. For nonnegativity we ask the mean of the posterior to be nonnegative (via a ReLU) \footnotemark, but we \textit{do not} add a norm constraint as the VAE loss already one in its KL term between the Gaussian posterior and the Gaussian prior.
\footnotetext{Using a nonnegative posterior mean is odd when the prior is Gaussian, but it allows for easier comparison.}
While state-of-the-art results are not our aim (instead we wish to elucidate that simple biological constraints lead to disentangling), our disentangling results (Fig. \ref{fig:subspace_vae}B) are competitive and often better than those in the literature (comparing to results of many models shown in \citet{locatello_challenging_2019}) even though those models explicitly ask for a factorised aggregate posterior. The particular baseline model we show here is \(\beta\)-VAE (Fig. \ref{fig:subspace_vae}D). We see that (1) our constraints lead to disentangling (Fig. \ref{fig:subspace_vae}B-C); (2) including a ReLU improves \(\beta\)-VAE disentangling (as predicted by nonnegativity arguments above, Fig. \ref{fig:subspace_vae}D); and (3) our constraints give results in the Goldilocks region of high disentangling and high reconstruction accuracy (Fig. \ref{fig:subspace_vae}E).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.99\linewidth]{figures/VAE.pdf}
\end{center}
\caption{\textbf{Learning data generative factors with variational autoencoders.}
\textbf{A)} We train on the Shapes3D dataset, with two example images shown. These images have 6 underlying factors.
\textbf{B)} MIG scores are higher with higher weight regularization, and generally higher than any \(\beta\)-VAE (panel D).
\textbf{C)} Mutual information matrix for a high scoring model.
\textbf{D)} \(\beta\)-VAE MIG scores. Adding a ReLU improves MIG scores.
\textbf{E)} MIG score against \(R^2\) shows models with our constraints lie in the Goldilocks region of high disentangling and high reconstruction. All learning curves show mean and standard error from 5 random seeds. Results from an additional dataset in Fig. \ref{fig:subspace_vae_dsprites}}
\label{fig:subspace_vae}
\end{figure}
\section{Disentangling in brains: A theory of cell types}
We next turn our attention to neuroscience, which is indeed the inspiration for our biological constraints. While we hope our general theory of neural representations will be useful for explaining representations across tasks and brain areas, for reasons stated below, we choose our first example from spatial processing in the hippocampal formation. We show our biological constraints lead to separate neural populations (modules) coding for separate task variables, but \textbf{only} when task variables correspond to independent factors of variation. Importantly, the modules consist of distinct functional cell types with similar firing properties, resembling grid \citep{hafting_microstructure_2005} and object-vector cells \citep{hoydal_object-vector_2019} (GCs and OVCs).
We choose to focus on spatial representations for two reasons. Firstly, there is a significant puzzle about why neurons deep in the brain, synaptically far from the sensorimotor periphery, almost miraculously develop single cell representations for human-interpretable factors (e.g. GCs for location in space, and OVCs for relative location to objects). Such observations are not easily accounted for by standard neural network accounts that argue that representations are unlikely to be human-interpretable \citep{richards_deep_2019}. Secondly, whilst these bespoke spatial representations are commonly observed to factorise into single cells, there are situations in which selectivity spans across multiple task variables \citep{boccara_entorhinal_2019,hardcastle_multiplexed_2017}. For example, sometimes spatial firing patterns of GCs are warped by reward \citep{boccara_entorhinal_2019} and sometimes they are not \citep{butler_remembered_2019}. There is no theory for explaining why and when this happens.
\textbf{A factorised task for rodents.} We consider a task in which rodents must know where they are in space, but must also approach one of multiple objects. If objects appear in different places in different contexts, the task is factorised into two independent factors (Fig. \ref{fig:GridOVC}A): `Where am I in allocentric spatial coordinates?' and `Where am I in object-centric coordinates?'. By contrast, if objects always appear in the same locations, the task is not factorised (as spatial location can predict object location). Our theory says that solving this task requires two neural sub-spaces - one for allocentric location and one for location relative to objects - and that these sub-spaces should be represented in separate neural populations when the task is factorised, but not otherwise.
The standard neuroscience finding appears factorised with two distinct modules of non-overlapping cell populations: (1) GCs \citep{hafting_microstructure_2005} which represent allocentric space via hexagonal firing patterns (Fig. \ref{fig:GridOVC}B left); and (2) OVCs \citep{hoydal_object-vector_2019} which represent relative location to objects through firing fields at specific relative distances and orientations (Fig. \ref{fig:GridOVC}B right).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.96\linewidth]{figures/GridOVC.pdf}
\end{center}
\caption{\textbf{Modules of distinct cell types form with nonnegativity and factorised tasks.}
\textbf{A)} Top: Task schematic of an environment with objects that move location in different contexts. Bottom: We train a representation to predict 1) spatial location, 2) object location, and 3) correct action at every location.
\textbf{B)} When rodents navigate such environments, GCs encode location in physical space, while OVCs encode relative location to objects. These plots are ratemaps; the average firing of a given cell at every location.
\textbf{C-D)} Model ratemaps when the representation has a ReLU activation function. We see separate modules of GCs that do not change across tasks, and OVCs that move around to track objects.
\textbf{E)} To quantify `module-ness', we see how much each cell's ratemaps changes between tasks (via a spatial correlation). Some cells don't change (GCs), other cells do (OVCs).
\textbf{F-G)} There are no clear modules without a ReLU activation.
\textbf{H)} As another demonstration of disentangling, we compute the cosine distance between the population's contribution to spatial versus object decoding.}
\label{fig:GridOVC}
\end{figure}
\textbf{Model with additional structural constraint.}
Predicting allocentric spatial locations from egocentric self-motion cues is known as path integration \citep{burak_accurate_2009}, and is believed to be a fundamental function of entorhinal cortex (where GCs and OVCs are found). GCs naturally emerge from training RNNs to path integrate under several additional biological constraints \citep{sorscher_unified_2019,sorscher_unified_2020, banino_vector-based_2018, cueva_emergence_2018}. Hence to model this task (with locations \textit{and} objects) we could train an RNN, \( {\bm{z}} \), that predicts (1) what the spatial location, \( {\bm{x}} \), will be and (2) whether we will encounter an object, after an action, \({\bm{a}}\), from the current location, and (3) what the expected action, \({\bm{a}}\), will be. However, here we adopt a far more general framework that does not limit future applications of our approach simply to sequential integration problems.
In particular, it was recently shown \citep{gao_path_2021, dorrell_actionable_2022} that path integration constraints can be applied directly on the representation by adding a new constraint in the loss imposing
\[
{\bm{z}}({\bm{x}}) = f ( {\bm{W}}_{{\bm{a}}} {\bm{z}}({\bm{x}} - {\bm{a}}) ).
\label{eq:structconstr}
\]
Here \(f(\cdot)\) is an activation function and \( {\bm{W}}_{{\bm{a}}} \) is a weight matrix that depends on the action \( {\bm{a}} \). This surrogate constraint imposes potential path integration by ensuring that a motion \({\bm{a}} \) in space \({\bm{x}} \) imposes a lawful change in neural representation \({\bm{z}} \), thereby transforming the sequential path integration problem into the problem of directly estimating neural representations of space. Thus we minimise:
\[
\mathcal{L} = \\
\underbrace{\mathcal{L}_{\text{nonneg}} + \mathcal{L}_{\text{activity}} + \mathcal{L}_{\text{weight}}}_{\text{Biological constraints}} + \\
\underbrace{\mathcal{L}_{\text{location}} + \mathcal{L}_{\text{actions}} + \mathcal{L}_{\text{objects}}}_{\text{Functional constraints}} + \\
\underbrace{\mathcal{L}_{\text{path integration}}}_{\text{Structural constraints}}
\]
These are the same biological constraints as above, but now the functional constraints involve predicting location, object, and action, and an additional structural constraint imposes \eqref{eq:structconstr}. Interestingly, the structural constraint leads to a pattern forming optimisation dynamics (see Appendix \ref{Appendix:Cell_type_optimisation} for mathematical details).
\textbf{Modules of distinct cell types when tasks are factorised and representations are nonnegative.} Just as our theory predicts, when training on tasks where objects and space are factorised (i.e. objects can be anywhere in space), under our biological constraints of nonnegativity and energy efficiency, distinct neural modules emerge, each selective for a single task factor (Fig. \ref{fig:GridOVC}C-D). We see GC-like neurons that consistently represent space independent of object locations, and OVC-like neurons that recenter their representations around the moving objects or are inactive if no objects are present (further cells are shown in Fig. \ref{fig:ExtraCells}). To quantify whether the population really has two distinct cell types (two modules) we analyse the consistency of a cell's representation by taking its average spatial correlation between many different object configurations (Fig. \ref{fig:GridOVC}E); some cells change a lot (i.e. OVCs) some cells don't change (i.e. GCs). Thus the representation has two disentangled modules. Without the nonnegativity constraint, the neural representation is entangled with mixed-selective neurons and no clear distinction between cells that change and those that don't (Fig. \ref{fig:GridOVC}F-G).
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.96\linewidth]{figures/Warping.pdf}
\end{center}
\caption{\textbf{Entangled tasks lead to entangled representations and grid cell warping.} We show a representative selection of cells from a model with \textbf{A)} a factorised task and \textbf{B)} an entangled task. The phases of the firing fields in the entangled task are locked to object location - they have warped their firing fields. This is not the case for the factorised task (asides from object specific cells).
\textbf{C)} To quantify phase locking for each of the task/model variant, we compute the average spatial correlation of patches around objects. Only the entangled task shows high correlation, i.e. the cell representations have warped around objects. Each point is a model trained from a random seed.}
\label{fig:Warping}
\end{figure}
\textbf{Grid cell warping and mixed-selectivity when tasks are entangled.} Experimental results show GCs sometimes warp their firing fields towards rewarded locations \citep{boccara_entorhinal_2019} and sometimes don't \citep{butler_remembered_2019}. Intriguingly, in the warping situation, the rodents exhibited stereotyped behaviour around consistent rewards. We now explain these neuroscience observations as a consequence of space becoming entangled with objects/rewards.
Modelling factorised versus entangled tasks (objects changing locations versus always staying in the same locations), produces very different GC behaviours. In the factorised case, grid fields are unrelated to objects (Fig. \ref{fig:Warping}A), whereas in the entangled task grid fields warp to objects (Fig. \ref{fig:Warping}B). We quantify this by measuring the average spatial correlation of patches around each object. Only when the task is entangled are fields consistently warped towards objects (Fig. \ref{fig:Warping}C). Thus we have an explanation for GC warping; they warp when behaviour becomes stereotyped around consistently placed objects/rewards.
\section{Discussion}
We have proven that simple biological constraints like nonnegativity and energy efficiency lead to disentangling, and empirically verified this in machine learning and neuroscience tasks, leading to a new understanding of functional cell types. We now consider some more neuroscience implications.
\textbf{Representing categories.} Our theory additionally says that representations of individual categories should be encoded in separate neural populations (disentangled; see Appendix \ref{Appendix:Categorical} \footnotemark.). This provides a potential explanation for "grandmother cells" that selectively represent specific concepts or categories \citep{quiroga_invariant_2005}, and have long puzzled proponents of distributed representations. It further potentially explains situations where animals have been trained on multiple tasks, and different neurons are found to engage in each task \citep{rainer_selective_1998, roy_prefrontal_2010, asaad_task-specific_2000, lee_task_2022, flesch_orthogonal_2022}.
\footnotetext{We note this phenomena could also be accounted for with a sparsity constraint.}
\textbf{When to disentangle?} Our theory speaks to situations in which brains or networks must generalise to new combinations of learnt factors. In this situation, if the input-output (or input-latent-output) transformations are linear, biological constraints will cause complete disentangling of the networks. When the mappings are nonlinear, we show empirically that mixed selectivity exists, but gradually de-mixes as layers approach the output (in supervised), or latent (in unsupervised) network layers.
Optimising for low firing contrasts with previous ideas which instead optimise for linear read-out. This latter situation is akin to kernel regression, where mixed selectivity through random expansion increases the dimensionality of neural representations, allowing simple linear read-outs in the high dimensional space to perform arbitrary nonlinear operations on the original low dimensional space.
\textbf{Mixed-selective cells in the brain.} Mixed selectivity does clearly exist in the brain. For example, Kenyon cells in the Drosophila mushroom body increase the dimensionality of their inputs by an order of magnitude by close to random projections \citep{aso_neuronal_2014}. This may allow linear read-out to behaviour via simple dopamine gating. Similarly rodent hippocampal cells encode conjunctions of spatial and sensory variables to allow rapid formation of new memories \citep{komorowski_robust_2009}]. More recently it has been suggested that PFC neurons have this same property, for the same reason \citep{rigotti_importance_2013}. However, it is less clear that this is a general property of representations in associative cortex (including PFC), which can separate into different neuronal representations of different interpretable factors \citep{hirokawa_frontal_2019, bernardi_geometry_2020} or tasks \citep{lee_task_2022, flesch_orthogonal_2022}.
One possibility is that in overtrained situations with only a relatively small number of categories or trial-types (where mixed selectivity has been observed), the task can effectively be solved by categorising the current trial into one of a few previous experiences. By contrast in tasks where combinatorial generalisation is required, the factored solution may be preferred.
\textbf{A program to understand how brain representations structure themselves.} This work is one piece of the puzzle. It tells us when neural circuits systems should represent different factors in different neurons. It does not tell us, however, how each factor itself should be represented. For example it does not tell us why GCs and OVCs look the ways they do. We believe that the same principles of nonnegativity, minimising neural activity, and representing structure, will be essential components obtaining this more general understanding.
Indeed in a companion paper, we use the same constraints, along with formalising structure/path-integration using group and representations theory, to mathematically understand why grid cells look like grid cells \citep{dorrell_actionable_2022}.
Similarly, our current understanding is limited to the optimal solution for factorised representations, but we anticipate similar ideas will be applicable to neural dynamics \citep{driscoll_flexible_2022}.
\section{Conclusion}
We introduced constraints inspired by biological neurons - nonnegativity and energy efficiency (w.r.t. either activity or weights) - and proved these constraints lead to linear factorised codes being disentangled. We empirically verified this in simulation, and showed the same constraints lead to disentangling with both non-linear data and nonlinear networks. We even achieve competitive disentangling scores on a baseline disentangling task, even though this was not our specific aim. We showed these biological constraints explain why neuroscientists observe bespoke cell types, e.g. GCs \citep{hafting_microstructure_2005}, OVCs \citep{hoydal_object-vector_2019}, border vector cells \citep{solstad_representation_2008, lever_boundary_2009}, since space, boundaries, and objects appear in a factorised form (i.e. occur in any independent combination), and so are optimally represented by different neural populations for each factor. These same principles explain why neurons in inferior temporal cortex are axis aligned to underlying factors of variation that generate the data they represent \citep{chang_code_2017, bao_map_2020, higgins_unsupervised_2021}, why visual cortex neurons are decorrelated \citep{ecker_decorrelated_2010}, or why neurons in parietal cortex only selective for specific tasks \citep{lee_task_2022}. Lastly, we also explained the confusing finding of grid fields warping towards rewards \citep{boccara_entorhinal_2019} as the space and rewards becoming entangled.
This work bridges the gap between single neuron and population responses, and offers an understanding of properties of neural representations in terms of task structure above and beyond just dimensionality. Additionally it demonstrates the utility of neurobiological considerations in designing machine learning algorithms. Overall, we hope this work demonstrates the promise of a unified research program that more deeply connects the neuroscience and machine learning communities to help in their combined quest to both understand and learn neural representations. Such a unified approach spanning brains and machines could help both sides, offering neuroscientists a deeper understanding of how cortical representations structure themselves, and offering machine learners novel ways to control and understand the representations their machines learn.
\section*{Acknowledgements}
We thank Emile Mathieu and Andrew Saxe for helpful comments and advice on our manuscript. We thank the following funding sources: Sir Henry Wellcome Post-doctoral Fellowship (222817/Z/21/Z) to J.C.R.W.; the Gatsby Charitable Foundation to W.D.; the James S. McDonnell, Simons Foundations, NTT Research, and an NSF CAREER Award to S.G.; Wellcome Principal Research Fellowship (219525/Z/19/Z), Wellcome Collaborator award (214314/Z/18/Z), and Jean-François and Marie-Laure de Clermont-Tonnerre Foundation award (JSMF220020372) to T.E.J.B.; the Wellcome Centre for Integrative Neuroimaging and Wellcome Centre for Human Neuroimaging are each supported by core funding from the Wellcome Trust (203139/Z/16/Z, 203147/Z/16/Z).
\clearpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,864 |
Sophie L Morgan
Sophie is an award-winning disability advocate, inclusion consultant and one of the first, and only, female disabled television hosts in the world.
Sophie is determined to channel her adversity into opportunity and she sees her challenges as a unique chance for creativity, she has become the ultimate agent for change, voted in the Top 10 most influential people with a disability in the UK three years in a row.
She is best known for her role as one of the lead presenters of the Paralympic Games (Channel 4) for nearly a decade. In addition to sports broadcasting, she fronts her own travel series (Living Wild), consumer and current affairs documentaries (Dispatches & Unreported World) and is a well-known television personality (ITV's 'Loose Women').
Sophie is Global Ambassador for brands such as Toyota and Can-Am, and has consulted for international companies such as Target, Virgin Media, Air BnB, John Lewis and more. She is also patron of several charities (Scope, Back Up), Global Ambassador for Women's Rights and Inclusive Education for Leonard Cheshire and on the Special Advisory Committee for Human Rights Watch, on behalf of whom she spoke at the United Nations.
Books by Sophie L Morgan
Driving Forwards: A journey of resilience and empowerment after life-changing injury | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 958 |
Спољашње везе
Списак улица Глогоња
Улице у Панчеву | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,521 |
Q: How to make GridLayout row fill availible space? I have a GridLayout. I have two rows. I want one row to take up 500 pixels. I want the other row to take up the rest of the space.
How can I do this?
A: That's not a property of the GridLayout, but of the contained widgets' layout data. Say your composite contains two widgets:
parent.setLayout( new GridLayout() );
Button upper = new Button( parent, SWT.PUSH );
GridData upperData = new GridData( SWT.FILL, SWT.TOP, true, false );
upperData.heightHint = 500;
upper.setLayoutData( upperData );
Button lower = new Button( parent, SWT.PUSH );
GridData lowerData = new GridData( SWT.FILL, SWT.FILL, true, true );
lower.setLayoutData( lowerData );
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,261 |
'use strict';
/**
* BatchAI properties
*
*/
class BatchAIProperties {
/**
* Create a BatchAIProperties.
* @member {string} [vmSize] Virtual Machine Size
* @member {string} [vmPriority] Virtual Machine priority
* @member {object} [scaleSettings] Scale settings for BatchAI
* @member {number} [scaleSettings.maxNodeCount] Max number of nodes to use
* @member {number} [scaleSettings.minNodeCount] Min number of nodes to use
* @member {boolean} [scaleSettings.autoScaleEnabled] Enable or disable auto
* scale
*/
constructor() {
}
/**
* Defines the metadata of BatchAIProperties
*
* @returns {object} metadata of BatchAIProperties
*
*/
mapper() {
return {
required: false,
serializedName: 'BatchAI_properties',
type: {
name: 'Composite',
className: 'BatchAIProperties',
modelProperties: {
vmSize: {
required: false,
serializedName: 'vmSize',
type: {
name: 'String'
}
},
vmPriority: {
required: false,
serializedName: 'vmPriority',
type: {
name: 'String'
}
},
scaleSettings: {
required: false,
serializedName: 'scaleSettings',
type: {
name: 'Composite',
className: 'ScaleSettings'
}
}
}
}
};
}
}
module.exports = BatchAIProperties;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,407 |
Media & Telecoms
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,407 |
package com.google.android.exoplayer2.drm;
import android.annotation.TargetApi;
import android.net.Uri;
import android.text.TextUtils;
import com.google.android.exoplayer2.C;
import com.google.android.exoplayer2.drm.ExoMediaDrm.KeyRequest;
import com.google.android.exoplayer2.drm.ExoMediaDrm.ProvisionRequest;
import com.google.android.exoplayer2.upstream.DataSourceInputStream;
import com.google.android.exoplayer2.upstream.DataSpec;
import com.google.android.exoplayer2.upstream.HttpDataSource;
import com.google.android.exoplayer2.upstream.HttpDataSource.InvalidResponseCodeException;
import com.google.android.exoplayer2.util.Assertions;
import com.google.android.exoplayer2.util.Util;
import java.io.IOException;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.UUID;
/**
* A {@link MediaDrmCallback} that makes requests using {@link HttpDataSource} instances.
*/
@TargetApi(18)
public final class HttpMediaDrmCallback implements MediaDrmCallback {
private static final int MAX_MANUAL_REDIRECTS = 5;
private final HttpDataSource.Factory dataSourceFactory;
private final String defaultLicenseUrl;
private final boolean forceDefaultLicenseUrl;
private final Map<String, String> keyRequestProperties;
/**
* @param defaultLicenseUrl The default license URL. Used for key requests that do not specify
* their own license URL.
* @param dataSourceFactory A factory from which to obtain {@link HttpDataSource} instances.
*/
public HttpMediaDrmCallback(String defaultLicenseUrl, HttpDataSource.Factory dataSourceFactory) {
this(defaultLicenseUrl, false, dataSourceFactory);
}
/**
* @param defaultLicenseUrl The default license URL. Used for key requests that do not specify
* their own license URL, or for all key requests if {@code forceDefaultLicenseUrl} is
* set to true.
* @param forceDefaultLicenseUrl Whether to use {@code defaultLicenseUrl} for key requests that
* include their own license URL.
* @param dataSourceFactory A factory from which to obtain {@link HttpDataSource} instances.
*/
public HttpMediaDrmCallback(String defaultLicenseUrl, boolean forceDefaultLicenseUrl,
HttpDataSource.Factory dataSourceFactory) {
this.dataSourceFactory = dataSourceFactory;
this.defaultLicenseUrl = defaultLicenseUrl;
this.forceDefaultLicenseUrl = forceDefaultLicenseUrl;
this.keyRequestProperties = new HashMap<>();
}
/**
* Sets a header for key requests made by the callback.
*
* @param name The name of the header field.
* @param value The value of the field.
*/
public void setKeyRequestProperty(String name, String value) {
Assertions.checkNotNull(name);
Assertions.checkNotNull(value);
synchronized (keyRequestProperties) {
keyRequestProperties.put(name, value);
}
}
/**
* Clears a header for key requests made by the callback.
*
* @param name The name of the header field.
*/
public void clearKeyRequestProperty(String name) {
Assertions.checkNotNull(name);
synchronized (keyRequestProperties) {
keyRequestProperties.remove(name);
}
}
/**
* Clears all headers for key requests made by the callback.
*/
public void clearAllKeyRequestProperties() {
synchronized (keyRequestProperties) {
keyRequestProperties.clear();
}
}
@Override
public byte[] executeProvisionRequest(UUID uuid, ProvisionRequest request) throws IOException {
String url = request.getDefaultUrl() + "&signedRequest=" + new String(request.getData());
return executePost(dataSourceFactory, url, new byte[0], null);
}
@Override
public byte[] executeKeyRequest(UUID uuid, KeyRequest request) throws Exception {
String url = request.getDefaultUrl();
if (forceDefaultLicenseUrl || TextUtils.isEmpty(url)) {
url = defaultLicenseUrl;
}
Map<String, String> requestProperties = new HashMap<>();
// Add standard request properties for supported schemes.
String contentType = C.PLAYREADY_UUID.equals(uuid) ? "text/xml"
: (C.CLEARKEY_UUID.equals(uuid) ? "application/json" : "application/octet-stream");
requestProperties.put("Content-Type", contentType);
if (C.PLAYREADY_UUID.equals(uuid)) {
requestProperties.put("SOAPAction",
"http://schemas.microsoft.com/DRM/2007/03/protocols/AcquireLicense");
}
// Add additional request properties.
synchronized (keyRequestProperties) {
requestProperties.putAll(keyRequestProperties);
}
return executePost(dataSourceFactory, url, request.getData(), requestProperties);
}
private static byte[] executePost(HttpDataSource.Factory dataSourceFactory, String url,
byte[] data, Map<String, String> requestProperties) throws IOException {
HttpDataSource dataSource = dataSourceFactory.createDataSource();
if (requestProperties != null) {
for (Map.Entry<String, String> requestProperty : requestProperties.entrySet()) {
dataSource.setRequestProperty(requestProperty.getKey(), requestProperty.getValue());
}
}
int manualRedirectCount = 0;
while (true) {
DataSpec dataSpec =
new DataSpec(
Uri.parse(url),
data,
/* absoluteStreamPosition= */ 0,
/* position= */ 0,
/* length= */ C.LENGTH_UNSET,
/* key= */ null,
DataSpec.FLAG_ALLOW_GZIP);
DataSourceInputStream inputStream = new DataSourceInputStream(dataSource, dataSpec);
try {
return Util.toByteArray(inputStream);
} catch (InvalidResponseCodeException e) {
// For POST requests, the underlying network stack will not normally follow 307 or 308
// redirects automatically. Do so manually here.
boolean manuallyRedirect =
(e.responseCode == 307 || e.responseCode == 308)
&& manualRedirectCount++ < MAX_MANUAL_REDIRECTS;
url = manuallyRedirect ? getRedirectUrl(e) : null;
if (url == null) {
throw e;
}
} finally {
Util.closeQuietly(inputStream);
}
}
}
private static String getRedirectUrl(InvalidResponseCodeException exception) {
Map<String, List<String>> headerFields = exception.headerFields;
if (headerFields != null) {
List<String> locationHeaders = headerFields.get("Location");
if (locationHeaders != null && !locationHeaders.isEmpty()) {
return locationHeaders.get(0);
}
}
return null;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 765 |
During our previous Kickstarter campaign for Mysthea, we received a ton of requests from our backers wondering if there was a chance to tell the story of the Volarees Guild's Dragon and this is exactly what we accomplished.
Now we are proud to show you how we developed the dreadful Volfyirion!
At first we thought Volfyirion should be represented with its head up right, to better showcase its design and shape. Yet, after working on it a bit, we decided that it wasn't dynamic enough for a dreadful dragon. We tried to move the head aside, but it didn't enhance all the features of the dragon.
The dragon design was solid, so we tried to work more on the pose. At first we tried to point Volfyirion's face upwards with all its tail down on the ground and its claws up in the air. We felt it was cool, yet it wasn't much of a dragon posture. Lastly, we tried to take advantage of its peculiar shape and we developed the last pose on the right of the second sketch.
We gathered a lot of feedback and we understood that this pose was really appreciated. We believe that Volfyirion with its tail wrapped around its body and the menacing look is the one that represents it better and we decided to go down this route. This pose showcases all its might and how dangerous it would be to face it.
The miniature is really big in size, precisely 120mm tall and presents all the majestic details you would expect from a fearsome dragon.
The miniature is also available as a standalone miniature separated from the card game. We decided to create a whole new set of rules for Mysthea, making a mini expansion that can be bought by itself if backers just want to add the dragon to their collection. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,336 |
Q: How do I differentiate between two WhatsApp messages? First, how did I read WhatsApp messages from notifications?
Well, I researched and have been able to implement a NotificationListenerService with proper permissions(no underhanded strategies whatsoever) and listen to notifications messages from WhatsApp. Credit goes to this guy for giving me this idea.
Anyway, I am able to read messages now but the problem is that WhatsApp apparently sends out the same message multiple times sometimes but doesn't display multiple notifications for it. How do I know?
I logged the notification received by my listener and found several identical messages. The general structure of notification looks something like this:
StatusBarNotification.toString():
> 10-18 23:43:24.236 16159-16181/com.company D/WhatsApNotifListService:
> StatusBarNotification(pkg=com.whatsapp user=UserHandle{0} id=1
> tag=null score=0 key=0|com.whatsapp|1|null|10170: Notification(pri=0
> contentView=com.whatsapp/0x1090078 vibrate=null sound=null
> defaults=0x0 flags=0x200 color=0xffe65100 category=msg
> groupKey=group_key_messages vis=PRIVATE
> publicVersion=Notification(pri=0 contentView=com.whatsapp/0x1090078
> vibrate=null sound=null defaults=0x0 flags=0x0 color=0xffe65100
> category=msg vis=PRIVATE)))
Notification.extras(A Bundle) structure:
> 10-18 23:42:46.199 16159-16211/com.company D/Util: android.title
> Manish (java.lang.String)
>
> 10-18 23:42:46.199 16159-16211/com.company D/Util: android.subText
> null (null)
>
> 10-18 23:42:46.201 16159-16211/com.company D/Util:
> android.car.EXTENSIONS Bundle[mParcelledData.dataSize=1852]
> (android.os.Bundle)
>
> 10-18 23:42:46.201 16159-16211/com.company D/Util:
> android.showChronometer false (java.lang.Boolean)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util: android.icon
> 2130840435 (java.lang.Integer)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util: android.text Bdbdjd
> (java.lang.String)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util: android.progress 0
> (java.lang.Integer)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util: android.progressMax
> 0 (java.lang.Integer)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util: android.showWhen
> true (java.lang.Boolean)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util:
> android.rebuild.applicationInfo ApplicationInfo{1a7615bc com.whatsapp}
> (android.content.pm.ApplicationInfo)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util: android.largeIcon
> android.graphics.Bitmap@a4eb945 (android.graphics.Bitmap)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util: android.infoText
> null (null)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util:
> android.wearable.EXTENSIONS Bundle[mParcelledData.dataSize=668]
> (android.os.Bundle)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util:
> android.originatingUserId 0 (java.lang.Integer)
>
> 10-18 23:42:46.202 16159-16211/com.company D/Util:
> android.progressIndeterminate false (java.lang.Boolean)
Is there any way like Id or something that can be extracted to differentiate two messages?
A: I can't believe I solved my problem. The timestamp was the answer. Credit goes to @xenolion for suggesting time. My mind had preconceived notion about time being different but I just checked the timestamp of the messages and the identical messages have the same timestamp.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,081 |
Jeff Francis Adjusting To New Role
CBN's Melissa Couto talks Jeff Francis and his switch to the bullpen, Justin Morneau and his RBI milestone, Brett Lawrie's favourite ballparks, Travis Snider's Wednesday night pitching debut, and more in this week's ThrowinSmoke column
For a starting pitcher, being relegated to the bullpen can be taken as a slap in the face.
Jeff Francis, however, doesn't see it that way at all.
Designated for assignment by the Cincinnati Reds after his first and only start of the season last month, the left-hander from North Delta, B.C., was picked up two days later by the Oakland Athletics, who intended on using him as a long reliever.
On a recent trip to Toronto, Francis said he didn't mind the move -- and he wasn't using his new role as a stepping stone toward earning a spot in the starting rotation.
"The aim is to just give the team innings," Francis said after making his first appearance of the season for Oakland -- one scoreless frame in the A's 5-2 loss to the Blue Jays. "There's no goal other than to go out there and get outs. There's a role for me to fill here and I'm happy to do that."
Since his Rogers Centre appearance on May 24, Francis has pitched 6 2/3 innings in four outings for Oakland, giving up seven runs, nine hits and two walks, while striking out four.
The 33-year-old was drafted in the first round (ninth overall) by the Colorado Rockies in 2002, and won 17 games for them in 2007, helping lead the Rockies to the World Series that year. The following season, Francis pitched through shoulder soreness and finished with a 4-10 record and a 5.01 ERA before undergoing surgery that sidelined him for all of 2009.
Since 2010, Francis has gone 19-36 through 104 games (87 starts), with stints in the minor leagues along the way.
Though he's pitched out of the bullpen before, the Canadian says he hasn't quite gotten used to the difference in preparation.
"I'm still trying to figure that one out," Francis said. "It's not really straightforward. You just have to be ready at all times and you have to do what you can to help your team.
"Even if you go out there and pitch one day, they could need you the next day. That's different, but it's not something I haven't done before. I'm still learning the ropes though, that's for sure."
GET TO KNOW: BLUE JAYS 3B BRETT LAWRIE (Langley B.C.)
My favourite ballpark to play in: "Wow, I've been to a few really cool ballparks and they all have their distinct thing, so it's hard to pick. [Lawrie-headshot] I've always liked going to Yankee Stadium, but I played in Pittsburgh this year for the first time and that was awesome.
"Pittsburgh, the background setting and whatnot, it's a beautiful ballpark and it's really cool to play in front of that. For the scenery part, I like Pittsburgh, but baseball-wise and the heroics of it all, I'd say Yankee Stadium, Fenway Park, places like that. At Fenway, you walk down the same tunnel as all the greats from the 1900s and 1800s, so there's different histories with those parks and I like that."
The strangest thing I've ever seen on the field: "Oh wow, I've seen some crazy things. There have been crazy plays in the field, streakers, fans coming on to the field. You see great plays every day that you think 'I can't believe he made that catch,' or whatever it is, but these guys we play against are the best at what they do. You just hope you're not the one with the bat when stuff like that happens."
BLUE JAYS NOTES: Toronto has yet to sign its two first round draft picks, RHP Jeff Hoffman and C Max Pentecost, though general manager Alex Anthopoulos says he's confident deals with both players will be reached shortly. The Jays failed to sign their first round pick in two of the last three drafts. ... CF Colby Rasmus was back in the Blue Jays lineup on Wednesday for the first time in over a month. Rasmus was placed on the 15-day DL with right hamstring tightness on May 13. He was 1-for-4 with an RBI in the 7-3 loss.
CANADIANS IN THE MAJOR LEAGUES
After a slight cold streak last week, Colorado first baseman Justin Morneau has found his stroke again. [Justin Morneau]
The New Westminster, B.C., native is hitting .381 in his last seven games, with 11 hits -- including three doubles and a home run -- and eight RBIs. He had four multi-hit games in a row from June 11-14.
As the Canadian Baseball Network's Neil Munro points out, Morneau surpassed Fredericton's Matt Stairs on the all-time Canadian RBI list by driving in his 900th career run (Stairs had 899) in Colorado's 8-2 win over the Atlanta Braves last Wednesday. Since then, the 33-year-old has picked up seven more RBIs, bringing his total to 907.
Larry Walker of Maple Ridge, B.C., leads all Canadians in that category with 1,311 RBIs over 17 major league seasons.
NOTES: Seattle LHP James Paxton (Ladner, B.C.) is still rehabbing the strained left lat muscle he suffered in early April. The Mariners expect to have him back around the all-star break. ... Former Blue Jays OF Travis Snider pitched the ninth inning of Pittsburgh's game against Cincinnati on Wednesday, giving up two runs but striking out Reds 1B Joey Votto (Etobicoke, Ont.). Votto is hitting .320 with five RBIs in his last six games.
CANADIANS IN THE MINOR LEAGUES
The Vancouver Canadians have begun their quest for a four-peat.
Winners of three straight Northwest League championships, the short season-A affiliate of the Blue Jays had their home opener against the Spokane Indians Wednesday at Nat Bailey Stadium. The Canadians fell 5-2, evening their record at 3-3 after beginning the year on a five-game road trip.
Right-handed pitcher Andrew Case of Saint John, N.B., is the lone Canadian on Vancouver's roster. Case was signed by the Blue Jays last September, following his stellar performance at Tournament 12 -- where he pitched a no-hitter for the Maritimes team in the semifinal of the Toronto showcase.
The 21-year-old Case made his professional debut June 13 against the Salem-Keizer Volcanoes in Keizer, Ore., walking two batters and striking out one through two scoreless innings. He had been in extended spring training prior to the start of the Northwest League season.
NOTES: C/RF Rowan Wick (North Vancouver, B.C.) is the Canadian Baseball Network minor league Player of the Week after hitting .529 for the class-A State College Spikes through the first five games of the New York-Penn League season.
WHEN HOCKEY MEETS CANADIAN BASEBALL
Members of the NHL champion Los Angeles Kings made a couple of Canadian baseball players smile when they brought the Stanley Cup to Dodger Stadium prior to L.A.'s game against the Colorado Rockies on Tuesday.
Dodgers outfielder Jamie Romak (London, Ont.) and Rockies first baseman Morneau had their photos taken with the famous trophy.
-- Follow Melissa Couto on Twitter @ThrowinSmoke
Melissa Couto June 20, 2014 Comment
Drew Hutchison A Keen Student
Canadians in the MajorsMelissa Couto June 28, 2014
Taveras Has Canadian Past
Melissa Couto June 13, 2014 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,914 |
\section{Introduction}
Orbital-free density functional theory (OFDFT) is a lucrative path to linear scaling DFT methods. The modern OFDFT theory is mostly investigated in the setting of non-local or generalized gradient approximation (GGA)-style kinetic functionals. Potential functionals, such as those used in Berthold-Georg Englert and Julian Schwinger's statistical atom model \cite{statistical_atom_1,statistical_atom_2,statistical_atom_3}, have fallen out of favor. Yet, it has been shown that at least GGA-style orbital-free functionals usually suffer from a theoretical flaw: the atomic nucleus is not well described, as evidenced by the singular Pauli potential \cite{pauli_potential_properties,pauli_potential_functionals,ofdft_issues}. The Pauli potential describes a positive effective repulsion arising from the Pauli exclusion principle, and its violation implies a fundamentally wrong solution of the electronic problem. Potential functionals allow linear-scaling DFT to sidestep this flaw and our intention here is to quantify the potential functionals' self-consistent accuracy by benchmarking Englert and Schwinger's model \cite{statistical_atom_2,statistical_atom_3} to more well-known OFDFT models.
A flurry of recent activity uses potential as a variable instead of density. Much of the work is focused especially on the Pauli potentials' role and especially in atoms \cite{pauli_potential_differential_equation,pauli_potential_functionals,Finzel2016}. The potential variable has even been combined with machine learning\cite{potential_machine_learning}, which also has had some applications for the search of kinetic energy functionals \cite{ofdft_neural_networks,ofdft_machine_learning_functionals}.
Pseudopotentials \cite{ofdft_pseudopotential,ofdft_issues} have been useful in conventional approaches to extended systems with OFDFT, where an active effort to improve local pseudopotentials' transferability is ongoing \cite{LPP_EA,LPP_OEP}. While this solution works remarkably well \cite{ofdft_pseudopotential}, a few questions remain for systematic improvement. In general, how much of the overall accuracy is because of the better ion-energy interaction description or the better electronic kinetic energy functional? Even more importantly, in the case of fitted pseudopotentials, when is a pseudopotential overfitted compared to the kinetic energy functional? One approach is to investigate the full potential solution to OFDFT equations, as we did by using the projector-augmented-wave method\cite{ofdft_paw}, but one of the problems here is the near-nucleus singularity of the Pauli potential, which brings the solution's plausibility into question.
The problem near the atomic nucleus is not a new one, and it motivated the development of the statistical atom model \cite{statistical_atom_1}. Englert and Schwinger's approach, however, is based on the effective potential instead of the density, which allows natural energy scale separation of the problematic core electrons from the bulk. In particular, we are interested in the model proposed in two papers \cite{statistical_atom_2,statistical_atom_3} and later refined in Englerts' book, {\it Semiclassical Theory of Atoms} \cite{semiclassical_atom}.
The perturbative results from the statistical atom are of amazing accuracy, but because our focus is on extended systems, self-consistent solutions of atoms are more interesting to us. We therefore explore a self-consistent solution to Englert and Schwinger's model and assess its quality by comparing it to two other representative orbital-free models that belong to the Thomas-Fermi-Dirac (TFD)-$\lambda$vW family of orbital-free models. Results show that the ES model generally offers the same accuracy as the well-known TFD-$\frac{1}{5}$vW model, but it works better for few quantities, especially for the Pauli potential. We also discerned that the model's current limitation is the inability to model atoms with a low atomic number (<12).
\section{The Model}
We briefly present the potential functional formalism in DFT and show how it can be used to correct the Thomas-Fermi description of atoms.
\subsection{Formalism}
We start with the usual DFT formalism, where energy is separated as
\begin{align}
\label{eq:dft_functional}
E[n] = E_{\text{kin}}[n] + E_\text{int}[n] + \int d\mathbf{r} V_\text{ext}(\mathbf{r})n(\mathbf{r}),
\end{align}
where $E_\text{int}$ is the functional containing electron–electron interaction. We enforce the particle number $N$ restriction via Lagrange multiplier $\zeta$ and apply the Legendre transformation to the kinetic energy
\begin{align*}
E_1[V + \zeta] = E_\text{kin}[n] + \int d\mathbf{r} \left( V(\mathbf{r}) + \zeta \right)n(\mathbf{r}),
\end{align*}
where we introduced an effective single particle potential $V$.
By incorporating this into functional \eqref{eq:dft_functional}, we arrive at a joint functional where the variables $n, V$ and $\zeta$ are treated as independent variables \cite{leading_gradient_correction_2d}
\begin{align*}
E[V,n,\zeta] = E_1[V + \zeta]
- \int (d\mathbf{r})\left[ V(\mathbf{r}) - V_\text{ext} \right]n(\mathbf{r})
+ E_\text{int}[n] - \zeta N.
\end{align*}
The variations of this functional lead us to the relations
\begin{align}
\label{eq:variation_n}
n(\mathbf{r}) &= \frac{\delta}{\delta V}E_1[V + \zeta], \\
\label{eq:variation_N}
N &= \frac{\delta}{\delta \zeta} E_1[V + \zeta] = \int d\mathbf{r} n(\mathbf{r}), \\
\label{eq:variation_V}
V &= V_\text{ext} + \frac{\delta}{\delta n} E_\text{int}[n] = V_\text{ext} + V_\text{int},
\end{align}
where $V_\text{int}$ is the single particle interaction potential. Here we use the binding energy $\zeta$ instead of the more common chemical potential $\mu = -\zeta$. The energy $E_1[V + \zeta]$ is possible to approximate semiclassically \cite{1Airy-averaged_corrections_2d,semiclassical_atom}.
We emphasize here that working with this joint functional is still working within the DFT framework given by the Hohenberg-Kohn theorems, although in this formalism we acknowledge effective potential and chemical potential as independent variables. For simple cases like the Thomas-Fermi theory, it is possible to reduce the potential functional version of $E_1[V + \zeta]$ to density functionals and vice versa \cite{semiclassical_atom}. Because we are working with a single particle potential and $E_1$ will be approximated with the help of a single particle potential, the kinetic energy in this model is non-interacting kinetic energy, which Kohn-Sham and other OFDFT schemes also use.
\subsection{Semiclassical $E_1$}
Englert and Schwinger derived a semiclassical Airy-average expression for $E_1$; this can be understood as a Thomas-Fermi expression that contains quantum corrections. The semiclassical expression is given in terms of Airy-average functions $F_m$, which are closely related to Airy functions.
The derivation is based on the approximation that each electron moves in a local harmonic potential. Then this is expanded in terms of $\mathbf{r}, \mathbf{p}$ commutators to a first order (Thomas-Fermi corresponds to a zeroth-order approximation; i.e., the position and momentum commute). \cite{semiclassical_atom}.
The main quantity in the quantum corrected theory is
\begin{align*}
y = 2(V + \zeta)|2\nabla V|^{-2/3},
\end{align*}
and the Airy-average functions are functions of this variable
\begin{align*}
F_0(y) &= \left[\text{Ai}(y)\right]^2, \\
F_1(y) &= -y\left[\text{Ai}(y)\right]^2 + \left[\text{Ai}'(y)\right]^2, \\
F_{-1}(y) &= -2\text{Ai}(y)\text{Ai}'(y).
\end{align*}
We can obtain the remaining Airy-average functions recursively with the rule
\begin{align*}
(m - \frac{1}{2})F_m(y) = \frac{1}{4}F_{m-3}(y) - yF_{m-1}(y).
\end{align*}
The quantum-corrected Thomas-Fermi expression for $E_1[V + \zeta]$ in terms of Airy-average functions is
\begin{align}
\label{eq:es_energy}
E_1[V + \zeta] = -\frac{1}{4\pi}\int d\mathbf{r}\left[ |2\nabla V|^{5/3}F_3(y) - \frac{1}{3}\nabla^2 V |2\nabla V|^{1/3} F_1(y) \right],
\end{align}
with approximations to the second order in potential $O(\nabla^2 V)$. Reducing this to a kinetic energy functional \cite{semiclassical_atom} results in a Thomas-Fermi kinetic energy functional plus one-ninth of the von Weizsäcker kinetic energy functional, which is more widely known as the gradient-corrected Thomas-Fermi.
\subsection{Partition of $E_1[V + \zeta]$}
We note that the semiclassical evaluation is not valid near the atomic nucleus, which is why it is reasonable to split the evaluation of energy into two parts: Electrons described well by the Thomas-Fermi theory and strongly bound electrons (SBE) that are better described as hydrogenic states \cite{schwinger_leading_correction}. This is the main motivation for the joint functional treatment, as similar treatment is unavailable in terms of density functionals.
Energy $E_1$ is the trace over the non-interacting atomic Hamiltonian
\begin{align*}
E_1[V + \zeta] = \text{Tr}(H + \zeta)\eta(-H-\zeta),
\end{align*}
where the Hamiltonian is $H = \frac{1}{2}\mathbf{p}^2 + V(\mathbf{r})$ and $\eta$ is the Heaviside function. We can split this into two parts using an energy scale given by $\zeta_s$, so that the electrons with energy levels above $\zeta_s$ are treated semiclassically and electrons with energy below $\zeta_s$ are treated with discrete quantum states. We accomplish this by adding an intelligent zero based on $\zeta_s$. This results in
\begin{align*}
E_1 = \underbrace{E_1[V + \zeta] - E_1[V + \zeta_s] + (\zeta - \zeta_s)N[V + \zeta_s]}_{E_{\zeta\zeta_s}} + E_S,
\end{align*}
where the functional $E_{\zeta\zeta_s}$ is approximated semiclassically and $E_S$ is evaluated with some other quantum mechanical method. Formally this can be thought of as the evaluation of the semiclassical approximation from which we subtract semiclassical evaluation on energy scales below $\zeta_s$ and finally adding the correct treatment for electrons below energy $\zeta_s$.
The critical part of the ES model is to treat the exact part $E_S$ as hydrogenic states. It is assumed that the potential near the nucleus is quite close to Coulombic potential $-\frac{Z}{r}$ so that perturbative evaluation of the states is accurate enough
\begin{align*}
E_S = Z^2n_s + \int d\mathbf{r} (V + \frac{Z}{r})n_\text{SBE},
\end{align*}
where $n_s$ is the uppermost electron shell treated with hydrogenic states and $n_\text{SBE}$ is the density of the electrons treated with hydrogenic states. By shell, we mean the collection of all hydrogenic states with same energy; thus the shells are tabulated by the quantum number $n$. The density $n_\text{SBE}$ is obtained from spherically averaged hydrogenic states.
\begin{align*}
n_\text{SBE} = \sum_{i = 1}^{i \leq n_s}|\psi_i|_\text{av}^2,
\end{align*}
where $|\psi_i|_\text{av}$ is the spherically averaged wavefunction of $i$th hydrogenic shell.
For justification and details, see \cite{schwinger_leading_correction,statistical_atom_1}. Originally, Englert and Schwinger investigated corrections to the Thomas-Fermi model—and as they pointed out, the cut-off energy $\zeta_s$ has certain ambiguity because we are trying to patch a continuous semiclassical model with a discrete quantum model. The most consistent way to achieve this for the Thomas-Fermi model is to average over two electronic shells with equal weights on an energy scale. This is also a good choice for the Thomas-Fermi model with corrections used here.
More concretely, this means
\begin{align*}
E_{\zeta\zeta_s} \longrightarrow \sum_{i = 1}^2 \frac{1}{2}E_{\zeta\zeta_i},
\end{align*}
where $\zeta_i$ is the corrected binding energy of $i$th electronic shell
\begin{align}
\label{eq:def_zeta_i}
\zeta_i = \frac{Z^2}{2n_i^2} - \int d\mathbf{r} (V + \frac{Z}{r})|\psi_i|_\text{av}^2.
\end{align}
For practical purposes, these corrections can be embedded inside the Airy-average functions $F_m$; thus they become corrected $F_m$. The correction induces their own $y$ variables
\begin{align*}
y_i = 2(V + \zeta_i)|2\nabla V|^{-2/3},
\end{align*}
and by doing replacement
\begin{align*}
F_m \longrightarrow \sum_{i = 1}^2\frac{1}{2}w_j\left[F_m(y) - F_m(y_i) - (y_j - y)F_{m-1} \right],
\end{align*}
we can use the energy expression \eqref{eq:es_energy}. The corrections for strongly bound electrons are inside the functions $F_m$.
Our numerical results indicate that for all atoms of the relevant size ($Z < 100$), the only energetically believable correction is the one where only the lowest shell (i.e., 1s electrons) is treated exactly, and average occurs over shells 1 and 2. This approximation naturally breaks down when the number of total electrons is low, namely $Z \sim 12$.
\subsection{Semiclassical Density}
The density can be calculated via relation $n = \frac{\delta}{\delta V}E_1$ \eqref{eq:variation_n}. The resulting density splits straightforwardly into two contributions
\begin{align*}
n = \frac{\delta}{\delta V}E_1 = \underbrace{\left(\frac{\partial e_1}{\partial V} - \nabla \cdot \frac{\partial e_1}{\partial \nabla V} + \nabla^2 V\frac{\partial e_1}{\partial \nabla^2 V} \right)}_{\tilde{n}} +
\frac{\delta \zeta_s}{\delta V}\frac{\partial}{\partial \zeta_s}E_1 + n_\text{SBE}
,
\end{align*}
where $e_1$ is the energy density of $E_1$, $\tilde{n}$ is the semiclassical contribution, and
the two last terms are the contribution of the strongly bound electrons
. Note that the contribution from strongly bound electrons contains the derivative with respect to the core binding energy $\zeta_s$, because it depends on the potential via \eqref{eq:def_zeta_i}. The derivative is
\begin{align}
\label{eq:def_Q}
\frac{\delta \zeta_s}{\delta V}\frac{\partial}{\partial \zeta_s}E_1
= \sum_{i = 1}^2 \frac{1}{2} \underbrace{(\zeta_i - \zeta)\left[\int d\mathbf{r} \frac{1}{\pi} |2\nabla V|^{1/3} F_1(y_j) - \frac{1}{3}|2\nabla V|^{-1}\nabla^2 V F_{-1}(y_j)\right]}_{:=\; Q_i}|\psi_i|_\text{av}^2,
\end{align}
so there will be contributions from the hydrogenic states that depend on the potential.
The semiclassical contribution $\tilde{n}$ in spherical symmetry is
\begin{align}
\label{eq:full_density}
\tilde{n} =& \frac{1}{2\pi}|2\nabla V| F_2 - \frac{1}{6\pi}|2\nabla V|^{1/3}\nabla^2 V F_0 \\
\nonumber
&+ \frac{1}{r}\left[
\frac{1}{36\pi}|2\nabla V|^{2/3}F_{-2}
- \frac{1}{9\pi}|2\nabla V|^{-2/3}\nabla^2 VF_{-3}
- \frac{1}{108\pi} |2\nabla V|^{-2/3}\nabla^2 VF_{-5}\right] \\
\nonumber
&+ \frac{1}{r^2}\left[
\frac{1}{36\pi}|2\nabla V|^{1/3}F_{-2} + \frac{1}{108\pi}|2\nabla V|^{1/3}F_{-5}
\right],
\end{align}
where the first term $\frac{1}{2\pi}|2\nabla V| F_2$ contains the Thomas-Fermi limit. The expression is valid only for neutral atoms and positive ions, as it assumes that the potential's gradient is positive.
Theoretically, using a simpler density expression is also a viable option, which is only the first two terms of \eqref{eq:full_density}. For our benchmarking, we use only the aforementioned density, but later we discuss the self-consistent accuracy of the simpler density.
As in the case of energy, we can include the correction for strongly bound electrons by using the corrected Airy-average functions $F_m$.
\subsection{Interaction}
\label{sec:interaction}
So far, we have detailed how to calculate $E_1$ and electronic density, which are independent of any possible interaction. Now we detail the electronic interaction included via the effective potential $V$.
The interaction term $E_\text{int}$ is separated into an electrostatic term (Hartree) and into an exchange term
\begin{align*}
E_\text{int} = E_\text{H} + E_\text{ex}.
\end{align*}
The corresponding interaction potential is then
\begin{align*}
V_\text{int} = \frac{\delta E_\text{int}}{\delta n} = V_\text{H} + V_\text{ex}.
\end{align*}
The Hartree term is well known
\begin{align*}
V_\text{H} = \int d\mathbf{r}' \frac{n(\mathbf{r})}{|\mathbf{r} - \mathbf{r}'|},
\end{align*}
and its connection to density will be used to achieve a self-consistent solution. We do not add correlation effects to the model, as the effect of correlation in atoms is below the accuracy of semiclassical $E_1$ \cite{semiclassical_atom}.
Before the model is complete, we must comment on how to add the exchange effects. The aim is to include local exchange effects, as described by the Dirac exchange functional \cite{lda_dirac}
\begin{align}
\label{eq:dirac_ex}
E_\text{ex} = - \frac{1}{4\pi^3}\int d\mathbf{r} (3\pi^2n(\mathbf{r}))^{4/3}.
\end{align}
To be consistent, we discard exchange effects on strongly bound electrons, because we already approximated them to be non-interacting (although they will receive a marginal contribution via potential). The exchange potential is necessary for self-consistency, given by
\begin{align}
\label{eq:V_ex_n}
V_\text{ex} = \frac{\delta}{\delta n}E_\text{ex} = -\frac{1}{\pi}(3\pi^2n(\mathbf{r}))^{1/3}.
\end{align}
The most straightforward way to add exchange effects is simply by replacing $n \rightarrow \tilde{n}$ so that only the smooth part is treated with exchange. Testing showed us that while this method produced exchange effects of the correct magnitude, it is not the most accurate way to include the exchange (while discarding contributions of the strongly bound electrons).
The second route taken by Englert and Schwinger is to start from the Thomas-Fermi theory to arrive at a potential description of the exchange. At the Thomas-Fermi level, the exchange potential can be calculated from $V_\text{ex} = \pi \frac{\partial}{\partial V}n$, and the exchange energy density $\epsilon_\text{ex}$ from relation $\frac{\epsilon_\text{ex}}{\partial y} = \frac{1}{2\pi}|2\nabla V|^{4/3}V_\text{ex}^2$. To include exchange effects on the Thomas-Fermi level, the obvious choice for $n$ here is the Thomas-Fermi part of the density $\frac{1}{2\pi}|2\nabla V|F_2$, resulting in the potential
\begin{align}
\label{eq:V_ex_1}
V_\text{ex} = -|2\nabla V|^{1/3}F_1
\end{align}
and the exchange energy
\begin{align*}
E_\text{ex} = \int d\mathbf{r} \; \frac{1}{2\pi}|2\nabla V|^{4/3}\left( -\frac{1}{4}F_1F_{-1} + \frac{1}{8}F_0^2 + \frac{1}{2}yF_1^2\right).
\end{align*}
We found the description of the exchange potential inadequate at larger distances in the atoms. Thus, we also use another version of the exchange potential \cite{statistical_atom_2}, which uses the density expression $\tilde{n}$ where the Laplacian of the potential has been approximated out. The resulting exchange potential is
\begin{align}
\label{eq:V_ex_2}
V_\text{ex} = -|2\nabla V|^{1/3}(F_1 - \frac{1}{6}F_{-2}),
\end{align}
which we will use in our implementation. We expect the previous exchange energy description to be accurate enough for this potential. Later we discuss different exchange approximations that are meaningful.
\section{Implementation}
We obtained the self-consistent solution via relation $n = \frac{\delta}{\delta V}E_1$ and the connection of a single particle potential $V$ to the electrostatic potential $V_H$. For a given effective potential $V$, we find the corresponding binding energy $\zeta$ with relation $N = \int d\mathbf{r} n[V(\mathbf{r})] + \zeta]$. This yields the density $n[V(\mathbf{r}) + \zeta]$, which we can use to find a new Hartree potential $V_H$ through a Poisson equation. The new effective potential is now obtained from this Hartree potential with $V = V_H + V_\text{ext} + V_\text{ex}[n, V]$, where in $V_\text{ex}$ we use the old potential and density. This is iterated until the change in potential and binding energy is small enough. With a good initial guess, this procedure gives the self-consistent solution. Here the Thomas-Fermi potential is a sufficient initial guess.
We describe this idea in a bit more detail for spherically symmetrical atoms. With relation \eqref{eq:variation_V} and the connection of an electrostatic potential $V_H$ with a Poisson equation, we arrive at
\begin{align*}
-\frac{1}{4\pi}\nabla^2\left(V + \frac{Z}{r}\right) = n(\mathbf{r}) - \nabla^2 V_\text{ex},
\end{align*}
with boundary conditions $rV(\mathbf{r} \longrightarrow 0) \longrightarrow -Z$ and $rV(r \longrightarrow \infty) = -Z + N$, where zero potential has been assigned to infinitely far away from the nucleus. After we use the fact that our system is spherically symmetric and introduce the auxiliary quantity $V(\mathbf{r}) = -\frac{Z}{r}\Phi(r)$, we have the differential equation
\begin{align}
\label{eq:differential_eq}
Z\frac{\partial^2}{\partial r^2}\Phi(r) &= 4\pi rn(r) - \frac{\partial^2}{\partial r^2}rV_\text{ex},\quad \text{with boundary conditions} \\
\nonumber
\Phi(0) &= 1, \\
\nonumber
\Phi(\infty) &= 1 - \frac{N}{Z}.
\end{align}
After obtaining the effective potential $V$, we find the corresponding binding energy with Newton's method from the relation $N -\int d\mathbf{r} n(V + \zeta) = 0$.
We use a non-uniform grid that is denser near the nucleus, where the change in values is greater than in the tail (the grid is correspondingly sparser in the tail).
\subsection{Numerical Method}
Englert and Schwinger's original paper \cite{statistical_atom_3} uses the shooting method to solve the resulting differential equation. We solve the resulting differential with a simple 1D finite element method with linear elements. For the finite element method, we derive the weak form of the differential equation, which is
\begin{align*}
-Z\int dr \; \frac{\partial \Phi(r)}{\partial r}\cdot\frac{\partial v}{\partial r}
= \int dr \; 4\pi rn(r) \cdot v + \frac{\partial\; rV_\text{ex}(r)}{\partial r}\cdot\frac{\partial v}{\partial r},
\end{align*}
where $v$ is the test function. During testing we noted that the exchange potential is the most sensitive quantity from a numerical point of view. In the numerical study \cite{statistical_atom_3}, the solution for neutral atoms was not obtained because the effective potential's long-range behavior is unknown.
We obtain the neutral atom solution simply by setting the boundary condition to zero, as dictated by \eqref{eq:differential_eq}, and then we converge the result with respect to grid size until the errors (due to the grid's finite size) are below the error threshold.
The differential equation is a bit different than Englert and Schwinger's book \cite{semiclassical_atom}. We use the more straightforward expression derived from a Poisson equation than they mention in the book \cite{semiclassical_atom}, where all the terms containing the Laplacian of effective potential are moved to the left side of the equation.
\section{Results}
We benchmark the numerical results against perturbative results calculated by Englert and Schwinger, the Kohn-Sham results, and the TFD-$\lambda$W model. First we evaluate our results against Englert and Schwinger's \cite{semiclassical_atom}, and then compare the self-consistent results for the following methods:
\begin{itemize}
\item[ES] The Englert-Schwinger model with exchange potential \eqref{eq:V_ex_2} and density expression \eqref{eq:full_density}.
\item[KS-LDA] Spherically symmetric Kohn-Sham atom, where the xc-functional is the Dirac exchange \eqref{eq:dirac_ex} and no correlation (KS-LDA stands for Kohn-Sham local density approximation).
\item[TFD-$\frac{1}{9}$W] The OFDFT model, which contains Dirac exchange and Thomas-Fermi plus one-ninth of a von Weizsäcker term as a kinetic energy functional, where the von Weizsäcker factor is derived as a quantum correction to the Thomas-Fermi functional.
\item[TFD-$\frac{1}{5}$W] The OFDFT model, which contains the Dirac exchange and Thomas-Fermi plus one-fifth of a von Weizsäcker term as the kinetic energy functional, where the von Weizsäcker factor is fitted rather than derived.
\end{itemize}
All of these models are based on DFT. In each model, we treat the exchange effect with the Dirac exchange \eqref{eq:dirac_ex}. Thus we choose to benchmark against Kohn-Sham as the most accurate of the approximations. Originally, Englert and Schwinger opted to benchmark against Hartree-Fock data.
The exact difference between the orbital-free models is a bit more complex. Both orbital-free and ES are (in a sense) approximating the energy $E_1$. The ES model approximates it directly, including both kinetic and potential terms, while TFD-$\lambda$W models approximate it via approximating the kinetic energy density functional only. From a formal point of view TFD-$\frac{1}{9}$W and the ES model both expand the semiclassical trace to the same order in $\hbar$, but we must remember that the TFD-$\frac{1}{9}$W model disregards the strongly bound electron correction completely. In model TFD-$\frac{1}{5}$W, the von Weizsäcker fraction is fitted to produce best results for atoms \cite{tfdvw_atom_fitted_lambda}.
The atomic Kohn-Sham solver used is available in a grid-based implementation of the projector-augmented waves (GPAW) DFT package \cite{gpaw}.
The kinetic energy density functional methods are implemented within this same Kohn-Sham solver. The implementation details are described elsewhere \cite{ofdft_paw,ofdft_atoms}.
\subsection{Comparison to Englert-Schwinger Reference Numerical Data}
We first study the model presented in the book {\it Semiclassical Theory of Atoms} \cite{semiclassical_atom}. Thus, we use density expression \eqref{eq:full_density} and exchange potential \eqref{eq:V_ex_1} and solve the resulting differential equation.
We compare the numerical results against the ones provided in \cite{semiclassical_atom} in Table \ref{tab:numerical_ES_comparison} for Krypton electronic configuration. The numbers correspond to fair accuracy. The biggest error is the binding energy of $Z$=38, charge=2 where we have a deviation of 3.3\%. Deviation in other values are below 2\%. Core binding energies $\zeta_1$ and $\zeta_2$ show strong similarity, which is to be expected as the potential should have a $-\frac{Z}{r}$ shape near the hydrogenic states.
The neutral atom results by Englert and Schwinger are the result of extrapolation, so they are omitted from the comparison.
\begin{table}[!tbp]
\begin{tabular}{c |c | l | l | l | l | l | l |}
Z & charge & Source & $\zeta$ & $\zeta_1$ & $\zeta_2$ & $Q_1$ & $Q_2$\\
\hline
36 & 0
& This work & 0.0315 & 496.31 & 59.98 & 0.8190 & 4.044 \\
\hline
37& 1
& This work & 0.3311 (1.1) & 526.62 (0.5) & 64.816 (1.9) & 0.8225 (0.3) & 4.0824 (1.3) \\
& & Ref \cite{semiclassical_atom} & 0.3347 & 529.42 & 66.103 & 0.8251 & 4.137 \\
\hline
38 & 2
&This work & 0.7432 (3.3) & 558.96 (0.4) & 69.703 (0.2) & 0.8244 (0.4) & 4.127 (1.0) \\
& & Ref \cite{semiclassical_atom} & 0.7687 & 561.29 & 71.377 & 0.8276 & 4.168 \\
\end{tabular}
\caption{Comparison of numerical parameters. Error compared to reference values \cite{statistical_atom_3} are in parenthesis (in percent).
}
\label{tab:numerical_ES_comparison}
\end{table}
\subsection{Effect of Exchange Potential}
Exchange potential and energy have few possible approximations, as indicated in section \ref{sec:interaction}. To choose the best one, we take a look at Krypton to see the effects of different exchange functionals. We look at the total energy, binding energy $\zeta$, and averages over densities $\langle \frac{1}{r} \rangle$, $\langle r \rangle$ and $\overline{r}^2$. The last one is defined by
\begin{align*}
\overline{r}^2 = \frac{1}{N}\int d\mathbf{r} \; r^2n(\mathbf{r}).
\end{align*}
The results for different exchange expressions for neutral Krypton are tabulated in Table \ref{tab:ex_potentials}.
From the change $\overline{r}^2$ we can see that the choice of exchange potential has a strong effect near the atom's edge, which is quite natural \cite{semiclassical_atom}.
As the exchange energy is negative and the corresponding potential is attractive, we would expect the correct inclusion of exchange to make the atomic size smaller.
The energy difference is not that useful on a semiclassical scale if we compare it to a highly accurate Englert-Schwinger prediction for semiclassical energy, which is $-2747.64$ Hartrees for Krypton. The deviation from this value is 0.7\%, 0.1\% and 0.2 \% for methods \eqref{eq:V_ex_n}, \eqref{eq:V_ex_1}, and \eqref{eq:V_ex_2}, respectively. Our choice, then, should be based on the quality of density at the atom's outer reaches. The experimental value for $\overline{r}^2$ can be obtained via diamagnetic susceptibilities \cite{statistical_atom_3}, which are provided by the {\it CRC Handbook of Chemistry and Physics} \cite{experimental_data}. The experimental value of $\overline{r}^2$ for Krypton is $1.010$ Bohr$^2$. This indicates that all methods are a bit insufficient, but that \eqref{eq:V_ex_2} is clearly the best for $\overline{r}^2$. Finally, we want to note that there is a deviation of 0.05 - 0.1 Bohr$^2$ to the reported $\overline{r}^2$values\cite{statistical_atom_3} so that the deviation to the experimental value might already be at the level of precision of the chosen methods.
\begin{table}[!tbp]
\begin{tabular}{l | c | c | c | c | c | c |}
Exchange potential $V_\text{ex}$ & $E$ & $\zeta$ & $\langle \frac{1}{r} \rangle$ & $\langle r \rangle$ & $\overline{r}^2$ & $r_\text{classical}$ \\
\hline
\eqref{eq:V_ex_n} $-\frac{1}{\pi}(3\pi^2\tilde{n}(\mathbf{r}))^{1/3}$ & -2765.852 & 0.08867 & 186.618 & 29.0069 & 1.3718 & 4.7613 \\
\eqref{eq:V_ex_1} $-|2\nabla V|^{1/3}F_1$ & -2744.064 & 0.03152 & 187.351 & 28.6654 & 1.3650 & 4.9333 \\
\eqref{eq:V_ex_2} $-|2\nabla V|^{1/3}(F_1 - \frac{1}{6}F_{-2})$ & -2742.266 & 0.02609 & 187.744 & 27.3852 & 1.1655 & 4.1455
\end{tabular}
\caption{Krypton $Z=36$ with different exchange potentials with full density expression \eqref{eq:full_density}. Classical radius is defined by $V(r_\text{classical}) + \zeta = 0$. }
\label{tab:ex_potentials}
\end{table}
We therefore determined that for the self-consistent ES atom model, the exchange potential \eqref{eq:V_ex_2} is preferred. Still, the model's accuracy in the atom's outer reaches is somewhat limited by the exchange approximation, and it remains unclear how the exchange effect should be treated for strongly bound electrons. We intend to address this point in future work.
In the following, we use exchange potential \eqref{eq:V_ex_2}.
\subsection{Assessment of ES Improvements in Neutral Atoms}
We compute the energies and a few other descriptive quantities for the ES model and compare them to other models for neutral systems to assess the quality of the self-consistent solution. The total energy as a quantity is not so important, as the real predictive power lies in the energy differences, but having this information is somewhat useful, because it informs the quality of approximations made. As previously mentioned, though, here we consider Kohn-Sham as the "ground truth."
Most of our quantities depend on density to indicate the shape and quality. The near-nucleus area is probed by averaging $\langle 1/r \rangle$, which is related to the shielding of the nuclear magnetic moment \cite{semiclassical_atom}. The average over $r$ is related to electric polarizability \cite{semiclassical_atom}. Finally, as we mentioned in the previous section, we measure $\langle r^2 \rangle$, which probes the density at atoms' outer edges and is related to diamagnetic susceptibility.
For atoms, the energy difference is tested only by calculating ionization potential and comparing the total energies of charged and neutral systems. As the semiclassical model incorporates no information of the valence electron shells, our primary interest is in the ionization potential of alkali metals, where the semiclassical approximation could produce good results.
First we show the general trends over the whole $Z$ to get a sense of what is or is not a reasonable comparison. Figure \ref{fig:total_energy_error} shows the relative error when compared to Kohn-Sham energies. It is reassuring to see the error is generally small, and it goes down for both methods when $Z$ goes higher, which means that the methods are capturing the essence of the Thomas-Fermi theory. The larger deviation for small-$Z$ for ES can be explained by the strongly bound electron correction, with averaging over shells not as valid for small-$Z$. For high-$Z$, one possible explanation is that we are correcting for a tad too few electrons here, but adding a second shell of strongly bound electrons does not improve the situation here, as then we are overcorrecting.
The expectation value $\langle \frac{1}{r} \rangle$ in Figure \ref{fig:invr_Z} is quite smooth for all models, as mainly strongly bound electrons contribute to the density near the nucleus. The ES model handles them explicitly, while the TFD-$\lambda$vW model seems to handle them implicitly with the von Weizsäcker term.
In Figure \ref{fig:r_Z}, we start to see the so-called shell oscillation for $\langle r \rangle$. The main comment here is that we should not take this value too seriously when doing comparisons, as long as the semiclassical model has a decent average over the shell effects. As is obvious, both models satisfy this requirement.
Next we look at the general trend of $\overline{r}^2$ in Figure \ref{fig:r2_Z}. The shell oscillations already present in the case of $\langle r \rangle$ are magnified, as we are probing even farther reaches of the atoms. We see again that both OFDFT models are reasonable.
Comparing the ES model with KS-LDA and experimental numbers should only be taken seriously for inert atoms, which have a closed shell structure.
In Xenon ($Z$=54) we see that experimental value is closer to semiclassical value than KS-LDA. The ES model is actually in better agreement with the experimental values as it assigns smaller sizes for almost all atoms\cite{[{See supplemental note}] fn1}.
The effect is mostly due to the exchange term as seen from Table \ref{tab:ex_potentials}.
The origin of these shell oscillations in the Kohn-Sham model is obvious: some outer orbitals are more delocalized than others.
\begin{figure}[ !tbp]
\includegraphics[width=.7\textwidth]{models_relative_energy.eps}
\caption{Total energy error compared to the Kohn-Sham energy as a function of $Z$.}
\label{fig:total_energy_error}
\end{figure}
\begin{figure}[ !tbp]
\includegraphics[width=.7\textwidth]{models_invr.eps}
\caption{Expectation value $\langle \frac{1}{r}\rangle$ as a function of $Z$.}
\label{fig:invr_Z}
\end{figure}
\begin{figure}[ !tbp]
\includegraphics[width=.7\textwidth]{models_r.eps}
\caption{Expectation value $\langle r \rangle$ as a function of $Z$.}
\label{fig:r_Z}
\end{figure}
\begin{figure}[ !tbp]
\includegraphics[width=.7\textwidth]{models_exp_r2.eps}
\caption{Expectation value $\langle r^2 \rangle$ as a function of $Z$. The experimental values are from \cite{experimental_data}.}
\label{fig:r2_Z}
\end{figure}
We can see next in the comparison of different atomic models the absolute value of the numbers presented as general trends in Figures \ref{fig:total_energy_error} through \ref{fig:r2_Z}.
We feel it is informative enough to focus on three representative closed-shell systems: Argon (low-$Z$, where the strongly bound electron approximation is still valid), Krypton (medium-$Z$), and Xenon (high-$Z$).
\begin{table}[ !tbp]
\centering
\begin{tabular}{l|c|c|c|c|}
Quantity & KS-LDA & TFD+$\frac{1}{9}$vW & TFD+$\frac{1}{5}$vW & ES \\
\hline
E & -524.51 & -561.80& -524.75 & -518.60\\
$\langle r \rangle$ & 16.205 & 16.202 & 16.267 & 16.729 \\
$\overline{r}^2$ & 1.489 & 1.61 & 1.630 & 1.611 \\
$\langle \frac{1}{r} \rangle$ & 69.567 & 73.796 & 69.566 & 74.228
\end{tabular}
\caption{Energies and averages of different models for Argon $Z=18$. Results are in atomic units.}
\label{tab:model_quantities_Ar}
\end{table}
\begin{table}[ !tbp]
\centering
\begin{tabular}{l|c|c|c|c|}
Quantity & KS-LDA & TFD+$\frac{1}{9}$vW & TFD+$\frac{1}{5}$vW & ES \\
\hline
E & -2746.649 & -2895.528 & -2744.153 & -2742.266 \\
$\langle r \rangle$ & 26.37 & 27.69 & 27.72 & 27.81 \\
$\overline{r}^2$ & 1.120 & 1.274 & 1.277 & 1.166 \\
$\langle \frac{1}{r} \rangle$ & 182.63 & 190.14 & 181.63 & 186.34
\end{tabular}
\caption{Energies and averages of different models for Krypton $Z=36$. Results are in atomic units.}
\label{tab:model_quantities_Kr}
\end{table}
\begin{table}[ !tbp]
\centering
\begin{tabular}{l|c|c|c|c|}
Quantity & KS-LDA & TFD+$\frac{1}{9}$vW & TFD+$\frac{1}{5}$vW & ES \\
\hline
E & -7223.567 & -7556.458 & -7208.302 & -7207.0235 \\
$\langle r \rangle$ & 39.131 & 37.610 & 37.606 & 38.650 \\
$\overline{r}^2$ & 1.172 & 1.092 & 1.092 & 1.000 \\
$\langle \frac{1}{r} \rangle$ & 317.670 & 330.690 & 317.60 & 323.931
\end{tabular}
\caption{Energies and averages of different models for Xenon $Z=54$. Results are in atomic units.}
\label{tab:model_quantities_Xe}
\end{table}
\begin{table}[ !tbp]
\begin{tabular}{c | c | c}
Atom & ES & KS \\ \hline
K & 0.1310 & 0.1364 \\
Rb & 0.1202 & 0.1319 \\
Cs & 0.1133 & 0.1223
\end{tabular}
\caption{Ionization potential in Hartrees calculated by the energy difference $E(Z) -E(Z-1)$ for ES and KS models. KS results are calculated with GPAW DFT code\cite{gpaw}, where we used Dirac exchange functional.}
\label{tab:ionization_potential}
\end{table}
From Figures \ref{fig:total_energy_error} through \ref{fig:r2_Z} and Tables \ref{tab:model_quantities_Ar} through \ref{tab:model_quantities_Xe}, we can draw some conclusions. First is the remarkable and well-known accuracy of the TFD+$\frac{1}{5}$vW model and the fact that ES is much better than TFD+$\frac{1}{9}$vW, which has a functional that is formally expanded to the same order in $\hbar$, but is missing the corrections for strongly bound electrons. We did not include TFD+$\frac{1}{9}$vW in Figures \ref{fig:total_energy_error} through \ref{fig:r2_Z}, because the accuracy is substantially lower than for other models. The deviation of TFD+$\frac{1}{9}$vW from KS-LDA for Argon, Krypton, and Xenon is 7.1 \%, 5.4 \%, 4.6 \%, which is substantially higher than for TFD+$\frac{1}{5}$vW or ES, as seen from Figure \ref{fig:total_energy_error}. Overall, it seems that after $Z > 40$, TFD+$\frac{1}{5}$vW and the ES model offer relatively similar accuracy, with TFD+$\frac{1}{5}$vW being slightly better.
For all models, the density-dependent quantities $\langle r \rangle$, $\overline{r}^2$ and $\langle \frac{1}{r} \rangle$ are quite reasonable, surprisingly even for TFD+$\frac{1}{9}$vW. This reflects the fact that all the models have a reasonable density average over the shell effects in KS-LDA densities, which are shown in Figure \ref{fig:densities}.
Near the nucleus, TFD+$\frac{1}{5}$vW has the best average description compared to KS-LDA if we look at quantity $\langle \frac{1}{r} \rangle$. The worst deviation for TFD+$\frac{1}{5}$vW is for Krypton with 0.6 \% from KS-LDA, while for TFD+$\frac{1}{9}$vW and for ES the worst case is Argon, with deviations of 6.1 \% and 6.7 \% respectively.
The first quantity to contain shell effects is $\langle r \rangle$. For these values, all the models give surprisingly similar results. The worst case for ES is Argon, where we have a deviation of 3.2 \%, while the worst case for kinetic energy functionals is Krypton, where the deviation is $\sim 5.1 \%$. Again we note that shell oscillation plays a role here and that the Argon value for ES is probably particularly bad: in Figure \ref{fig:r_Z} we see a bump in the ES values for small $Z$. This is most likely a slight artifact caused by the averaging procedure.
The quantity $\overline{r}^2$ is an interesting one. For Argon, all the semiclassical models give similar results, which are significantly above the KS-LDA, up to $\sim 10 \%$ deviation for TFD+$\frac{1}{5}$vW. The case for Krypton is similar, except for ES, which is closer to KS-LDA than kinetic energy functionals. Finally, for Krypton, ES is below all the other models, which is a good thing if we look at the experimental values in Figure \ref{fig:r2_Z}.
For inert atoms there is a trend: all semiclassical values start above KS-LDA values in Argon and end up below KS-LDA in Xenon. However, we should not read too much into this, as the quantity is quite dependent on the outer orbitals of the particular element, as Figure \ref{fig:r2_Z} shows. The ES model clearly wins in the description of the atom's outer reaches when considering experimental values.
The ionization potential calculated with a difference of total energies $E(N) - E(N-1)$ and the results are in Table \ref{tab:ionization_potential}. We only calculated it for alkali metals, as the ES model will fail to provide reasonable potential for other elements because the shell effects are missing.
A deviation in the ionization potential emerges (up to 9 \% deviation for Cesium), but the trend is similar for both methods.
\begin{figure}[ !tbp]
\includegraphics[width=.8\textwidth]{density_plots.eps}
\caption{Semiclassical and Kohn-Sham densities of inert atoms.}
\label{fig:densities}
\end{figure}
Atomic densities for closed-shell atoms are shown in \ref{fig:densities}. We also plot Kohn-Sham densities as a reference. We can see that the ES densities are not completely structureless; they do contain some structure due to averaging over the hydrogenic shells. Yet, obviously, they do not contain the shell structure of Kohn-Sham due to single particle states.
Previous results prove that ES model does not lose to the density functional models and is even better than Kohn-Sham in some special cases.
But what the ES model mostly supplies is theoretical clarity and rigor. One thorny issue with TFD+$\lambda$vW models and other GGA-based OFDFT models has been the Pauli potential $v_\Theta$'s negativity, which is defined as.
\begin{align*}
v_\Theta = \frac{\delta T_\Theta}{\delta n} = \frac{\delta T_s}{\delta n} - \frac{\delta T_\text{vW}}{\delta n} = V - \mu - \frac{\delta T_\text{vW}}{\delta n},
\end{align*}
where $T_s$ is the non-interacting kinetic energy and $T_\text{vW}[n]$ is the von Weizsäcker kinetic energy functional.
It has been shown that $v_\Theta$ should always be positive \cite{pauli_potential_properties}, but for TFD+$\lambda$vW and many other GGA kinetic functionals, it has been found to be negative near a nucleus \cite{pauli_potential_functionals}. This emerge because the semiclassical evaluation is not valid near a nucleus; see \cite{semiclassical_atom} for an excellent discussion. The ES model does not have this problem, because the problematic region is removed from the semiclassical evaluation.
\begin{figure}[ !tbp]
\includegraphics[width=.8\textwidth]{Kr_paulipotential.eps}
\caption{Pauli potential $v_\Theta$ of four atomic models.}
\label{fig:pauli_potential}
\end{figure}
Figure \ref{fig:pauli_potential} shows the self-consistent Pauli potentials. Clearly in the self-consistent calculations, the Pauli non-positivity problem is contained in a small region, but the Pauli potential is still qualitatively wrong. In comparison, the ES model's Pauli potential is qualitatively better, although naturally missing the shell effects. The ES model has oscillations due to the averaging over shells, but this is more of an artifact than a result of the oscillations being in the wrong region.
This raises the question of how the TFD+$\frac{1}{5}$vW's energetics can be so good (while having reasonable geometric properties, too) if it completely ignores the correction for strongly bound electrons? The answer must be related to the dual nature of the von Weizsäcker term, because it has two roles as a density functional: it is a gradient correction to Thomas-Fermi, but it also is exact for up to two non-interacting particles. These two roles naturally differ by a constant factor, but the form is the same.
Kinetic energy functionals of GGA type can be built around the dual nature by interpolating between these two extremes \cite{pauli_potential_functionals,ofdft_vt84f,kinetic_energy_perdew}. Here we are inclined to support the view of Englert and Schwinger: it is more straightforward to exclude the strongly bound electrons from the semiclassical evaluation than try to modify the functional to include them via the von Weizsäcker term. The near nucleus Pauli potential singularity is also avoided by the use of pseudopotentials as the strongly bound electrons are excluded from the OFDFT calculation. The non-negativity constraint might be a non-issue for valence density\cite{[{Authors are not aware of any kinetic energy functional that breaks this condition for valence densities}]fn2}.
We also must remember that the $\lambda$ value of $\frac{1}{5}$ is obtained by fitting, while the strongly bound electron correction is better motivated. If the ES model is compared to the gradient expansion in atoms, then it is clearly preferable.
\subsection{Accuracy of Self-Consistent Density Approximations}
As mentioned earlier, it is not necessary to calculate the density with the full expression \eqref{eq:full_density} to get the same semiclassical accuracy. Only the first line of \eqref{eq:full_density} is necessary, as Englert argued \cite{semiclassical_atom}. The rest of the expression contains full divergence—i.e., it integrates to zero so it does not contribute to the number of electrons, only to the distribution. We are interested in the approximation for future use, if the method is extended to larger systems. We also want to see the effect on $\overline{r}^2$ to see how sensitive it is to density approximations, because we already established the effect of exchange.
The simple density approximation is obtained simply by discarding the higher-order variations. The resulting expression is
\begin{align}
\label{eq:simple_density}
\tilde{n}_\text{approx} \approx \frac{\partial e_1}{\partial V} = \frac{1}{2\pi}|2\nabla V| F_2 - \frac{1}{6\pi}|2\nabla V|^{-1/3}\nabla^2 V F_0.
\end{align}
We compare the difference of two density expressions \eqref{eq:simple_density} and \eqref{eq:full_density} in self-consistent calculations. We focus on Krypton for simplicity, but the trends are reproduced over a range of $Z$. As both integrate to the electron number, the difference is purely just a redistribution of the density.
\begin{table}[!tbp]
\begin{tabular}{c | c | c | c | c}
Atom & Density & Energy & $\overline{r}^2$ & $\langle \frac{1}{r} \rangle$ \\
\hline
Ar & $\tilde{n}$ \eqref{eq:full_density} & -518.591 & 1.611 & 74.228 \\
Ar & $\tilde{n}_\text{approx}$ \eqref{eq:simple_density} & -519.082 & 1.652 & 73.301 \\
\hline
Kr & $\tilde{n}$ \eqref{eq:full_density} & -2742.320 & 1.163 & 187.744 \\
Kr & $\tilde{n}_\text{approx}$ \eqref{eq:simple_density} & -2743.970 & 1.193 & 186.344 \\
\hline
Xe & $\tilde{n}$ \eqref{eq:full_density}& -7207.024 & 1.000 & 323.598 \\
Xe & $\tilde{n}_\text{approx}$ \eqref{eq:simple_density} & -7209.345 & 1.014 & 322.737 \\
\end{tabular}
\caption{Results for inert atoms with density expressions \eqref{eq:simple_density} and \eqref{eq:full_density}.}
\label{tab:density_comparison}
\end{table}
From Table \ref{tab:density_comparison} we can see that the simple density assigns more density near the nucleus while the full expression localizes the density more, which is seen in $\overline{r}^2$. Full density has a bit better energy when compared to KS-LDA . For energy, the maximal deviation between density expressions is 0.4 \% percent in Xenon, and for $\overline{r}^2$ the maximal deviation is 3\% percent in Argon. On a semiclassical scale, these differences are quite small. Overall the differences show that the full density \eqref{eq:full_density} is better, as expected, but the simpler expression \eqref{eq:simple_density} is still valid.
\section{Conclusion}
Having benchmarked the self-consistent ES model, we found that it compares favorably against the simple TFD-$\lambda$vW model. While the self-consistent model does not achieve the accuracy of Englert and Schwinger's perturbative work, still yet the performance is good and could lead to a viable means of obtaining physically plausible full solutions to OFDFT equations. An important point to remember here is that self-consistent atoms are not valuable themselves, as they are just a stepping stone toward larger systems.
The problematic points of the self-consistent ES model are that it cannot be applied to light elements (according to our tests, the strongly bound electron approximation breaks down numerically at $Z \sim 12$) and its non-ambiguous inclusion of exchange effects. Given this model's potential, however, such issues are worth considering and investigating with rigor; these issues can and will be addressed in future work.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,724 |
only the previous ctb line is needed for the various prediction mechanisms.
changes. And the margin of this mail is too small to explain it.
the list of MVs does not need to be fully derived, so don't waste time on it. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,357 |
Gynecology articles |
Nov 22, 2005 Viewed: 2279
A couple is said to be infertile if pregnancy does not result after 1 year of normal sexual activity without contraceptives. About 25% of couples experience infertility at some point in their reproductive lives; the incidence of infertility increases with age. The male partner contributes to about 40% of cases of infertility, and a combination of factors is common.
Diagnostic Survey
During the initial interview, the clinician can present an overview of infertility and discuss a plan of study. Separate private consultations are then conducted, allowing appraisal of psychosexual adjustment without embarrassment or criticism. Pertinent details (eg, sexually transmitted disease or prior pregnancies) must be obtained. The ill effects of cigarettes, alcohol, and other recreational drugs on male fertility should be discussed. Prescription drugs that impair male potency should be discussed as well. The gynecologic history should include queries regarding the menstrual pattern. The present history includes use and types of contraceptives, douches, libido, sex techniques, frequency and success of coitus, and correlation of intercourse with time of ovulation. Family history includes repeated abortions and maternal DES use.
General physical and genital examinations are performed on both partners. Basic laboratory studies include complete blood count, urinalysis, cervical culture for chlamydia, serologic test for syphilis, rubella antibody determination, and thyroid function tests. Tay-Sachs screening should be offered if both parents are Jews and sickle cell screening if both parents are black.
The patient is instructed to chart her basal body temperature orally daily on arising and to record on a graph episodes of coitus and days of menstruation. Self-performed urine tests for the midcycle LH surge enhance temperature observations relating to ovulation. Couples should be advised that coitus resulting in conception occurs during the 6-day period ending with the day of ovulation.
The male partner is instructed to bring a complete ejaculate for analysis. Sexual abstinence for at least 3 days before the semen is obtained is emphasized. A clean, dry, wide-mouthed bottle for collection is preferred. Condoms should not be employed, as the protective powder or lubricant may be spermicidal. Semen should be examined within 1-2 hours after collection. Semen is considered normal with the following minimum values: volume, 1.5-5 mL; concentration, 20 million sperm per milliliter; motility, 60%; and normal forms, 35%. If the sperm count is abnormal, further evaluation includes a search for exposure to environmental and workplace toxins, alcohol or drug abuse, and hypogonadism.
A. First Testing Cycle
While the contribution of cervical factors to infertility is controversial, most gynecologists include a postcoital test in their workup. The test is scheduled for just before ovulation (eg, day 12 or 13 in an expected 28-day cycle). Preovulation timing can be enhanced by serial urinary LH tests. The patient is examined within 6 hours after coitus. The cervical mucus should be clear, elastic, and copious owing to the influence of the preovular estrogen surge. (The mucus is scantier and more viscid before and after ovulation.) A good spinnbarkeit (stretching to a fine thread 4 cm or more in length) is desirable. A small drop of cervical mucus should be obtained from within the cervical os and examined under the microscope. The presence of five or more active sperm per high-power field constitutes a satisfactory postcoital test. If no spermatozoa are found, the test should be repeated (assuming that active spermatozoa were present in the semen analysis). Sperm agglutination and sperm immobilization tests should be considered if the sperm are immotile or show ineffective tail motility.
The presence of more than three white blood cells per high-power field in the postcoital test suggests cervicitis in the woman or prostatitis in the man. When estrogen levels are normal, the cervical mucus dried on the slide will form a fern-like pattern when viewed with a low-power microscope. This type of mucus is necessary for normal sperm transport.
The serum progesterone level should be measured at the midpoint of the secretory phase (21st day); a level of > 3 ng/mL confirms ovulation.
B. Second Testing Cycle
Hysterosalpingography using an oil dye is performed within 3 days following the menstrual period. This x-ray study will demonstrate uterine abnormalities (septa, polyps, submucous myomas) and tubal obstruction. A repeat x-ray film 24 hours later will confirm tubal patency if there is wide pelvic dispersion of the dye. This test has been associated with an increased pregnancy rate by some observers. If the woman has had prior pelvic inflammation, one should give doxycycline, 100 mg twice daily, beginning immediately before and for 7 days after the x-ray study.
C. Further Testing
1. Gross deficiencies of sperm (number, motility, or appearance) require repeat analysis. Zona-free hamster egg penetration tests are available to evaluate the ability of human sperm to fertilize an egg.
2. Obvious obstruction of the uterine tubes requires assessment for microsurgery or in vitro fertilization.
3. Absent or infrequent ovulation requires additional laboratory evaluation. Elevated FSH and LH levels indicate ovarian failure causing premature menopause. Elevated LH levels in the presence of normal FSH levels confirm the presence of polycystic ovaries. Elevation of blood prolactin (PRL) levels suggests pituitary microadenoma.
4. Ultrasound monitoring of folliculogenesis may reveal the occurrence of unruptured luteinized follicles.
5. Endometrial biopsy in the luteal phase associated with simultaneous serum progesterone levels will rule out luteal phase deficiency.
D. Laparoscopy
Approximately 25% of women whose basic evaluation is normal will have findings on laparoscopy explaining their infertility (eg, peritubal adhesions, endometriotic implants).
A. Medical Measures
Fertility may be restored by appropriate treatment in many patients with endocrine imbalance, particularly those with hypo- or hyperthyroidism. Antibiotic treatment of cervicitis is of value. In women with abnormal postcoital tests and demonstrated antisperm antibodies causing sperm agglutination or immobilization, condom use for up to 6 months may result in lower antibody levels and improved pregnancy rates.
Women who engage in vigorous athletic training often have low sex hormone levels; fertility improves with reduced exercise and some weight gain.
B. Surgical Measures
Excision of ovarian tumors or ovarian foci of endometriosis can improve fertility. Microsurgical relief of tubal obstruction due to salpingitis or tubal ligation will reestablish fertility in a significant number of cases. In special instances of cornual or fimbrial block, the prognosis with newer surgical techniques has become much better. Peritubal adhesions or endometriotic implants often can be treated via laparoscopy or via laparotomy immediately following laparoscopic examination if prior consent has been obtained.
With varicocele in the male, sperm characteristics are often improved following surgical treatment.
C. Induction of Ovulation
1. Clomiphene citrate
Clomiphene citrate stimulates gonadotropin release, especially LH. Consequently, plasma estrone (E1) and estradiol (E2) also rise, reflecting ovarian follicle maturation. If E2 rises sufficiently, an LH surge occurs to trigger ovulation.
After a normal menstrual period or induction of withdrawal bleeding with progestin, one should give 50 mg of clomiphene orally daily for 5 days. If ovulation does not occur, the dosage is increased to 100 mg orally daily for 5 days. If ovulation still does not occur, the course is repeated with 150 mg daily and then 200 mg daily for 5 days, with the addition of chorionic gonadotropin, 10,000 units intramuscularly, 7 days after clomiphene.
The rate of ovulation following this treatment is 90% in the absence of other infertility factors. The pregnancy rate is high. Twinning occurs in 5% of these patients, and three or more fetuses are found in rare instances (< 0.5% of cases). An increased incidence of congenital anomalies has not been reported. Painful ovarian cyst formation occurs in 8% of patients and may warrant discontinuation of therapy. Several studies have suggested a two- to threefold increased risk of ovarian cancer with the use of clomiphene for more than 1 year.
In the presence of increased androgen production (DHEA-S > 200 ug/dL), the addition of dexamethasone, 0.5 mg, or prednisone, 5 mg, at bedtime, improves the response to clomiphene. Dexamethasone should be discontinued after pregnancy is confirmed.
2. Bromocriptine
Bromocriptine is used only if PRL levels are elevated and there is no withdrawal bleeding following progesterone administration (otherwise, clomiphene is used). To minimize side effects (nausea, diarrhea, dizziness, headache, fatigue), bromocriptine should be taken with meals. The initial dosage is 2.5 mg once daily, increased to two or three times daily in increments of 1.25 mg. The drug is discontinued once pregnancy has occurred.
3. Human menopausal gonadotropins (hMG)
hMG or recombinant FSH is indicated in cases of hypogonadotropism and most other types of anovulation (exclusive of ovarian failure). Because of the complexities, laboratory tests, and expense associated with this treatment, these patients should be referred to a specialist.
4. Gonadotropin-releasing hormone (GnRH)
Hypothalamic amenorrhea unresponsive to clomiphene will be reliably and successfully treated with subcutaneous pulsatile gonadotropin-releasing hormone (GnRH). Use of this substance will avoid the dangerous ovarian complications and the 25% incidence of multiple pregnancy associated with hMG, though the overall rate of ovulation and pregnancy is lower than when hMG is used.
D. Treatment of Endometriosis
E. Treatment of Inadequate Transport of Sperm
Intrauterine insemination of concentrated washed sperm has been used to bypass a poor cervical environment associated with scant or hostile cervical mucus. The sperm must be handled by sterile methods, washed in sterile saline or tissue culture solutions, and centrifuged. A small amount of fluid (0.5 mL) containing the sperm is then instilled into the uterus.
F. Artificial Insemination in Azoospermia
If azoospermia is present, artificial insemination by a donor usually results in pregnancy, assuming female function is normal. The use of frozen sperm is currently preferable to fresh sperm because the frozen specimen can be held pending cultures and blood test results for sexually transmitted diseases, including HIV infection.
G. Assisted Reproductive Technologies
Couples who have not responded to traditional infertility treatments, including those with tubal disease, severe endometriosis, oligospermia, and immunologic or unexplained infertility, may benefit from in vitro fertilization (IVF), gamete intrafallopian transfer (GIFT), and zygote intrafallopian transfer (ZIFT). These techniques are complex and require a highly organized team of specialists. All of the procedures involve ovarian stimulation to produce multiple oocytes, oocyte retrieval by TVS-guided needle aspiration, and handling of the oocytes outside the body. With IVF, the eggs are fertilized in vitro and the embryos transferred to the uterine fundus. GIFT involves the placement of sperm and eggs in the uterine tube by laparoscopy or minilaparotomy. With ZIFT, fertilization occurs in vitro, and the early development of the embryo occurs in the uterine tube after transfer by laparoscopy or minilaparotomy. The later two procedures are used infrequently, and the rate of live births per retrieval for 383 programs in the United States in 2000 was 31%. Age is an important determinant of success - for couples under the age of 35, the average rate of live birth was 38% per retrieval, while the rate for women over 42 was 6%. In 2000, 53% of pregnancies were multiple.
A recent development is intracytoplasmic sperm injection (ICSI), which allows fertilization with a single sperm. It was originally intended for couples with male factor infertility, but it was used in approximately half of the IVF procedures in 2000.
The prognosis for conception and normal pregnancy is good if minor (even multiple) disorders can be identified and treated; it is poor if the causes of infertility are severe, untreatable, or of prolonged duration (over 3 years).
It is important to remember that in the absence of identifiable causes of infertility, 60% of couples will achieve a pregnancy within 3 years. Couples with unexplained infertility who do not achieve pregnancy within 3 years should be offered ovulation induction or assisted reproductive technology. Also, offering appropriately timed information about adoption is considered part of a complete infertility regimen.
Rosene-Montella K et al: Evaluation and management of infertility in women: the internist's role. Ann Intern Med 2000; 132:973.
Wright VC et al: Assisted reproductive technology surveillance - United States, 2000. MMWR Surveill Summ 2003;52:1.
Revision date: June 20, 2011
Last revised: by Amalia K. Gagarina, M.S., R.D.
Iatrogenic Ovarian Failure
Hyperandrogenism in Adolescent Girls
Delayed Puberty Epidemiological Aspects
Classification of Polycystic Ovary Syndrome (PCOS)
Primary Amenorrhea with Developed Secondary Sex Characteristics
Non-Classic Adrenal Hyperplasia
Non-Tumoral Acquired Organic Lesions of CNS
Checklist for a Healthy Pelvic Floor Before & After…
I've Had a Cesarean in the Past. What is the…
How Can I Prepare My Pelvic Floor, To Avoid Problems After… | {
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{"url":"http:\/\/www.iumpa.upv.es\/news\/conferences\/conferencia-thomas-kalmes","text":"Queridos compa\u00f1eros.\n\nEste pr\u00f3ximo mi\u00e9rcoles 19 de junio a las 12h. en el Seminario del IUMPA\nse impartir\u00e1 la siguiente conferencia:\n\nConferenciante: Thomas Kalmes\n\nT\u00edtulo: \u201cAn approximation theorem of Runge type for kernels of certain non-elliptic partial differential operators\u201d\n\nFrom Runge\u2019s classical theorem on rational approximation it follows that for open subsets $X_1\\subseteq X_2$ of the complex plane $\\mathbb{C}$ every function holomorphic in $X_1$ can be approximated uniformly on compact subsets of $X_1$ by functions which are holomorphic in $X_2$ if and only if $\\mathbb{C}\\backslash X_1$ has no compact connected component which is contained in $X_2$. This approximation theorem has been generalized independently by Lax and Malgrange from holomorphic functions, i.e.\\ functions in the kernel of the Cauchy-Riemann operator, to kernels of elliptic constant coefficient partial differential operators $P(D)$.\nWe report on a recent generalization of the above approximation theorem to constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schr\\\u201dodinger operator as well as non-degenerate parabolic differential operators like the heat operator. Under the additional hypothesis that $P(D)$ is surjective on both $C^\\infty (X_1)$ and $C^\\infty(X_2)$, we characterize when solutions of the equation $P(D)u=0$ in $X_1$ can be approximated by solutions of the same equation in $X_2$. As part of our result we prove that for a large class of non-elliptic operators $P(D)$ there are non-trivial smooth solutions $u$ to the equation $P(D)u=0$ in $\\mathbb{R}^d$ with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for $P(D)$.","date":"2021-03-02 16:33:20","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\": 0.9284531474113464, \"perplexity\": 254.28931670838898}, \"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-2021-10\/segments\/1614178364027.59\/warc\/CC-MAIN-20210302160319-20210302190319-00472.warc.gz\"}"} | null | null |
namespace blink {
scoped_refptr<PaintRecordPattern> PaintRecordPattern::Create(
sk_sp<PaintRecord> record,
const gfx::RectF& record_bounds,
RepeatMode repeat_mode) {
return base::AdoptRef(
new PaintRecordPattern(std::move(record), record_bounds, repeat_mode));
}
PaintRecordPattern::PaintRecordPattern(sk_sp<PaintRecord> record,
const gfx::RectF& record_bounds,
RepeatMode mode)
: Pattern(mode),
tile_record_(std::move(record)),
tile_record_bounds_(record_bounds) {
// All current clients use RepeatModeXY, so we only support this mode for now.
DCHECK(IsRepeatXY());
// FIXME: we don't have a good way to account for DL memory utilization.
}
PaintRecordPattern::~PaintRecordPattern() = default;
sk_sp<PaintShader> PaintRecordPattern::CreateShader(
const SkMatrix& local_matrix) const {
return PaintShader::MakePaintRecord(
tile_record_, gfx::RectFToSkRect(tile_record_bounds_),
SkTileMode::kRepeat, SkTileMode::kRepeat, &local_matrix);
}
} // namespace blink
| {
"redpajama_set_name": "RedPajamaGithub"
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RSO Records (skrót od Robert Stigwood Organisation Records) – brytyjska wytwórnia płytowa, założona w 1973 roku przez Roberta Stigwooda, a zlikwidowana w 1983 roku. Wylansowała takie gwiazdy rocka i muzyki pop jak: Bee Gees, Cream, Blind Faith, Eric Clapton i Yvonne Elliman. Sprzedała w wielomilionowych nakładach albumy Saturday Night Fever: The Original Movie Sound Track i Grease: The Original Soundtrack from the Motion Picture.
Historia
Reaction Records (1966–1967)
Założyciel RSO Records, Robert Stigwood, urodził się w Australii. Po przyjeździe do Wielkiej Brytanii pracował jako niezależny producent. Po poznaniu Briana Epsteina został współdyrektorem jego wytwórni, NEMS Enterprises, a w 1966 roku założył własną wytwórnię, Reaction Records, której dystrybucją zajmował się Polydor Records. Wśród pierwszych wydawnictw tej wytwórni znalazły się single The Who: "Substitute", "I'm A Boy" i "Happy Jack" oraz album zespołu, A Quick One a także dwa pierwsze albumy Cream: Fresh Cream i Disraeli Gears. 24 lutego 1967 roku Stigwood, zachęcony sukcesem Cream podpisał kontrakt z innym zespołem, Bee Gees, którego członkowie, bracia Gibb przybyli do Wielkiej Brytanii, podobnie jak on, z Australii, Wyprodukował szereg singli tego zespołu, poczynając od "New York Mining Disaster 1941" (12. miejsce na liście przebojów w Wielkiej Brytanii i 14. w Stanach Zjednoczonych) i "To Love Somebody" (17. miejsce na liście przebojów w Wielkiej Brytanii).
RSO (1967–1972)
Po śmierci Epsteina w 1967 roku Stigwood założył firmę Robert Stigwood Organisation, która wypromowała takich artystów jak David Bowie i Rod Stewart. Dystrybucją płyt RSO w Europie zajmował się również Polydor, a w Stanach Zjednoczonych Atco Records z adnotacją: "na mocy porozumienia z RSO". Gdy w 1968 roku zespół Cream rozpadł się, Stigwoodowi udało się przekonać Claptona do utworzenia supergrupy Blind Faith, a po jej rozpadzie pokierować jego solową karierą.
RSO Records (1973–1983)
W 1973 roku Stigwood założył własną wytwórnię płytową, RSO Records (Robert Stigwood Organisation Records). Decyzja zapadła podczas pobytu z zespołem The Who w Japonii. Projektanci opracowali logo nowej wytwórni, ale Stigwood nie był z niego zadowolony. Dopiero gdy kilku z jego japońskich przyjaciół podarowało mu wykonaną z masy papierowej krowę, która w kulturze japońskiej jest symbolem dobrego zdrowia i szczęścia, zaakceptował nowe logo. Do współpracy pozyskał producenta Al Coury'ego, pracującego przedtem dla Capitol Records; Al Coury został prezesem nowego wydawnictwa. W jego ramach intensywnie współpracował z Bee Gees, flagową grupą wytwórni i kierował marketingiem albumów zespołu ze ścieżkami dźwiękowymi Saturday Night Fever: The Original Movie Sound Track i Grease: The Original Soundtrack from the Motion Picture, przyczyniając do tego, iż stały się one jednymi z najlepiej sprzedających się albumów wszech czasów.
Pierwszym albumem wydanym pod szyldem nowej wytwórni był Derek and the Dominos in Concert (styczeń 1973), a jednym z pierwszych sukcesów – szlagier Erica Claptona, "I Shot the Sheriff" (cover przeboju Boba Marleya), pochodzący z albumu 461 Ocean Boulevard. Utwór osiągnął 14 września 1974 roku 1. miejsce na liście przebojów Hot 100 tygodnika Billboard. W tym samym roku Stigwood wyprodukował filmową wersję rock-opery Tommy zespołu The Who, obsadzonej przez takie gwiazdy rocka, jak Eric Clapton, Elton John i Tina Turner; Rock-opera odniosła sukces, a album ze ścieżką dźwiękową z niej znalazł się na szczycie list przebojów.
Jesienią 1977 roku Stigwood wyprodukował film Gorączka sobotniej nocy z Johnem Travoltą w roli głównej. Piosenki do filmu napisał zespół Bee Gees. Cztery z nich zdobyły pierwsze miejsca na listach przebojów; trzy wykonane przez zespół: "How Deep Is Your Love", "Stayin' Alive" i "Night Fever" oraz czwarta, "If I Can't Have You", wykonana przez Yvonne Elliman. Album ze ścieżką dźwiękową, Saturday Night Fever: The Original Movie Sound Track, sprzedano w Stanach Zjednoczonych w ponad 10 milionach egzemplarzy, a na świecie – w około 25 milionach. Był to największy sukces w karierze Stigwooda oraz najlepiej sprzedający się album w tamtym czasie.
Latem 1978 roku Stigwood zaprezentował się na Broadwayu filmową wersję musicalu Grease, z Johnem Travoltą i Olivią Newton-John w rolach głównych. Był to jego kolejny sukces komercyjny. Album ze ścieżką dźwiękową filmu został sprzedany w liczbie 25 milionów egzemplarzy.
W 1978 roku dziewięć kolejnych singli RSO Records znalazło się na pierwszych miejscach amerykańskich list przebojów.
Na początku lat 80., wraz z przeminięciem mody na muzykę disco, dochody RSO Records spadły. W 1983 roku Robert Stigwood sprzedał swoją wytwórnię PolyGramowi, który od listopada 1981 roku zajmował się dystrybucją jego wydawnictw. Przedtem zadanie to realizowały wytwórnie: Atlantic (marzec 1973 – grudzień 1975) i Polydor (styczeń 1976 – grudzień 1977). Od stycznia 1978 do października 1981 RSO Records sprzedawała samodzielnie własne wydawnictwa.
Artyści
Zobacz też
Przypisy
Linki zewnętrzne
RSO Records na Discogs
RSO Records
Wytwórnie muzyczne w Londynie
Popowe wytwórnie muzyczne
Rockowe wytwórnie muzyczne | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,168 |
Jean-Claude Leclercq es un antiguo ciclista francés, nacido el 22 de julio de 1962 en Abbeville, que fue profesional de 1984 a 1993. Descubierto por Jean de Gribaldy, se convirtió en campeón de Francia en ruta en 1985. En 1987, ganó la Flecha Valona, una de las clásicas más prestigiosas.
Resultados
1984
1 etapa del Tour de Limousin
1985
Campeonato de Francia en Ruta
1 etapa del Trofeo Joaquim Agostinho
1986
1 etapa de la Vuelta a Suiza
2º en el Campeonato de Francia de Ciclismo en Ruta
1987
Flecha Valona
1988
1 etapa de la Vuelta a Suiza
1990
2 etapas de la Tirreno-Adriático
1991
1 etapa de la Vuelta a Suiza
1 etapa del Critérium Internacional
1 etapa del Tour de Romandía
1992
1 etapa de la Dauphiné Libéré
Resultados en las grandes vueltas
Tour de Francia
1986 : 56.º
1987 : 50.º
1988 : 58.º
1989 : 68.º
1990 : 139.º
1992 : abandono
Giro de Italia
1993 : abandono
Enlaces externos
Jean-Claude Leclercq en siteducyclisme.net
Ciclistas de Francia
Nacidos en Abbeville (Somme) | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,455 |
Q: Inserting events into google calendar My project is of creating chatbots. And I have an appointment component in it. So when visitor selects any time slots and date of appointment it should get insert in google calendar. I have integrated google calendar before this using Oath login. So this is working fine. Problem comes when I select 3rd April 12AM slot it gets inserted into 2nd April 12AM. And this problem is only from 12AM to 2.30AM. After 2.30AM, all the time slot is inserted perfectly into 3rd April. Please help me out with this issue.
I am sharing my code as well:
include_once 'google-api-php-client/vendor/autoload.php';
$client = new Google_Client();
define("SCOPE",Google_Service_Calendar::CALENDAR_READONLY);
define("APP_NAME","Google Calendar");
$client->setClientId('xyz.apps.googleusercontent.com');
$client->setClientSecret('abc');
$client->setRedirectUri(base_url('dashboard/Dashboard'));
$client->setApplicationName(APP_NAME);
$client->setScopes([SCOPE]);
$client->setAccessType('offline');
$client->setPrompt('select_account consent');
$client->setAccessToken($google_access_token);
$service = new Google_Service_Calendar($client);
$user_timezone = $service->calendars->get('primary')->getTimezone();
$appointment_data=$this->db->query("select * from {PRE}_master_bot_question where question_id='".$response_data['question_id']."' ")->row_array();
$answer_text=explode(",",$this->input->post('answer', true));
$from_time = $answer_text[1];
$datetime_from = date("c", strtotime($date.$from_time)); //Convert datetime into RFC3339 format
$newDate = date('H:i:s', strtotime($answer_text[1]. ' +'.$appointment_data['duration'].'minutes'));
$to_time = date("g:iA", strtotime($newDate));
$datetime_to = date("c", strtotime($date.$to_time)); //Convert datetime into RFC3339 format
// Get user calendar timezone
$user_timezone = $service->calendars->get('primary')->getTimezone();
$event = new Google_Service_Calendar_Event(array(
'summary' => 'Appointment confirmation',
'location' => 'Online Meeting',
'start' => array(
'dateTime' => $datetime_from,
'timeZone' => $user_timezone
),
'end' => array(
'dateTime' => $datetime_to,
'timeZone' => $user_timezone
),
"guestsCanInviteOthers" => false,
"guestsCanModify" => false,
"guestsCanSeeOtherGuests" => false,
));
$calendarId = 'primary';
$event = $service->events->insert($calendarId, $event);
Thanks
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,912 |
Fuel/Friends exclusive: Two new songs from Pete Yorn
Pete Yorn nominated Fuel/Friends this weekend as the virtual digs where you can hear and download his two newest songs! Thanks Pete. He must fondly remember the time I essentially trapped him in a parking lot and made him talk to me for 45 minutes. Nicest guy ever.
"Shotgun" is from Pete's new full length, produced by Mike Mogis and recorded in Omaha. This is fresh from the mastering lab, and feels both darkly brooding and soaring at the same time.
Shotgun – Pete Yorn
…And "Sans Fear" is a highly winsome tune from the Frank Black (Pixies) produced EP, which was recorded in Salem, Oregon.
Sans Fear – Pete Yorn
It's been all quiet since the Westerns EP and Nightcrawler full-length, both in 2006. Now we have two new Yorn releases slated for 2009 and touring to follow. I feel good again.
Previously: Yorn's American Blues, Vol. 1
Tagged with frank black, pete yorn.
Pete Yorn's American Blues (with Frank Black)
That scruffy troubadour in the Omaha studio with producer Mike Mogis (Bright Eyes/Saddle Creek) isn't Jim Morrison in the Paris years, nor is it Jesus, we don't think. That is one Pete Yorn back recording new music. The formidable Frank Black of the Pixies has also been joining him in recent months for "new explorations in music," and all the results are expected in the form of an album rumored to be called American Blues.
American Blues, Vol. 1 – Pete Yorn
Tagged with black francis, frank black, pete yorn.
New Pete Yorn & Kinky: "Use Me" (Bill Withers)
Pete Yorn wants you to use him (and keep on using him until you use him up), and to that end he's crooning along with Mexican electronica/rock band Kinky, covering Bill Withers' soulful classic.
Use Me (Bill Withers cover) – Pete Yorn & Kinky
This comes from the new release from producer Robin Danar, which features a bunch of other artists: Inara George, Paul Buchanan (The Blue Nile), Jesca Hoop, Gary Jules and The Section Quartet, Lisa Loeb and Steve Reynolds, Jim Bianco, Minibar, Rachael Yamagata, Julian Coryell, Quincy Coleman, Julianna Raye and Nic Harcourt.
It's called Altered States and it came out yesterday on Shanachie Records.
Tagged with pete yorn.
Your eyes make a circle :: WAZ & "I Will Follow"
In my estimation, the covers that are really worth their salt take a song that you think you know, and then go right ahead to completely re-imagine it. My favorite revisions unearth a hidden nugget of emotional truth, or get at something that you might have missed the first time around. As The Bangles say, "I see you in a different light."
WAZ is a musician from Southern California who first rose to prominence as the guitarist for Pete Yorn during the musicforthemorningafter era, but now has a solo career in his own right. Waz is a shortening of his last name (no it's not Mike Wazowski) and everyone 'cept his momma calls him that. His cover of U2′s "I Will Follow" is bittersweetly intimate and stopped me in my tracks.
I never knew that this song was originally written about the death of Bono's mother; it has such a huge anthemic blazing image to me, as does much of the U2 from that early era. The original is majestic and almost defiant; it wants to be sung from a thousand enormous arenas. But this version aches, a meandering confessional in front of a backdrop of delicate strings. I love what WAZ does here to reach a different place with this song.
I Will Follow – WAZ
WAZ just opened for Frank Black of the Pixies on some solo shows, culminating at the rad Hotel Utah in SF. More tour dates are anticipated in 2008 to go along with the self-release of his full-length album coming out at the end of January. That album will be released without this track due to some legal wrangling, but he gives his blessing for you to enjoy it here. So do.
Tagged with covers, pete yorn, u2, waz.
I can feel the earth begin to move, I hear my needle hit the groove
[2003 Glastonbury photo credit]
Two nights ago I watched the 2003 Britpop documentary Live Forever (more on that later), which begins by laying a foundation of the music scene in Nineties England from the initial impact of the Stone Roses — so I smiled today when this fantastic cover version came up on a mix I'd made.
Yorn: "So like I said, this is hot shit for us to be over here at Glastonbury. We come from the U.S. of A and this is a very exotic festival that we love and we're happy to be here and we're huge fans of the music over here and blah blah blah . . . This is from Manchester, okay?"
She Bangs The Drums (Stone Roses cover, live at Glastonbury 2003) – Pete Yorn
(apparently this is encoded at a rate that streaming doesn't agree with. Until I can fix it, if you download it, it sounds fine; if you click the blue arrow, you get Alvin & The Chipmunks singing the Stone Roses, which is actually a whole different kind of interesting)
Speaking of she bangs the drum, I could not stop my own personal rhythm section pattered out onto my legs last night at the screening of the Pearl Jam documentary. Seeing and hearing Immagine in Cornice on the big screen with all the glorious surround-sound was an immense experience of live PJ fabulousness. My personal highlights were the renditions of Blood (ugh, love that song), Come Back (sheerly absurdly gorgeous), and a compelling ending of Rockin' In The Free World with every single Italian audience member's hands raised in the air, clapping along in unison.
In addition to the beautifully-done cinematic treatment of their live shows, the documentary also offered some very interesting behind-the-scenes glimpses: the urgent reorganization of the encore setlist backstage while the crowd screams for more, Jeff skateboarding at some deserted Italian skatepark, Ed and his daughter Olivia talking on the tour bus (and how cute is she?), a bunch of Italian kids sitting on the street belting out a passionate acoustic rendition of Porch. Stone barely made an appearance (it's all Stone's fault) and not surprisingly I would have liked for it to be longer so they could have shown more of what goes on that we don't see onstage. But overall, solid A. If I can't see PJ live this year, heck I'll settle for last night. Thanks to all who came out for an awesome experience, it was moltissimo fun.
Finally, the road to Denver will again be my buddy tonight as I head back up to see the Ike Reilly Assassination at the Larimer Lounge. Last time he was here it was acoustic and still mind-blowing, so I am very excited to get the full band baptism. I highly recommend this show if you can make it out.
Tagged with concerts, films, ike reilly, pearl jam, pete yorn, stone roses.
Yorn joins the Swedish Invasion
I find this cover toe-tappingly good ("Everyone keeps asking me if this is my song, so I decided that I have to play it," Yorn says) — and coincidental, considering that the first time I remember enjoying the original song was at an intermission before Pete Yorn took the stage in Denver. Thanks to Stereogum for the cover from two nights ago, and to You Ain't No Picasso for digging up the video:
Young Folks (Peter, Bjorn, and John cover) – Pete Yorn
Live at the Bottle & Cork, Dewey Beach, DE 7/29/07
Now if only we frickin knew who is duetting with him on "Shampoo", I'd be one completely satisfied girl.
Tagged with covers, pete yorn, peter bjorn and john.
New tune and a mystery from Pete Yorn: "Shampoo"
Pete Yorn posted a new song to his MySpace page a week or two ago called "Shampoo" and I've been neglecting my Official Ambassador of Yorn duties in passing along the good news (I hope they don't, uh, fire me).
This track revisits some of the fantastic male/female harmonies (it's what made "The Man" the jam of my summer last year) with a vocalist that I swear I know. Who is that singing along with him? It's driving me a bit nuts trying to place her voice.
Of this delightful new confection, Pete said, "[There's a] batch that is ten songs that I'm just putting together that is kind of a concept record, based on a character from a movie that I like a lot. There's a character called George Roundie that Warren Beatty played in a film called Shampoo and all the songs seem to relate to his life or the character's life. And so it's a soundtrack, I made my own soundtrack to that movie I guess. I'll put that out at a later date."
Shampoo – Pete Yorn
I've gotten the following three bits of information about this song from a source quite close to Sir Yorn. He says:
a little more info. i will give you three items on which to chew:
1. the "george roundy" concept album went away.
2. the song "shampoo" is part of a completely different concept album which does exist and is finished.
3. there may or may not be more tantalizing information regarding this project but i am not at liberty to disperse it.
More tantalizing information. How mysterious. I can't wait.
It's been a long time since I compiled one of these odds and ends posts, but there were several little things today that caught my eye:
Ûž Brian Deck is on board to produce the new Counting Crows record, according to Adam:
Rehearsals have been going really well the past few days. I'm pretty excited about the 2nd half of this record. I really dig the producer we've chosen. His name's Brian Deck. He produced "The Moon and Antarctica" for Modest Mouse, "Our Endless Numbered Days" for Iron and Wine, "The Animal Years" for Josh Ritter, and this album I love by the Fruit Bats called "Mouthfuls". We're getting really cool weird twisted folksy sounds.
Ûž Paul McCartney is set to release a new album this summer, the inaugural release for new Starbucks label.
I drink Starbucks. I love McCartney. But why does this just feel so dirty and somehow depressing?
Ûž Mason Jennings has a new blog post that starts with the sentence, "Did you ever just get so high that you wrote on your arm never to smoke weed again? Me neither." It goes on to discuss music he likes and life in general lately for him, but opening sentences don't get much more engaging than that one.
Ûž I truly love the new Hold Steady video for "Stuck Between Stations." That is a dang fine song, and since I haven't caught them live yet, I've never seen it performed, seen the way they jolt out their music.
Incidentally, I think their piano player may actually be Oliver, Kat's husband from Miami Ink. Rock the 'stache, dude.
Ûž SPIN tries to deconstruct the method behind Ryan Adams' crazy, internet-facilitated, musical-diarrhea madness.
Ûž Pete Yorn's cousin/merch man/video whiz Maxx updates Pete's MySpace friends with setlists and excellent pictures from the road. The most recent post has a haiku to match each photograph, and is a must-read. I laughed out loud at a few:
sid is funnier
when he's not wearing his clothes
but someone else's
simon is undead
he will eat your flesh
even from the stage
Ûž SXSW. Most of the SXSW coverage from my fellow bloggers seems like drinking out of a firehose, and I am not able to fully absorb all of it yet (although I am trying). This, however, was one show that I had read about and found video for — very cool. Pete Townshend was at the fest to speak at a panel and joined British buzz band The Fratellis for a cool little cover of The Who's "The Seeker":
And the best picture that I've seen so far from SXSW was taken by my friend Brian H., who has been regularly updating me with more pics and details than you can shake a stick at (thanks!). I don't know the story behind this shot, but I thought it was cool how it speaks to the environment of total musical domination in Austin these past few days:
Rock 'n roll.
Tagged with counting crows, hold steady, mason jennings, paul mccartney, pete yorn, ryan adams.
Pete Yorn endorses your blog choice
Saw Pete Yorn on Friday night for what I thought was fantastic show with a full band backing (and if you squint and cross your eyes, you can see me in the audience, just three blurry heads to the left of Joe Kennedy's shoulder). Instead of the "acoustic" setting of the You and Me Tour last year, this was a loud and spirited affair with all the members of Minibar on stage (and at the end, the guy from Aqualung rattling his tambourine). This freed Pete up to do some heartwarmingly not-smooth dance moves while singing. We love you Pete.
Everyone was in fine spirits (and enjoying fine spirits), and Pete was animated and friendly. From the opening beats of 'Black,' the guys played a superb setlist, although it was shorter than I liked (having to sit through three opening bands. Yes, three). Highlights for me were a gospel-ly rockin' version of 'Golden Road' that was very different from the Westerns EP, as well as a fun singalong cover of the Stones' 'Dead Flowers.' I debated bringing my camera in and recording that for you since I knew he's been adding it to the shows lately, but I decided to go bulkfree and not lug it. So here's another few guys I like covering it instead…
Dead Flowers – Willie Nelson, Ryan Adams, Hank Williams III, and Keith Richards
The encore ended with a soaring (as usual) version of Crystal Village that made everyone want to kind of drape their arms around their neighbor. Okay, well maybe only me. But those lyrics: "Take my hand, come with me, I see the lights so brightly, and we'll fall as if we never really mattered. . ." It's a fantastic song, rips off Wilco. That's okay.
Crystal Village (live in Jersey) – Pete Yorn
After the show, I talked for a few minutes with Pete, and the first thing he said (if I may lapse into a bit of restrained girlish squealing) is that he liked my blog, that I did a good job with it. So hoo-WAH! Straight from the horse's mouth. Do catch this tour if you can — they've still lots of shows to go, and are sounding great.
Springsteen tribute show planned at Carnegie Hall
Bruce Springsteen is set to get the tribute treatment from the musical community on April 5 at Carnegie Hall in New York. If you were lucky enough to click on the ticket sale website on January 31, then you may have already snagged yourself a pair of tickets (in what concert producer Michael Dorf is calling a "premature leak." Those are always a bit embarassing). Tickets legitimately went on sale Monday (and seem to be sold out) with the proceeds benefitting the Music For Youth program, as with previous tributes to Dylan & Joni Mitchell.
The benefit show will feature appearances from Badly Drawn Boy, Pete Yorn, Steve Earle, Chris Isaak and Josh Ritter, among others. I can't find any recordings of Isaak ever covering Springsteen (some fan correct me if I am wrong), and same for Ritter (although there is plenty of press likening him to Springsteen's songwriting). But here's some hints of what the night may sound like . . .
Thunder Road – Badly Drawn Boy
Dancing In The Dark –> Murray (live) – Pete Yorn
State Trooper (live) – Steve Earle
And if I may, how awesome would it be to see these humble suggestions added to the lineup?
Hungry Heart – Jesse Malin
For You – The Format
No Surrender (live 9/30/05) – Eddie Vedder
(yeah, I've posted this before, but it's one of my absolute favorites)
Tagged with badly drawn boy, chris isaak, ed vedder, jesse malin, josh ritter, pete yorn, springsteen, steve earle, the format. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,288 |
\section{Introduction}
$E_6$ inspired models are well motivated extensions of the Standard
Model (SM). Indeed, supersymmetric (SUSY) models based on the $E_6$
gauge symmetry or its subgroup can originate from the ten--dimensional
heterotic superstring theory \cite{1}. Within this framework gauge and
gravitational anomaly cancellation was found to occur for the gauge
groups $SO(32)$ or $E_8\times E'_8$. However only $E_8\times E'_8$ can
contain the SM since it allows for chiral fermions while $SO(32)$ does
not. Compactification of the extra dimensions results in the breakdown
of $E_8$ up to $E_6$ or one of its subgroups in the observable sector
\cite{2}. The remaining $E'_8$ couples to the usual matter
representations of the $E_6$ only by virtue of gravitational
interactions and comprises a hidden sector that is thought to be
responsible for the spontaneous breakdown of local SUSY
(supergravity). At low energies the hidden sector decouples from the
observable sector of quarks and leptons, the gauge and Higgs bosons and
their superpartners. Its only manifest effect is a set of soft SUSY
breaking terms which spoil the degeneracy between bosons and fermions
within one supermultiplet \cite{3}. The scale of soft SUSY breaking
terms is set by the gravitino mass, $m_{3/2}$. In the simplest SUSY
extensions of the SM these terms also determine the electroweak (EW)
scale. A large mass hierarchy between $m_{3/2}$ and Planck scale can be
caused by the non--perturbative effects in the hidden sector that may
trigger the breakdown of supergravity (SUGRA) \cite{4}.
Since $E_6$ is a rank - 6 group the breakdown of $E_6$ symmetry may
result in low energy models based on rank - 5 or rank - 6 gauge groups,
with one or two additional $U(1)$ gauge group factors in comparison to
the SM. Indeed, $E_6$ contains the maximal subgroup $SO(10)\times
U(1)_{\psi}$ while $SO(10)$ can be decomposed in terms of the
$SU(5)\times U(1)_{\chi}$ subgroup \cite{5}--\cite{Langacker:2008yv}. By
means of the Hosotani mechanism \cite{6} $E_6$ can be broken directly to
$$
E_6\to SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}
$$
which has rank--6. This rank--6 model may be reduced further to an effective
rank--5 model with only one extra gauge symmetry $U(1)'$ which is a linear
combination of $U(1)_{\chi}$ and $U(1)_{\psi}$:
\begin{equation}
U(1)'=U(1)_{\chi}\cos\theta+U(1)_{\psi}\sin\theta\,.
\label{0}
\end{equation}
In the models based on rank - 6 or rank - 5 subgroups of $E_6$ the
anomalies are automatically cancelled if the low energy particle
spectrum consists of a complete representations of $E_6$.
Consequently, in $E_6$-inspired SUSY models one is forced
to augment the minimal particle spectrum by a number of exotics which,
together with ordinary quarks and leptons, form complete fundamental
$27$ representations of $E_6$. Thus we will assume that the
particle content of these models includes at least three fundamental
representations of $E_6$ at low energies. These multiplets decompose
under the $SU(5)\times U(1)_{\psi}\times U(1)_{\chi}$ subgroup of $E_6$
as follows: \begin{equation} \begin{array}{rcl} 27_i &\to &
\displaystyle\left(10,\,\dfrac{1}{\sqrt{24}},\,-\dfrac{1}{\sqrt{40}}\right)_i
+\left(5^{*},\,\dfrac{1}{\sqrt{24}},\,\dfrac{3}{\sqrt{40}}\right)_i
+\left(5^{*},\,-\dfrac{2}{\sqrt{24}},\,-\dfrac{2}{\sqrt{40}}\right)_i
\\[3mm] & + &
\displaystyle\left(5,\,-\dfrac{2}{\sqrt{24}},\,\dfrac{2}{\sqrt{40}}\right)_i
+\left(1,\,\dfrac{4}{\sqrt{24}},\,0\right)_i
+\left(1,\,\dfrac{1}{\sqrt{24}},\,-\dfrac{5}{\sqrt{40}}\right)_i\,. \end{array}
\label{1}
\end{equation} The first, second and third quantities in brackets are the $SU(5)$
representation and extra $U(1)_{\psi}$ and $U(1)_{\chi}$ charges
respectively, while $i$ is a family index that runs from 1 to 3. An
ordinary SM family, which contains the doublets of left--handed quarks
$Q_i$ and leptons $L_i$, right-handed up-- and down--quarks ($u^c_i$ and
$d^c_i$) as well as right--handed charged leptons $(e^c_i)$, is assigned
to $\left(10,\,\dfrac{1}{\sqrt{24}},\,-\dfrac{1}{\sqrt{40}}\right)_i$ +
$\left(5^{*},\,\dfrac{1}{\sqrt{24}},\,\dfrac{3}{\sqrt{40}}\right)_i$.
Right-handed neutrinos $N^c_i$ are associated with the last term in
Eq.~(\ref{1}),
$\left(1,\,\dfrac{1}{\sqrt{24}},\,-\dfrac{5}{\sqrt{40}}\right)_i$. The
next-to-last term, $\left(1,\,\dfrac{4}{\sqrt{24}},\,0\right)_i$,
represents new SM-singlet fields $S_i$, with non-zero $U(1)_{\psi}$
charges that therefore survive down to the EW scale. The pair of
$SU(2)_W$--doublets ($H^d_{i}$ and $H^u_{i}$) that are contained in
$\left(5^{*},\,-\dfrac{2}{\sqrt{24}},\,-\dfrac{2}{\sqrt{40}}\right)_i$
and $\displaystyle\left(5,\,-\dfrac{2}{\sqrt{24}},\,\dfrac{2}{\sqrt{40}}\right)_i$
have the quantum numbers of Higgs doublets. They form either Higgs or
Inert Higgs $SU(2)_W$ multiplets.~\footnote{We use the terminology
``Inert Higgs'' to denote Higgs--like doublets that do not develop
VEVs.} Other components of these $SU(5)$ multiplets form colour
triplets of exotic quarks $\overline{D}_i$ and $D_i$ with electric
charges $+ 1/3$ and $-1/3$ respectively. These exotic quark states carry
a $B-L$ charge $\left(\pm\dfrac{2}{3}\right)$ twice larger than that of
ordinary ones. In phenomenologically viable $E_6$ inspired models they
can be either diquarks or leptoquarks.
The presence of the $Z'$ bosons associated with extra $U(1)$ gauge
symmetries and exotic matter in the low-energy spectrum stimulated the
extensive studies of the $E_6$ inspired SUSY models over the years
\cite{5},~\cite{7}. Recently, the latest Tevatron and early
LHC $Z'$ mass limits in these models have been discussed in
\cite{Accomando:2010fz} while different aspects of phenomenology of
exotic quarks and squarks have been considered in \cite{Kang:2007ib}.
Also the implications of the $E_6$ inspired SUSY models have been
studied for EW symmetry breaking (EWSB)
\cite{Langacker:1998tc}--\cite{Daikoku:2000ep}, neutrino physics
\cite{Kang:2004ix}--\cite{Ma:1995xk}, leptogenesis
\cite{Hambye:2000bn}--\cite{King:2008qb}, EW baryogenesis
\cite{baryogen}, muon anomalous magnetic moment \cite{g-2}, electric
dipole moment of electron \cite{Suematsu:1997tv} and tau lepton
\cite{GutierrezRodriguez:2006hb}, lepton flavour violating processes
like $\mu\to e\gamma$ \cite{Suematsu:1997qt} and CP-violation in the
Higgs sector \cite{Ham:2008fx}. The neutralino sector in $E_6$ inspired
SUSY models was analysed previously in \cite{Keith:1997zb},
\cite{Suematsu:1997tv}--\cite{Suematsu:1997qt},
\cite{Suematsu:1997au}--\cite{E6neutralino-higgs}. Such models have
also been proposed as the solution to the tachyon problems of anomaly
mediated SUSY breaking, via $U(1)^\prime$ D-term contributions
\cite{Asano:2008ju}, and used in combination with a generation symmetry
to construct a model explaining fermion mass hierarchy and mixing
\cite{Stech:2008wd}. An important feature of $E_6$ inspired SUSY models
is that the mass of the lightest Higgs particle can be substantially
larger in these models than in the minimal supersymmetric standard model
(MSSM) and next-to-minimal supersymmetric standard model (NMSSM)
\cite{Daikoku:2000ep}, \cite{King:2005jy}--\cite{Accomando:2006ga}. The
Higgs sector in these models was examined recently in
\cite{E6neutralino-higgs}, \cite{King:2005jy}, \cite{E6-higgs}.
Within the class of rank - 5 $E_6$ inspired SUSY models, there is a
unique choice of Abelian $U(1)_{N}$ gauge symmetry that allows zero
charges for right-handed neutrinos and thus a high scale see-saw mechanism.
This corresponds to $\theta=\arctan\sqrt{15}$.
Only in this Exceptional Supersymmetric Standard Model (E$_6$SSM)
\cite{King:2005jy}--\cite{King:2005my} right--handed neutrinos may be
superheavy, shedding light on the origin of the mass hierarchy in the lepton sector
and providing a mechanism for the generation of the baryon asymmetry in the Universe
via leptogenesis \cite{Hambye:2000bn}--\cite{King:2008qb}. Indeed, the heavy Majorana
right-handed neutrinos may decay into final states with lepton number
$L=\pm 1$, thereby creating a lepton asymmetry in the early universe.
Since in the E$_6$SSM the Yukawa couplings of the new exotic particles
are not constrained by neutrino oscillation data, substantial values of
the CP--asymmetries can be induced even for a relatively small mass of
the lightest right--handed neutrino ($M_1 \sim 10^6\,\mbox{GeV}$) so
that successful thermal leptogenesis may be achieved without
encountering a gravitino problem \cite{King:2008qb}.
Supersymmetric models with an additional $U(1)_{N}$ gauge symmetry have
been studied in \cite{Ma:1995xk} in the context of non--standard
neutrino models with extra singlets, in \cite{Suematsu:1997au} from the
point of view of $Z-Z'$ mixing, in \cite{Keith:1997zb} and
\cite{Suematsu:1997au}--\cite{Keith:1996fv} where the neutralino sector
was explored, in \cite{Keith:1997zb}, \cite{King:2007uj} where the
renormalisation group (RG) flow of couplings was examined and in
\cite{Suematsu:1994qm}--\cite{Daikoku:2000ep} where EWSB was
studied. The presence of a $Z'$ boson and of exotic quarks predicted by
the Exceptional SUSY model provides spectacular new physics signals at
the LHC which were analysed in
\cite{King:2005jy}--\cite{Accomando:2006ga}, \cite{Howl:2007zi}. The
presence of light exotic particles in the E$_6$SSM spectrum also lead to
the nonstandard decays of the SM--like Higgs boson that were discussed
in details in \cite{Hall:2010ix}. Recently the particle spectrum and
collider signatures associated with it were studied within the
constrained version of the E$_6$SSM \cite{8}.
Although the presence of TeV scale exotic matter in $E_6$ inspired SUSY
models gives rise to specatucular collider signatures, it also causes
some serious problems. In particular, light exotic states generically
lead to non--diagonal flavour transitions and rapid proton decay. To
suppress flavour changing processes as well as baryon and lepton number
violating operators one can impose a set of discrete symmetries. For
example, one can impose an approximate $Z^{H}_2$ symmetry, under which
all superfields except one pair of $H^{d}_{i}$ and $H^{u}_{i}$ (say
$H_d\equiv H^{d}_{3}$ and $H_u\equiv H^{u}_{3}$) and one SM-type singlet
field ($S\equiv S_3$) are odd
\cite{King:2005jy}--\cite{King:2005my}. When all $Z^{H}_2$ symmetry
violating couplings are small this discrete symmetry allows to suppress
flavour changing processes. If the Lagrangian of the $E_6$
inspired SUSY models is invariant with respect to either a $Z_2^L$
symmetry, under which all superfields except leptons are even (Model I),
or a $Z_2^B$ discrete symmetry that implies
that exotic quark and lepton superfields are odd whereas the others
remain even (Model II), then the most dangerous baryon and lepton
number violating operators get forbidden and proton is sufficiently
longlived \cite{King:2005jy}--\cite{King:2005my}. The symmetries
$Z^{H}_2$, $Z_2^L$ and $Z_2^B$ obviously do not commute with $E_6$
because different components of fundamental representations of $E_6$
transform differently under these symmetries.
The necessity of introducing multiple discrete symmetries to ameliorate
phenomenological problems that generically arise due to the presence of
low mass exotics is an undesirable feature of these models.
In this paper we consider rank - 6 $E_6$ inspired SUSY models in which a
{\em single} discrete $\tilde{Z}^{H}_2$ symmetry serves to
simultaneously forbid tree--level
flavor--changing transitions and the most dangerous baryon and lepton
number violating operators. We consider models where the $U(1)_{\psi}$
and $U(1)_{\chi}$ gauge symmetries are spontaneously broken at some
intermediate scale so that the matter parity,
\begin{equation} Z_{2}^{M}=(-1)^{3(B-L)}\;,
\label{2}
\end{equation}
is preserved. As a consequence the low-energy spectrum of the models
will include {\em two} stable weakly interacting particles that
potentially contribute to the dark matter density of our Universe.
The invariance of the Lagrangian with respect to $Z_{2}^{M}$ and
$\tilde{Z}^{H}_2$ symmetries leads to unusual collider signatures
associated with exotic states that originate from $27$--plets.
These signatures have not been studied in details before. In
addition to the exotic matter multiplets that stem from the fundamental
$27$ representations of $E_6$ the considered models predict the
existence of a set of vector-like supermultiplets. In particular the
low-energy spectrum of the models involves either a doublet of
vector-like leptons or a triplet of vector-like down type quarks. If
these extra states are relatively light, they will manifest themselves
at the LHC in the near future.
The layout of this paper is as follows. In Section 2 we specify the
rank--6 $E_6$ inspired SUSY models with exact custodial symmetry.
In Section 3 we present five--dimensional ($5D$) and six--dimensional
($6D$) orbifold Grand Unified theories (GUTs) that lead to the rank--6
$E_6$ inspired SUSY models that we propose. In Sections 4 and 5 the
RG flow of gauge couplings and implications for collider phenomenology
and cosmology are discussed. Our results are summarized in Section 6.
\section{$E_6$ inspired SUSY models with exact custodial
$\tilde{Z}^{H}_2$ symmetry}
In our analysis we concentrate on the rank--6 $E_6$ inspired SUSY models with two
extra $U(1)$ gauge symmetries --- $U(1)_{\chi}$ and $U(1)_{\psi}$. In other words
we assume that near the GUT or string scale $E_6$ or its subgroup is broken
down to $SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$. In the
next section we argue that this breakdown can be achieved within orbifold GUT models.
We also allow three copies of 27-plets to survive to low energies so that anomalies get
cancelled generation by generation within each complete $27_i$ representation of $E_6$.
In $E_6$ models the renormalisable part of the superpotential comes from the
$27\times 27\times 27$ decomposition of the $E_6$ fundamental representation and
can be written as
\begin{equation}
\begin{array}{rcl}
W_{E_6}&=&W_0+W_1+W_2,\\[3mm]
W_0&=&\lambda_{ijk}S_i(H^d_{j}H^u_{k})+\kappa_{ijk}S_i(D_j\overline{D}_k)+
h^N_{ijk} N_i^c (H^u_{j} L_k)+ h^U_{ijk} u^c_{i} (H^u_{j} Q_k)+\\[2mm]
&&+h^D_{ijk} d^c_i (H^d_{j} Q_k) + h^E_{ijk} e^c_{i} (H^d_{j} L_k)\,,\\[3mm]
W_1&=& g^Q_{ijk} D_{i} (Q_j Q_k)+g^{q}_{ijk}\overline{D}_i d^c_j u^c_k\,,\\[3mm]
W_2&=& g^N_{ijk}N_i^c D_j d^c_k+g^E_{ijk} e^c_i
D_j u^c_k+g^D_{ijk} (Q_i L_j) \overline{D}_k\,.
\end{array}
\label{3}
\end{equation}
Here the summation over repeated family indexes ($i,j,k=1,2,3$) is implied.
In the considered models $B-L$ number is conserved automatically since the
corresponding global symmetry $U(1)_{B-L}$ is a linear superposition of
$U(1)_Y$ and $U(1)_{\chi}$. At the same time if terms in $W_1$ and $W_2$
are simultaneously present in the superpotential then baryon and lepton numbers
are violated. In other words one cannot define the baryon and lepton numbers of the
exotic quarks $D_i$ and $\overline{D}_i$ so that the complete Lagrangian is
invariant separately under $U(1)_{B}$ and $U(1)_{L}$ global
symmetries. In this case the Yukawa interactions in $W_1$ and $W_2$ give
rise to rapid proton decay.
Another problem is associated with the presence of three families of $H^u_{i}$
and $H^d_{i}$. All these Higgs--like doublets can couple to ordinary quarks
and charged leptons of different generations resulting in the phenomenologically
unwanted flavor changing transitions. For example, non--diagonal flavor
interactions contribute to the amplitude of $K^0-\overline{K}^0$ oscillations
and give rise to new channels of muon decay like $\mu\to e^{-}e^{+}e^{-}$.
In order to avoid the appearance of flavor changing neutral currents (FCNCs)
at the tree level and forbid the most dangerous baryon and lepton number violating
operators one can try to impose a single $\tilde{Z}^{H}_2$ discrete symmetry.
One should note that the imposition of additional discrete symmetry to stabilize
the proton is a generic feature of many phenomenologically viable SUSY models.
In our model building strategy we use $SU(5)$ SUSY GUT as a guideline.
Indeed, the low--energy spectrum of the MSSM, in addition to the complete $SU(5)$
multiplets, contains an extra pair of doublets from $5$ and $\overline{5}$ fundamental
representations, that play a role of the Higgs fields which break EW symmetry.
In the MSSM the potentially dangerous operators, that lead to the rapid proton decay,
are forbidden by the matter parity $Z_{2}^{M}$ under which Higgs doublets are even
while all matter superfields, that fill in complete $SU(5)$ representations,
are odd. Following this inspirational example we augment three
27-plets of $E_6$ by a number of components $M_{l}$ and $\overline{M}_l$ from
extra $27'_l$ and $\overline{27'}_l$ below the GUT scale. Because additional pairs
of multiplets $M_{l}$ and $\overline{M}_l$ have opposite $U(1)_{Y}$, $U(1)_{\psi}$
and $U(1)_{\chi}$ charges their contributions to the anomalies get cancelled
identically. As in the case of the MSSM we allow the set of multiplets $M_{l}$
to be used for the breakdown of gauge symmetry. If the corresponding set includes
$H^u\equiv H_u$, $H^d\equiv H_d$, $S$ and $N^c\equiv N^c_H$ then
the $SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$ symmetry can be
broken down to $U(1)_{em}$ associated with electromagnetism. The VEVs of $S$
and $N^c$ break $U(1)_{\psi}$ and $U(1)_{\chi}$ entirely while the
$SU(2)_W\times U(1)_Y$ symmetry remains intact. When the neutral components of
$H_u$ and $H_d$ acquire non--zero VEVs then $SU(2)_W\times U(1)_Y$ symmetry
gets broken to $U(1)_{em}$ and the masses of all quarks and charged leptons
are generated.
As in the case of the MSSM we assume that all multiplets $M_{l}$ are even
under $\tilde{Z}^{H}_2$ symmetry while three copies of the complete fundamental
representations of $E_6$ are odd. This forbids couplings in the superpotential
that come from $27_i \times 27_j \times 27_k$. On the other hand the $\tilde{Z}^{H}_2$
symmetry allows the Yukawa interactions that stem from $27'_l \times 27'_m \times 27'_n$,
and $27'_l \times 27_i \times 27_k$
The multiplets
$M_{l}$ have to be even under $\tilde{Z}^{H}_2$ symmetry because some of them are
expected to get VEVs. Otherwise the VEVs of the corresponding fields lead to the
breakdown of the discrete $\tilde{Z}^{H}_2$ symmetry giving rise to the baryon and
lepton number violating operators in general. If the set of multiplets $M_{l}$
includes only one pair of doublets $H_d$ and $H_u$ the $\tilde{Z}^{H}_2$ symmetry
defined above permits to suppress unwanted FCNC processes at the tree level since
down-type quarks and charged leptons couple to just one Higgs doublet $H_d$,
whereas the up-type quarks couple to $H_u$ only.
The superfields $\overline{M}_l$ can be either odd or even under this $\tilde{Z}^{H}_2$
symmetry. Depending on whether these fields are even or odd under $\tilde{Z}^{H}_2$
a subset of terms in the most general renormalizable superpotential can be written as
\begin{equation}
\begin{array}{c}
W_{\rm{total}}=Y'_{lmn} 27'_l 27'_m 27'_n + Y_{lij} 27'_l 27_i 27_j +
\tilde{Y}_{lmn} \overline{27'}_l \overline{27'}_m \overline{27'}_n +\\[2mm]
+ \mu'_{il} 27_i \overline{27'}_l + \tilde{\mu}'_{ml} 27'_m \overline{27'}_l...\,,
\end{array}
\label{4}
\end{equation}
where $Y'_{lmn}$ and $Y_{lij}$ are Yukawa couplings and $\mu'_{il}$ and $\tilde{\mu}'_{ml}$
are mass parameters. Also one should keep in mind that only $M_{l}$ and $\overline{M}_l$
components of $27'_l$ and $\overline{27'}_l$ appear below the GUT scale.
If $\overline{M}_l$ is odd under $\tilde{Z}^{H}_2$ symmetry then the term
$\tilde{\mu}'_{ml} 27'_m \overline{27'}_l$ and
$\tilde{Y}_{lmn}\overline{27'}_l \overline{27'}_m \overline{27'}_n $ are forbidden
while $\mu'_{il}$ can have non-zero values. When $\overline{M}_l$ is even $\mu'_{il}$
vanish whereas $\tilde{\mu}'_{ml} 27'_m \overline{27'}_l$ and
$\tilde{Y}_{lmn}\overline{27'}_l \overline{27'}_m \overline{27'}_n $ are allowed by
$\tilde{Z}^{H}_2$ symmetry. In general mass parameters $\mu'_{il}$ and $\tilde{\mu}'_{ml}$ are
expected to be of the order of GUT scale. In order to allow some of the $\overline{M}_l$ multiplets
to survive to low energies we assume that the corresponding mass terms are forbidden
at high energies and get induced at some intermediate scale which is much lower than $M_X$.
The VEVs of the superfields $N^c_H$ and $\overline{N}_H^c$ (that originate from
$27'_N$ and $\overline{27'}_N$) can be used not only for the breakdown of $U(1)_{\psi}$
and $U(1)_{\chi}$ gauge symmetries, but also to generate Majorana masses for the
right--handed neutrinos that can be induced through interactions
\begin{equation}
\Delta W_{N}=\displaystyle\frac{\varkappa_{ij}}{M_{Pl}}(27_i\,\overline{27'}_{N})(27_j\,\overline{27'}_{N})\,.
\label{8}
\end{equation}
The non--renormalizable operators (\ref{8}) give rise to the right--handed neutrino masses
which are substantially lower than the VEVs of $N^c_H$ and $\overline{N}_H^c$. Because
the observed pattern of the left--handed neutrino masses and mixings can be naturally
reproduced by means of seesaw mechanism if the right--handed neutrinos are superheavy,
the $N^c_H$ and $\overline{N}_H^c$ are expected to acquire VEVs
$<N_H^c>\simeq<\overline{N}_H^c>\lesssim M_X$. This implies that $U(1)_{\psi}\times U(1)_{\chi}$
symmetry is broken down to $U(1)_{N}$ near the GUT scale, where $U(1)_{N}$ symmetry
is a linear superposition of $U(1)_{\psi}$ and $U(1)_{\chi}$, i.e.
\begin{equation}
U(1)_N=\dfrac{1}{4} U(1)_{\chi}+\dfrac{\sqrt{15}}{4} U(1)_{\psi}\,,
\label{7}
\end{equation}
under which right-handed neutrinos have zero charges. Since $N^c_H$ and $\overline{N}_H^c$
acquire VEVs both supermultiplets must be even under $\tilde{Z}^{H}_2$ symmetry.
At the same time the VEVs of $N^c_H$ and $\overline{N}_H^c$ may break $U(1)_{B-L}$ symmetry.
In particular, as follows from Eq.~(\ref{3}) the VEV of $N^c_H$ can induce the bilinear
terms $M^{L}_{ij} (H^u_{i} L_{j})$ and $M^{B}_{ij} (D_i d^c_j)$ in the superpotential.
Although such breakdown of gauge symmetry might be possible
the extra particles tend to be rather heavy in the considered case and thus irrelevant for
collider phenomenology. Therefore we shall assume further that the couplings of $N^c_H$
to $27_i$ are forbidden. This, for example, can be achieved by imposing an extra discrete
symmetry $Z_n$. Although this symmetry can forbid the interactions of $N^c_H$ with
three complete $27_i$ representations of $E_6$ it should allow non--renormalizable
interactions (\ref{8}) that induce the large Majorana masses for right-handed neutrinos.
These requirements are fulfilled if Lagrangian is invariant under $Z_2$ symmetry transformations
$N^c_H\to -N^c_H$ and $\overline{N}_H^c\to -\overline{N}_H^c$. Alternatively, one can impose
$Z_n$ symmetry ($n>2$) under which only $N^c_H$ transforms. The invariance of the Lagrangian
with respect to $Z_n$ symmetry ($n>2$) under which only $N^c_H$ transforms implies that the
mass term $\mu_H N^c_H \overline{N}_H^c$ in the superpotential (\ref{4}) is forbidden.
On the other hand this symmetry allows non--renormalizable term in the superpotential
\begin{equation}
\Delta W_{N^c_{H}} = \varkappa\, \displaystyle\frac{(N^c_H \overline{N}_H^c)^n}{M^{2n-3}_{Pl}}\,,.
\label{5}
\end{equation}
In this case $N^c_H$ and $\overline{N}_H^c$ can develop VEVs along the $D$--flat direction so that
\begin{equation}
<N_H^c>\simeq<\overline{N}_H^c>\sim M_{Pl} \cdot
\biggl[ \displaystyle\frac{1}{\varkappa}\frac{M_S}{M_{Pl}}\biggr]^{\displaystyle\frac{1}{2n-2}}\,,
\label{6}
\end{equation}
where $M_S$ is a low--energy supersymmetry breaking scale. This mechanism permits to
generate $<N_H^c>\, \gtrsim 10^{14}\,\mbox{GeV}$ resulting in right-handed neutrino masses
of order of
$$
\varkappa_{ij} M_{Pl} \cdot \biggl[ \displaystyle\frac{1}{\varkappa}\frac{M_S}{M_{Pl}}\biggr]^{\displaystyle\frac{1}{n-1}}
\gtrsim 10^{11}\,\mbox{GeV}\,.
$$
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
& $27_i$ & $27_i$ &$27'_{H_u}$&$27'_{S}$&
$\overline{27'}_{H_u}$&$\overline{27'}_{S}$&$27'_N$&$27'_{L}$&$27'_{d}$\\
& & &$(27'_{H_d})$& &$(\overline{27'}_{H_d})$& &$(\overline{27'}_N)$&$(\overline{27'}_L)$&$(\overline{27'}_{d})$\\
\hline
&$Q_i,u^c_i,d^c_i,$&$\overline{D}_i,D_i,$ & $H_u$ & $S$ &
$\overline{H}_u$&$\overline{S}$&$N^c_H$&$L_4$&$d^c_4$\\
&$L_i,e^c_i,N^c_i$ & $H^d_{i},H^u_{i},S_i$& $(H_d)$ & &
$(\overline{H}_d)$&&$(\overline{N}_H^c)$&$(\overline{L}_4)$&$(\overline{d^c}_4)$\\
\hline
$\tilde{Z}^{H}_2$ & $-$ & $-$ & $+$ & $+$ &
$-$&$\pm$&$+$&$+$&$+$\\
\hline
$Z_{2}^{M}$ & $-$ & $+$ & $+$ & $+$ &
$+$&$+$&$-$&$-$&$-$\\
\hline
$Z_{2}^{E}$ & $+$ & $-$ & $+$ & $+$ &
$-$&$\pm$&$-$&$-$&$-$\\
\hline
\end{tabular}
\caption{Transformation properties of different components of $E_6$ multiplets
under $\tilde{Z}^H_2$, $Z_{2}^{M}$ and $Z_{2}^{E}$ discrete symmetries.}
\label{tab1}
\end{table}
The mechanism of the gauge symmetry breaking discussed above ensures that
the low--energy effective Lagrangian is automatically invariant under the
matter parity $Z_{2}^{M}$. Such spontaneous breakdown of the
$U(1)_{\psi}\times U(1)_{\chi}$ gauge symmetry can occur because $Z_{2}^{M}$
is a discrete subgroup of $U(1)_{\psi}$ and $U(1)_{\chi}$. This follows
from the $U(1)_{\psi}$ and $U(1)_{\chi}$ charge assignments presented in Eq.~(\ref{1}).
Thus in the considered case the VEVs of $N^c_H$ and $\overline{N}_H^c$ break
$U(1)_{\psi}\times U(1)_{\chi}$ gauge symmetry down to $U(1)_{N}\times Z_{2}^{M}$.
As a consequence the low--energy effective Lagrangian is invariant under both
$Z_{2}^{M}$ and $\tilde{Z}^{H}_2$ discrete symmetries. Moreover the $\tilde{Z}^{H}_2$
symmetry is a product of
\begin{equation}
\tilde{Z}^{H}_2 = Z_{2}^{M}\times Z_{2}^{E}\,,
\label{9}
\end{equation}
where $Z_{2}^{E}$ is associated with most of the exotic states. In other words
all exotic quarks and squarks, Inert Higgs and Higgsino multiplets
as well as SM singlet and singlino states that do not get VEV are odd
under $Z_{2}^{E}$ symmetry. The transformation properties of different
components of $27_i$, $27'_l$ and $\overline{27'}_l$ multiplets under
the $\tilde{Z}^{H}_2$, $Z_{2}^{M}$ and $Z_{2}^{E}$ symmetries are summarized
in Table~\ref{tab1}. Since the Lagrangian of the considered $E_6$ inspired
models is invariant under $Z_{2}^{M}$ and $\tilde{Z}^{H}_2$ symmetries it is
also invariant under the transformations of $Z_{2}^{E}$ symmetry.
Because $Z_{2}^{E}$ is conserved the lightest exotic state, which is odd
under this symmetry, is absolutely stable and contributes to the relic
density of dark matter.
It is also well known that in SUSY models the lightest supersymmetric
particle (LSP), i.e. the lightest $R$--parity odd particle
($Z_{2}^{R}=(-1)^{3(B-L)+2s}$), must be stable. If in the considered
models the lightest exotic state (i.e. state with $Z_{2}^{E}=-1$) has
even $R$--parity then the lightest $R$--parity odd state cannot decay
as usual. When the lightest exotic state is $R$--parity odd particle
either the lightest $R$--parity even exotic state or the next-to-lightest
$R$--parity odd state with $Z_{2}^{E}=+1$ must be absolutely stable.
Thus the considered $E_6$ inspired SUSY models contain at least two
dark-matter candidates.
The residual extra $U(1)_{N}$ gauge symmetry gets broken by the VEV
of the SM--singlet superfield $S$ (and possibly $\overline{S}$).
The VEV of the field $S$ induces the mass of the $Z'$ associated with
$U(1)_{N}$ symmetry as well as the masses of all exotic quarks and
inert Higgsinos. If $S$ acquires VEV of order $10-100\,\mbox{TeV}$
(or even lower) the lightest exotic particles can be produced at the LHC.
This is the most interesting scenario that we are going to focus on here.
In some cases the superfield $\overline{S}$ may also acquire non--zero
VEV breaking $U(1)_{N}$ symmetry as we will discuss later. If this is
a case then $\overline{S}$ should be even under the $\tilde{Z}^{H}_2$
symmetry. Otherwise the superfield $\overline{S}$ can be $\tilde{Z}^{H}_2$
odd.
The above consideration indicate that the set of multiplets $M_{l}$ has
to contain at least $H_u$, $H_d$, $S$ and $N^c_H$ in order to guarantee
the appropriate breakdown of the gauge symmetry in the rank--6 $E_6$
inspired SUSY models. However if the set of $\tilde{Z}^{H}_2$ even
supermultiplets $M_{l}$ involve only $H_u$, $H_d$, $S$ and $N^c_H$ then
the lightest exotic quarks are extremely long--lived particles. Indeed,
in the considered case the $\tilde{Z}^{H}_2$ symmetry forbids all
Yukawa interactions in $W_1$ and $W_2$ that allow the lightest exotic
quarks to decay. Moreover the Lagrangian of such model is invariant not
only with respect to $U(1)_L$ and $U(1)_B$ but also under $U(1)_D$
symmetry transformations
\begin{equation}
D\to e^{i\alpha} D\,,\qquad\qquad
\overline{D}\to e^{-i\alpha}\overline{D}\,.
\label{10}
\end{equation}
The $U(1)_D$ invariance ensures that the lightest exotic quark is very
long--lived. The $U(1)_L$, $U(1)_B$ and $U(1)_D$ global symmetries are expected
to be broken by a set of non--renormalizable operators which are suppressed
by inverse power of the GUT scale $M_X$ or $M_{Pl}$. These operators give rise
to the decays of the exotic quarks but do not lead to the rapid proton decay.
Since the extended gauge symmetry in the considered rank--6 $E_6$ inspired
SUSY models forbids any dimension five operators that break $U(1)_D$ global
symmetry the lifetime of the lightest exotic quarks is expected to be of
order of
\begin{equation}
\tau_D\gtrsim M_X^4/\mu_D^5\,,
\label{11}
\end{equation}
where $\mu_D$ is the mass of the lightest exotic quark. When
$\mu_D\simeq \mbox{TeV}$ the lifetime of the lightest exotic quarks
$\tau_D\gtrsim 10^{49}\,\mbox{GeV}^{-1}\sim 10^{17}\,\mbox{years}$, i.e.
considerably larger than the age of the Universe.
The long--lived exotic quarks would have been copiously produced
during the very early epochs of the Big Bang. Those lightest exotic
quarks which survive annihilation would subsequently have been confined
in heavy hadrons which would annihilate further. The remaining heavy
hadrons originating from the Big Bang should be present in terrestrial
matter. There are very strong upper limits on the abundances of
nuclear isotopes which contain such stable relics in the mass range
from $1\,\mbox{GeV}$ to $10\,\mbox{TeV}$. Different experiments set
limits on their relative concentrations from $10^{-15}$ to $10^{-30}$
per nucleon \cite{42}. At the same time various theoretical
estimations \cite{43} show that if remnant particles would exist in
nature today their concentration is expected to be at the level of
$10^{-10}$ per nucleon. Therefore $E_6$ inspired models with
very long--lived exotic quarks are ruled out.
To ensure that the lightest exotic quarks decay within a reasonable
time the set of $\tilde{Z}^{H}_2$ even supermultiplets $M_{l}$
needs to be supplemented by some components of $27$-plet that carry
$SU(3)_C$ color or lepton number. In this context we
consider two scenarios that lead to different collider signatures
associated with the exotic quarks. In the simplest case ({\bf scenario A})
the set of $\tilde{Z}^{H}_2$ even supermultiplets $M_{l}$ involves lepton
superfields $L_4$ and/or $e^c_4$ that survive to low energies. This implies
that $\overline{D}_i$ and $D_i$ can interact with leptons and quarks
only while the couplings of these exotic quarks to a pair of quarks are
forbidden by the postulated $\tilde{Z}^{H}_2$ symmetry. Then baryon
number is conserved and exotic quarks are leptoquarks.
In this paper we restrict our consideration to the $E_6$ inspired
SUSY models that lead to the approximate unification of the $SU(3)_C$,
$SU(2)_W$ and $U(1)_Y$ gauge couplings at some high energy scale $M_X$.
This requirement implies that in the one--loop approximation the
gauge coupling unification is expected to be almost exact. On the
other hand it is well known that the one--loop gauge coupling
unification in SUSY models remains intact if the MSSM particle content
is supplemented by the complete representations of $SU(5)$ (see for
example \cite{Hempfling:1995rb}). Thus we require that the extra matter
beyond the MSSM fill in complete $SU(5)$ representations.
In the {\bf scenario A} this requirement can be fulfilled if
$\overline{H}_u$ and $\overline{H}_d$ are odd under the $\tilde{Z}^{H}_2$
symmetry while $\overline{L}_4$ is $\tilde{Z}^{H}_2$ even
supermultiplet. Then $\overline{H}_u$ and $\overline{H}_d$ from the
$\overline{27'}_l$ can get combined with the superposition of the
corresponding components from $27_i$ so that the resulting
vectorlike states gain masses of order of $M_X$. The supermultiplets
$L_4$ and $\overline{L}_4$ are also expected to form vectorlike states.
However these states are required to be light enough to ensure that
the lightest exotic quarks decay sufficiently fast\footnote{Note that
the superfields $e^c_4$ and $\overline{e^c}_4$ are not allowed to
survive to low energies because they spoil the one--loop gauge coupling
unification.}. The appropriate mass term $\mu_L L_4\overline{L}_4$ in
the superpotential can be induced within SUGRA models just after the
breakdown of local SUSY if the K\"ahler potential contains an extra
term $(Z_L (L_4\overline{L}_4)+h.c)$\cite{45}.
The presence of the bosonic and fermionic components of $\overline{S}$
at low energies is not constrained by the unification of the $SU(3)_C$,
$SU(2)_W$ and $U(1)_Y$ gauge couplings since $\overline{S}$ is the SM
singlet superfield. If $\overline{S}$ is odd under the $\tilde{Z}^{H}_2$
symmetry then it can get combined with the superposition of the
appropriate components of $27_i$. The corresponding vectorlike states
may be either superheavy ($\sim M_X$) or gain TeV scale masses.
When $\overline{S}$ is $\tilde{Z}^{H}_2$ even superfield then its scalar
component is expected to acquire a non-zero VEV breaking $U(1)_N$
gauge symmetry.
Thus {\bf scenario A} implies that in the simplest case the low energy
matter content of the considered $E_6$ inspired SUSY models involves:
\begin{equation}
\begin{array}{c}
3\left[(Q_i,\,u^c_i,\,d^c_i,\,L_i,\,e^c_i,\,N_i^c)\right]
+3(D_i,\,\bar{D}_i)+2(S_{\alpha})+2(H^u_{\alpha})+2(H^d_{\alpha})\\[2mm]
+L_4+\overline{L}_4+N_H^c+\overline{N}_H^c+S+H_u+H_d\,,
\end{array}
\label{12}
\end{equation}
where the right--handed neutrinos $N^c_i$ are expected to gain masses
at some intermediate scale, while the remaining matter survives
down to the EW scale. In Eq.~(\ref{12}) $\alpha=1,2$ and $i=1,2,3$.
Integrating out $N^c_i$, $N^c_H$ and $\overline{N}_H^c$ as well as
neglecting all suppressed non-renormalisable interactions one gets
an explicit expression for the superpotential in the considered case
\begin{equation}
\begin{array}{c}
W_{A} = \lambda S (H_u H_d) + \lambda_{\alpha\beta} S (H^d_{\alpha} H^u_{\beta})
+ \kappa_{ij} S (D_{i} \overline{D}_{j}) + \tilde{f}_{\alpha\beta} S_{\alpha} (H^d_{\beta} H_u)
+ f_{\alpha\beta} S_{\alpha} (H_d H^u_{\beta}) \\[2mm]
+ g^D_{ij} (Q_i L_4) \overline{D}_j
+ h^E_{i\alpha} e^c_{i} (H^d_{\alpha} L_4) + \mu_L L_4\overline{L}_4
+ W_{MSSM}(\mu=0)\,.
\end{array}
\label{13}
\end{equation}
A second scenario, that allows the lightest exotic quarks to decay
within a reasonable time and prevents rapid proton decay, is realized
when the set of multiplets $M_{l}$ together with $H_u$, $H_d$, $S$ and
$N^c_H$ contains an extra $d^c_4$ superfield (instead of $L_4$) from $27'_{d}$.
If the $\tilde{Z}^{H}_2$ even supermultiplet $d^c_4$ survives to low
energies then exotic quarks are allowed to have non-zero Yukawa
couplings with pair of quarks which permit their decays. They can also
interact with $d^c_4$ and right-handed neutrinos. However if Majorana
right-handed neutrinos are very heavy ($\sim M_X$) then the interactions
of exotic quarks with leptons are extremely suppressed. As a consequence
in this {\bf scenario B} $\overline{D}_i$ and $D_i$ manifest themselves
in the Yukawa interactions as superfields with baryon number
$\left(\pm\dfrac{2}{3}\right)$.
Although in the {\bf scenario B} the baryon and lepton number violating
operators are expected to be suppressed by inverse powers of the masses
of the right--handed neutrinos they can still lead to the rapid proton
decay. The Yukawa interactions of the $\tilde{Z}^{H}_2$ even superfield
$d^c_4$ with other supermultiplets of ordinary and exotic matter
can be written in the following form
\begin{equation}
\Delta W_{d^c_4} = h^D_{i k} d^c_4 (H^d_{i} Q_k) +
g^{q}_{ij}\overline{D}_i d^c_4 u^c_j+ g^N_{ij} N_i^c D_j d^c_4\,.
\label{14}
\end{equation}
Integrating out Majorana right-handed neutrinos
one obtains in the leading approximation
\begin{equation}
\Delta W_{d^c_4} \to h^D_{i k} d^c_4 (H^d_{i} Q_k) +
g^{q}_{ij}\overline{D}_i d^c_4 u^c_j+
\frac{\tilde{\varkappa}_{ij}}{M_N} (L_i H_u) (D_j d^c_4)\,,
\label{15}
\end{equation}
where $M_N$ is an effective seesaw scale which is determined
by the masses and couplings of $N^c_i$ and
$\tilde{\varkappa}_{ij}\sim g^N_{ij}$. In the considered case
the baryon and lepton number violation takes place only when
all three terms in Eqs.~(\ref{14})--(\ref{15}) are present
in the superpotential. If $g^N_{ij}=0$
($\tilde{\varkappa}_{ij}=0$) or $g^{q}_{ij}=0$ the baryon and
lepton number conservation requires exotic quarks to be either
diquarks or leptoquarks respectively. When $h^D_{i k}$
vanish the conservation of the baryon and lepton numbers
implies that the superfields $D_i$, $\overline{D}_i$ and $d^c_4$
have the following $U(1)_L$ and $U(1)_B$ charges
$B_D=-B_{\overline{D}}=-B_{d^c_4}=-1/6$ and
$L_D=-L_{\overline{D}}=L_{d^c_4}=-1/2$. This consideration
indicates that in the case when all three terms are present
in Eqs.~(\ref{14})--(\ref{15}) the $U(1)_L$ and $U(1)_B$
global symmetries can not be preserved. It means that
in the leading approximation the proton decay rate is caused
by all three types of the corresponding Yukawa couplings and
has to go to zero when the Yukawa couplings of at least one
type of Yukawa interactions vanish. In practice, the proton
lifetime is determined by the one--loop box diagram that
leads to the dimension seven operator
\begin{equation}
\mathcal{L}_{p}\simeq \left(\frac{c_{ijkl}}{M_S^2}\right)
\left(\frac{\langle H_u \rangle}{M_N}\right)
\Biggl[\epsilon_{\alpha\beta\gamma}\overline{u^c}_{\alpha i} d_{\beta j}
\overline{\nu}_{k} d_{\gamma l}\Biggr]\,,
\label{16}
\end{equation}
where $\langle H_u \rangle = v_2/\sqrt{2}$ and
$c_{ijkl}\propto \tilde{\varkappa}\, g^{q}\, (h^D)^2$.
In Eq.~(\ref{16}) Greek indices denote the color degrees of freedom while
$SU(2)$ indices are suppressed. Here we assume that all
particles propagating in the loop have masses of the order of $M_S$.
For $M_N\gtrsim 10^{11}\,\mbox{GeV}$ and
$h^D_{i k} \sim g^{q}_{ij} \sim g^N_{ij}$ the appropriate suppression
of the proton decay rate can be achieved if the corresponding Yukawa
couplings are less than $10^{-5}$.
Once again, the requirement of the approximate unification of the
$SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ gauge couplings constrains the low
energy matter content in the {\bf scenario B}. The concept of gauge
coupling unification implies that the perturbation theory method
provides an adequate description of the RG flow of gauge couplings
up to the GUT scale $M_X$ at least. The requirement of the validity of
perturbation theory up to the scale $M_X$ sets stringent constraint
on the number of extra $SU(2)_W$ and $SU(3)_C$ supermultiplets that
can survive to low energies in addition to three complete fundamental
representations of $E_6$. For example, the applicability of perturbation
theory up to the high energies permits only one extra pair of $SU(3)_C$
triplet superfields to have mass of the order of TeV scale. The same
requirement limits the number of pairs of $SU(2)_W$ doublets to two.
Because in the {\bf scenario B} the $\tilde{Z}^{H}_2$ even supermultiplets
$d^c_4$ and $\overline{d^c}_4$ are expected to form vectorlike states
which have to have TeV scale masses the limit caused by the validity of
perturbation theory up to the scale $M_X$ is saturated. Then in order to
ensure that the extra matter beyond the MSSM fill in complete $SU(5)$
representations $\overline{H}_u$ and $\overline{H}_d$ should survive to
the TeV scale as well. As before we assume that these supermultiplets are
odd under the $\tilde{Z}^{H}_2$ symmetry so that they can get combined
with the superposition of the corresponding components from $27_i$ at
low energies forming vectorlike states. Again the superfield $\overline{S}$
may or may not survive to the TeV scale. It can be either even or
odd under the $\tilde{Z}^{H}_2$ symmetry. If $\overline{S}$ is
$\tilde{Z}^{H}_2$ even, it should survive to low energies
and its scalar component is expected to get a VEV.
Following the above discussion the low energy matter content in the
simplest case of the {\bf scenario B} may be summarized as:
\begin{equation}
\begin{array}{c}
3\left[(Q_i,\,u^c_i,\,d^c_i,\,L_i,\,e^c_i,\,N_i^c)\right]
+3(D_i,\,\bar{D}_i)+3(H^u_{i})+3(H^d_{i})+2(S_{\alpha})\\[2mm]
+d^c_4+\overline{d^c}_4+N_H^c+\overline{N}_H^c+H_u+\overline{H}_u
+H_d+\overline{H}_d+S\,.
\end{array}
\label{17}
\end{equation}
All states in Eq.~(\ref{17}) are expected to be considerably lighter than
the GUT scale $M_X$. Assuming that $N^c_i$, $N^c_H$ and $\overline{N}_H^c$
gain intermediate scale masses the renormalizable part of the
TeV scale superpotential associated with the {\bf scenario B}
can be written as
\begin{equation}
\begin{array}{c}
W_{B} = \lambda S (H_u H_d) + \lambda_{ij} S (H^d_{i} H^u_{j})
+ \kappa_{ij} S (D_{i} \overline{D}_{j}) + \tilde{f}_{\alpha i} S_{\alpha} (H^d_{i} H_u)
+ f_{\alpha i} S_{\alpha} (H_d H^u_{i}) \\[2mm]
+ g^{q}_{ij}\overline{D}_i d^c_4 u^c_j + h^D_{ij} d^c_4 (H^d_{i} Q_j)
+ \mu_d d^c_4\overline{d^c}_4 + \mu^u_{i} H^u_{i} \overline{H}_u
+ \mu^d_{i} H^d_{i} \overline{H}_d + W_{MSSM}(\mu=0)\,.
\end{array}
\label{18}
\end{equation}
The superpotential (\ref{18}) contains a set of the TeV scale mass
parameters, i.e. $\mu_d$, $\mu^u_{i}$, $\mu^d_{i}$. These are introduced
to avoid massless fermionic states associated with $d^c_4$, $\overline{d^c}_4$,
$\overline{H}_u$ and $\overline{H}_d$ supermultiplets and can be induced
after the breakdown of local SUSY as it has been discussed earlier.
On the other hand the superpotential (\ref{18}) also contains the Yukawa
couplings $g^{q}_{ij}$ and $h^D_{ij}$ which are expected to be small
in order to avoid rapid proton decay. The appropriate suppression of the
corresponding Yukawa couplings and mass parameters $\mu_d$, $\mu^u_{i}$
and $\mu^d_{i}$ can be achieved if the Lagrangian of the $E_6$ inspired
model is invariant under the discrete $Z_k$ symmetry which gets broken
spontaneously at the intermediate scale. As an example one can consider
the model with extra SM singlet superfield $\Phi$ which transforms
under the discrete $Z_k$ symmetry. For concreteness here we assume that
at high energies the Lagrangian of the model is invariant under the
$Z_6$ symmetry transformations
\begin{equation}
\Phi\to \omega\,\Phi\,,\qquad d^c_4\to \omega^5\, d^c_4\,,\qquad
\overline{d^c}_4\to \omega^3\, \overline{d^c}_4\,,\qquad
\overline{H}_u\to \omega^2\,\overline{H}_u\,,\qquad
\overline{H}_d\to \omega^2\,\overline{H}_d\,,
\label{19}
\end{equation}
where $\omega=e^{i\pi/3}$. Then the part of the superpotential
that depends on the $d^c_4$, $\overline{d^c}_4$, $\overline{H}_u$,
$\overline{H}_d$ and $\Phi$ takes the form
\begin{equation}
\begin{array}{rcl}
\Delta W_{Z_6} &=& \displaystyle\frac{\Phi}{M_{Pl}}\biggl[\sigma_{ij} d^c_4 (H^d_{i} Q_j) +
\tilde{\sigma}_{ij} \overline{D}_i d^c_4 u^c_j+ \hat{\sigma}_{ij} N_i^c D_j d^c_4\biggr]\\[3mm]
&+&\displaystyle\frac{\Phi^4}{M_{Pl}^{3}}\biggl[\eta_d d^c_4\overline{d^c}_4 +
\eta^u_{i} H^u_{i} \overline{H}_u + \eta^d_{i} H^d_{i} \overline{H}_d \biggr] +
\sigma\displaystyle\frac{\Phi^6}{M_{Pl}^{3}} + ...\,.
\end{array}
\label{20}
\end{equation}
At the intermediate scale the imposed $Z_6$ symmetry may be broken spontaneously
by the VEV of the superfield $\Phi$
\begin{equation}
<\Phi>\sim \biggl[\displaystyle\frac{M_S}{M_{Pl}}\biggr]^{1/4} M_{Pl}\simeq 10^{14}\,\mbox{GeV}
\label{21}
\end{equation}
inducing bilinear mass terms in the superpotential and small Yukawa
couplings of the $d^c_4$ supermultiplet to other superfields. The corresponding
Yukawa couplings and mass parameters are given by \footnote{The same mechanism
can be used for the generation of the mass term $\mu_L L_4\overline{L}_4$
in the scenario A.}
\begin{equation}
\mu_d \sim \mu^u_{i} \sim \mu^d_{i} \sim \dfrac{<\Phi^4>}{M_{Pl}^3}\simeq M_S\,,
\qquad
h^D_{i k} \sim g^{q}_{ij} \sim g^N_{ij} \lesssim \dfrac{<\Phi>}{M_{Pl}}\sim 10^{-4}\,.
\label{22}
\end{equation}
Although {\bf scenarios A} and {\bf B} discussed in this section
allow us to suppress baryon and lepton number violating operators and
non-diagonal flavor transitions they have at least one drawback.
Both scenarios imply that a number of incomplete $E_6$ multiplets
survive below the scale $M_X$. In fact, the number of incomplete $E_6$
multiplets tends to be larger than the number of generations.
Therefore the origin and mechanism resulting in the incomplete
$E_6$ representations requires further justification. The splitting
of GUT multiplets can be naturally achieved in the framework of
orbifold GUTs. In the next section we present
$5D$ and $6D$ orbifold GUT models that can lead to the
{\bf scenarios A} and {\bf B} just below the GUT scale.
\section{5D and 6D orbifold GUT models}
The structure of the $E_6$ inspired SUSY models discussed in the previous
section, its gauge group and field content, points towards an underlying
GUT model based on the $E_6$ or its subgroup. The breaking of these GUT groups
down to the $SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$
is in general rather involved and requires often large Higgs representations.
In particular, the splitting of GUT multiplets (like doublet-triplet splitting
within SU(5) GUT) requires either fine--tuning of parameters or additional,
sophisticated mechanisms \cite{Masiero:1982fe}--\cite{Altarelli:2000fu}.
Higher--dimensional theories offer new possibilities
to describe gauge symmetry breaking. A simple and elegant scheme is provided
by orbifold compactifications which have been considered for SUSY GUT models
in five dimensions \cite{Kawamura:2000ev}--\cite{Braam:2010sy} and six
dimensions \cite{5d+6d-susy-ogut}--\cite{Buchmuller:2004eg}. These models
apply ideas that first appeared in string--motivated work \cite{Candelas:1985en}:
the gauge symmetry is broken by identifications imposed on the gauge fields
under the spacetime symmetries of an orbifold. In these models many good
properties of GUT's like gauge coupling unification and charge quantization
are maintained while some unsatisfactory properties of the conventional breaking
mechanism, like doublet-triplet splitting, are avoided. Recently, orbifold
compactifications of the heterotic string have been constructed which can
account for the SM in four dimensions and which have five--dimensional or
six--dimensional GUT structures as intermediate step very similar to orbifold
GUT models \cite{Buchmuller:2005jr}. Hence, orbifold compactifications provide
an attractive starting point for attempts to embed the SM into higher dimensional
string theories.
\subsection{$SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$ model in five dimensions}
The simplest GUT group which unifies the gauge interactions of the SM is $SU(5)$
\cite{Georgi:1974sy}. Therefore we first analyze the higher dimensional SUSY GUT model
based on the $SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$ gauge group which is a rank--6
subgroup of $E_6$. For simplicity we consider a single compact extra dimension $S^1$,
$y (=x_5)$, and assume a fixed radius with size given by the GUT scale ($R\sim 1/M_X$).
The orbifold $S^1/Z_2$ is obtained by dividing the circle $S^1$ with a $Z_2$
transformation which acts on $S^1$ according to $y\to -y$. The components of the
$SU(5)$ supermultiplets that propagate in 5 dimensions transform under the specified
$Z_2$ action as $\Phi(x_{\mu}, -y) = P \Phi(x_{\mu}, y)$, where $P$ acts on each
component of the $SU(5)$ representation $\Phi$, making some components positive
and some components negative, i.e. $P=(+,+,...-, -,...)$. The Lagrangian should be
invariant under the $Z_2$ transformations\footnote{It is worth to point out that the
$Z_2$ invariance of the Lagrangian does not require that $P=\pm I$, where $I$
is the unit matrix. In general, matrix $P$ should satisfy the condition $P^2=I$.}.
The $Z_2$ transformation can be regarded as equivalence relation that allows to reduce
the circle $S^1$ to the interval $y\in [0, \pi R]$.
Here we consider a 5-dimensional space-time factorized into a product of the
ordinary $4D$ Minkowski space time $M^4$ and the orbifold $S^1/(Z_2\times Z'_2)$.
The orbifold $S^1/(Z_2\times Z'_2)$ is obtained by dividing $S^1/Z_2$ with
another $Z_2$ transformation, denoted by $Z'_2$, which acts as $y'\to -y'$,
with $y'\equiv y - \pi R/2$. Each reflection symmetry, $y\to -y$ and $y'\to -y'$,
has its own orbifold parity, $P$ and $P'$, which are defined by
\begin{equation}
\begin{array}{c}
\Phi(x,y)\to \Phi(x, -y) = P \Phi(x_{\mu}, y)\,,\\[2mm]
\Phi(x,y')\to \Phi(x, -y') = P' \Phi(x_{\mu}, y')
\end{array}
\label{23}
\end{equation}
where $\Phi(x,y)$ is an $SU(5)$ multiplet field living in the $5D$ bulk,
while $P$ and $P'$ are matrix representations of the two $Z_2$ operator
actions which have eigenvalues $\pm 1$. All interactions must be invariant
under $Z_2\times Z'_2$ symmetry.
Each reflection also introduces special points, $O$ and $O'$, located at
$y=0$ and $y=\pi R/2\equiv \ell$ which are fixed points of the transformations.
The equivalences associated with the two reflection symmetries allow to
work with the theory obtained by truncating to the physically irreducible
interval $y\in [0, \ell]$ with the two $4D$ walls (branes) placed at the
fixed points $y=0$ and $y=\ell$. These are only two inequivalent branes
(the branes at $y=\pi R$ and $y=-\pi R/2$ are identified with those at $y=0$
and $y=\pi R/2$, respectively). Thus physical space reduces to the
interval $[0, \ell]$ with a length of $\pi R/2$.
Denoting the $5D$ bulk field with $(P,\,P')=(\pm 1, \pm 1)$ by $\phi_{\pm\pm}$
one obtains the following Fourier expansions
\cite{Kawamura:2000ev}--\cite{Hebecker:2001wq}:
\begin{eqnarray}
\phi_{++}(x,y) =\sum_{n=0}^{\infty} \frac{1}{\sqrt{2^{\delta_{n,0}}}}
\sqrt{\frac{4}{\pi R}} \phi^{(2n)}_{++}(x)\cos\frac{2ny}{R}\,,
\label{24}
\end{eqnarray}
\begin{eqnarray}
\phi_{+-}(x,y) =\sum_{n=0}^{\infty} \sqrt{\frac{4}{\pi R}}
\phi^{(2n+1)}_{+-}(x)\cos\frac{(2n+1)y}{R}\,,
\label{25}
\end{eqnarray}
\begin{eqnarray}
\phi_{-+}(x,y) =\sum_{n=0}^{\infty} \sqrt{\frac{4}{\pi R}}
\phi^{(2n+1)}_{-+}(x)\sin\frac{(2n+1)y}{R}\,,
\label{26}
\end{eqnarray}
\begin{eqnarray}
\phi_{--}(x,y) =\sum_{n=0}^{\infty} \sqrt{\frac{4}{\pi R}}
\phi^{(2n+2)}_{--}(x)\sin\frac{(2n+2)y}{R}\,,
\label{27}
\end{eqnarray}
where $n$ is a non--negative integer. From the $4D$ perspective
the Fourier component fields $\phi^{(2n)}_{++}(x)$, $\phi^{(2n+1)}_{+-}(x)$,
$\phi^{(2n+1)}_{-+}(x)$ and $\phi^{(2n+2)}_{--}(x)$ acquire masses
$2n/R$, $(2n+1)/R$, $(2n+1)/R$ and $(2n+2)/R$ upon compactification.
Note that only $\phi_{++}(x,y)$ and $\phi_{+-}(x,y)$ can exist on
the $y=0$ brane. The fields $\phi_{++}(x,y)$ and $\phi_{-+}(x,y)$
are non--vanishing on the $y=\pi R/2$ brane, whereas the field
$\phi_{--}(x,y)$ vanishes on both branes.
Only $\phi_{++}(x,y)$ fields have zero--modes. Since full $SU(5)$
$5D$ multiplets $\Phi_i(x,y)$ can, in general, contain components
with even and odd parities, $P$ and $P'$, the matter content of the
massless sector can be smaller than that of the full $5D$ multiplet.
Unless all components of $\Phi(x,y)$ have common parities,
the gauge symmetry reduction occurs upon compactification.
As in the case of the simplest orbifold GUT scenarios
\cite{Kawamura:2000ev}--\cite{Hebecker:2001wq} we start
from the model with the minimal SUSY in $5D$ (with $8$ real supercharges,
corresponding to $N=2$ in 4D). We assume that the vector supermultiplets
associated with the $SU(5)$, $U(1)_{\chi}$ and $U(1)_{\psi}$ interactions
exist in the bulk $M^4\times S^1/(Z_2\times Z'_2)$. The $5D$ gauge
supermultiplets contain vector bosons $A_{M}$ ($M=0,1,2,3,5$) and
gauginos. The $5D$ gaugino is composed of two $4D$ Weyl fermions with
opposite $4D$ chirality, $\lambda$ and $\lambda'$. In addition
$5D$ vector supermultiplets have to involve real scalars $\sigma$
to ensure that the numbers of bosonic and fermionic degrees of freedom
are equal. Thus $5D$ gauge supermultiplets can be decomposed into
vector supermultiplets $V$ with physical components $(A_{\mu}, \lambda)$
and chiral multiplets $\Sigma$ with components
$\biggl((\sigma+i A_5)/\sqrt{2},\lambda'\biggr)$ under $N=1$
supersymmetry in $4D$. These two $N=1$ supermultiplets also form
$N=2$ vector supermultiplet in $4D$.
In addition to the $5D$ vector supermultiplets we assume the presence
of other $SU(5)$ representations as well as $SU(5)$ singlet superfields
that carry non--zero $U(1)_{\chi}$ and $U(1)_{\psi}$ charges in the $5D$
bulk. The corresponding representations also contain $5D$ fermions.
Since each $5D$ fermion state is composed of two $4D$ Weyl fermions,
$\psi$ and $\psi^c$, SUSY implies that each $5D$ supermultiplet includes
two complex scalars $\phi$ and $\phi^c$ as well. The states $\phi$, $\psi$,
$\phi^c$ and $\psi^c$ form one $4D$ $N=2$ hypermultiplet that consists
of two $4D$ $N=1$ chiral multiplets, $\hat{\Phi}\equiv (\phi,\,\psi)$
and $\hat{\Phi}^c \equiv (\phi^c,\,\psi^c)$, transforming as conjugate
representations with each other under the gauge group.
Taking into account that the derivative $\partial_5$ is odd under the
reflection $Z_2$ one can show that the 5D SUSY Lagrangian is invariant
under the following transformations \cite{Kawamura:2000ev}
\begin{equation}
\begin{array}{rcl}
A_{\mu}(x,y) & \to &A_{\mu}(x, -y) = P A_{\mu}(x,y) P^{-1}\,,\\[1mm]
A_{5}(x,y) & \to &A_{5}(x, -y) = - P A_{5}(x,y) P^{-1}\,,\\[1mm]
\sigma(x,y) & \to &\sigma(x, -y) = - P \sigma(x,y) P^{-1}\,,\\[1mm]
\lambda(x,y) & \to &\lambda(x, -y) = P \lambda(x,y) P^{-1}\,,\\[1mm]
\lambda'(x,y)& \to &\lambda'(x, -y) = - P \lambda'(x,y) P^{-1}\,,\\[1mm]
\phi_i(x,y) & \to &\phi_i(x, -y) = P \phi_i(x, y) \,,\\[1mm]
\psi_i(x,y) & \to &\psi_i(x, -y) = P \psi_i(x, y) \,,\\[1mm]
\phi_i^c(x,y)& \to &\phi_i^c(x, -y) = -P \phi_i^c(x, y) \,,\\[1mm]
\psi_i^c(x,y)& \to &\psi_i^c(x, -y) = -P \psi_i^c(x, y) \,,
\end{array}
\label{28}
\end{equation}
where index $i$ represents different $SU(5)$ supermultiplets
that exist in the bulk $M^4\times S^1/(Z_2\times Z'_2)$. In the case
of $SU(5)$ the components of the corresponding $N=2$ vector
supermultiplet in Eq.~(\ref{28}) are given by $V(x, y)=V^{A}(x, y) T^{A}$
and $\Sigma(x, y)=\Sigma^A(x, y) T^A$, where $T^A$ is the set of the
$SU(5)$ generators ($A=1,2,...,24$). The transformations in Eq.~(\ref{28})
are associated with the $Z_2$ reflection symmetry. By replacing $y$ and
$P$ by $y'$ and $P'$ in Eq.~(\ref{28}) one obtains $Z'_2$ transformations.
Note that mass terms for $\phi_i$, $\psi_i$, $\phi^c_i$ and $\psi^c_i$
are allowed by $N=2$ SUSY but these terms are not compatible with the
$P$ and $P'$ parity assignments as follows from Eq.~(\ref{28}).
Therefore the zero--modes of these fields do not receive a bulk
mass contribution.
It is convenient to choose the matrix representation of the parity
assignment $P$, expressed in the fundamental representation of $SU(5)$,
to be $P=\mbox{diag}(+1, +1, +1, +1, +1)$ so that
$V^{A}(x, -y) T^{A}=V^{A}(x, y) T^{A}$. This boundary condition does
not break $SU(5)$ on the $O$ brane at $y=0$. However $4D$ $N=2$ supersymmetry
gets broken by this parity assignment to $4D$ $N=1$ SUSY. This can be seen
explicitly by examining the masses of the Kaluza--Klein (KK) towers of the fields.
Indeed, according to the parity assignment $P$ only $A_{\mu}$, $\lambda$,
$\phi$ and $\psi$ are allowed to have zero--modes whereas other components
of the $N=2$ vector supermultiplet ($\sigma, \lambda'$) and $N=2$ hypermultiplets
$(\phi^c_i,\,\psi^c_i)$ with odd parity P do not possess massless modes.
For the $SU(5)$ gauge symmetry to provide an understanding of the quark and
lepton quantum numbers, the three families of $27_i$ representations of $E_6$
should reside on the $O$ brane where the $SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$
gauge symmetry and $N=1$ SUSY remains intact.
Then at low energies all ordinary quarks and leptons have to fill in complete
$SU(5)$ multiplets.
The $5D$ $SU(5)$ gauge symmetry is reduced to $4D$ $SU(3)_C\times SU(2)_W\times U(1)_Y$
gauge symmetry by choosing $P'=\mbox{diag}(-1, -1, -1, +1, +1)$ acting on
the fundamental representation of $SU(5)$. This boundary condition breaks not
only $SU(5)$ but also $4D$ $N=2$ SUSY to $4D$ $N=1$ SUSY on the $O'$ brane at $y=\ell$.
The parity assignment associated with the $Z'_2$ reflection symmetry leads to
the two types of the $SU(5)$ gauge generators $T^a$ and $T^{\hat{a}}$.
All generators of the SM gauge group satisfy the condition
\begin{equation}
P'\,T^a\, P' = T^a\,.
\label{29}
\end{equation}
Therefore the corresponding gauge fields $A^{a}_{\mu}(x,y)$ and gauginos $\lambda^a(x,y)$
are even under the reflections $Z_2$ and $Z'_2$ whereas $\sigma^a(x, y)$ and
$\lambda^{'a}(x,y)$ are odd. As a consequence the KK expansions of vector bosons
$A^{a}_{\mu}(x,y)$ and gauginos $\lambda^a(x,y)$ contain massless zero modes
$A^{a(0)}_{\mu}(x)$ and $\lambda^{a(0)}(x)$ corresponding to the unbroken
gauge symmetry of the SM. These zero modes form $4D$ $N=1$ vector supermultiplets.
The KK modes $A^{a(2n)}_{5}(x)$ are swallowed by $A^{a(2n)}_{\mu}(x)$ resulting in
the formation of vector boson state with mass $2n/R$. The KK gaugino modes
$\lambda^{a(2n)}(x)$ and $\lambda^{'a(2n)}(x)$ form $4D$ fermion state
with mass $2n/R$. The KK scalar mode $\sigma^{a(2n)}(x)$ also gains mass $2n/R$.
The other gauge generators $T^{\hat{a}}$ of $SU(5)$ obey the relationship
\begin{equation}
P'\,T^{\hat{a}}\, P' = - T^{\hat{a}}\,,
\label{30}
\end{equation}
which implies that $A^{\hat{a}}_{\mu}(x,y)$ and $\lambda^{\hat a}(x,y)$ are odd under
the $Z'_2$ symmetry while $\sigma^{\hat{a}}(x, y)$ and $\lambda^{'\hat{a}}(x,y)$ are even.
This means that all components of the $5D$ vector supermultiplet associated with
the broken $SU(5)$ generators $T^{\hat{a}}$ are odd either under the reflection $Z_2$
or $Z'_2$ so that their KK expansions does not possess massless modes.
The $Z_2$ and $Z'_2$ parity assignments for all components of the $5D$ bulk vector
supermultiplets are shown in Table~\ref{tab2}. The KK modes $A^{\hat{a}(2n+1)}_{\mu}(x)$,
$A^{\hat{a}(2n+1)}_{5}(x)$, $\sigma^{\hat{a}(2n+1)}(x)$, $\lambda^{\hat{a}(2n+1)}(x)$ and
$\lambda^{'\hat{a}(2n+1)}(x)$ form vector boson, scalar and fermion states
with masses $(2n+1)/R$.
\begin{table}[ht]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$5D$ fields &$SU(3)_C\times SU(2)_W$ &$Z_2\times Z'_2$& Mass\\
& quantum numbers & parity &\\
\hline
$A^a_{\mu},\,\lambda^a$ & $(8,1)+(1,3)+(1,1)$ & $(+,+)$ & $2n/R$\\
\hline
$A^{\hat{a}}_{\mu},\,\lambda^{\hat{a}}$ & $(3,2)+(\bar{3},2)$ & $(+,-)$ & $(2n+1)/R$\\
\hline
$A^a_{5},\,\sigma^a,\,\lambda^{'a}$ & $(8,1)+(1,3)+(1,1)$ & $(-,-)$ & $(2n+2)/R$\\
\hline
$A^{\hat{a}}_{5},\,\sigma^{\hat{a}},\,\lambda^{'\hat{a}}$ & $(3,2)+(\bar{3},2)$ & $(-,+)$ & $(2n+1)/R$\\
\hline
$A^{\chi}_{\mu},\,\lambda_{\chi}$ & $(1,1)$ & $(+,+)$ & $2n/R$\\
\hline
$A^{\chi}_{5},\,\sigma_{\chi},\,\lambda'_{\chi}$ & $(1,1)$ & $(-,-)$ & $(2n+2)/R$\\
\hline
$A^{\psi}_{\mu},\,\lambda_{\psi}$ & $(1,1)$ & $(+,+)$ & $2n/R$\\
\hline
$A^{\psi}_{5},\,\sigma_{\psi},\,\lambda'_{\psi}$ & $(1,1)$ & $(-,-)$ & $(2n+2)/R$\\
\hline
\end{tabular}
\caption{Parity assignments and KK masses of fields in the $5D$ bulk vector
supermultiplets associated with the $SU(5)$, $U(1)_{\psi}$ and $U(1)_{\chi}$
gauge interactions.}
\label{tab2}
\end{table}
At the fixed point $O'$ the gauge transformations generated by $T^{\hat{a}}$ as well as the
corresponding components of the $5D$ $SU(5)$ vector supermultiplet vanish. At the same time
at an arbitrary point in the bulk all generators of the $SU(5)$ gauge group are operative.
Thus orbifold procedure leads to a local explicit breaking of $SU(5)$ at the fixed point $O'$
due to the non--trivial orbifold quantum numbers of the gauge parameters.
The $Z_2$ and $Z'_2$ parity assignments for the components of the $U(1)_{\psi}$ and $U(1)_{\chi}$
bulk vector supermultiplets are such that the KK expansions of vector bosons $A^{\chi}_{\mu}(x,y)$
and $A^{\psi}_{\mu}(x,y)$ as well as the corresponding gaugino states $\lambda_{\chi}(x,y)$ and
$\lambda_{\psi}(x,y)$ contain massless zero modes $A^{\chi (0)}_{\mu}(x)$, $A^{\psi (0)}_{\mu}(x)$,
$\lambda^{(0)}_{\chi}(x)$ and $\lambda^{(0)}_{\psi}(x)$ associated with the unbroken $U(1)_{\psi}$
and $U(1)_{\chi}$ gauge symmetries (see Table~\ref{tab2}). Other KK modes form vector boson, scalar
and fermion states with masses $(2n+2)/R$ similar to the ones that appear in the case of unbroken
generators $T^{a}$ of $SU(5)$.
As in the simplest orbifold GUT scenarios
\cite{Kawamura:2000ev}--\cite{Altarelli:2001qj} we assume that all incomplete $SU(5)$
supermultiplets which are even under the custodial symmetry (the matter parity $Z_{2}^{M}$
in the case of the MSSM and the $\tilde{Z}^{H}_2$ symmetry in the case of the E$_6$SSM)
originate from the $5D$ bulk supermultiplets. In order to ensure that $H_u$ and
$\overline{H}_u$ as well as $H_d$ and $\overline{H}_d$ survive below the scale $M_X\sim 1/R$
we include two pairs of the $5D$ $SU(5)$ bulk supermultiplets $\Phi_{H_u}+\Phi_{\overline{H}_u}$
and $\Phi_{H_d}+\Phi_{\overline{H}_d}$ that decompose as follows
\begin{equation}
\Phi_{H_u} = \Phi_{\overline{H}_u}= \displaystyle\left(5,\,-\dfrac{2}{\sqrt{24}},\,\dfrac{2}{\sqrt{40}}\right),\qquad
\Phi_{H_d} = \Phi_{\overline{H}_d}= \displaystyle\left(5,\,\dfrac{2}{\sqrt{24}},\,\dfrac{2}{\sqrt{40}}\right),
\label{31}
\end{equation}
where first, second and third quantities in brackets are the $SU(5)$ representation,
extra $U(1)_{\psi}$ and $U(1)_{\chi}$ charges respectively. The multiplets $\Phi_{H_u}$ and
$\Phi_{\overline{H}_u}$ as well as $\Phi_{H_d}$ and $\Phi_{\overline{H}_d}$ transform
differently under $Z_2$ and $Z'_2$ (see Table~\ref{tab3}). Since $P'$ does not commute with
$SU(5)$ each $5D$ 5--plet is divided into four pieces associated with
different $N=1$ chiral supermultiplets:
\begin{equation}
5=(3,1,-1/3)+(1,2,1/2)+(\bar{3},1,1/3)+(1,2,-1/2)\,.
\label{311}
\end{equation}
In Eq.~(\ref{311}) first and second quantities in brackets are $SU(3)_C$ and $SU(2)_W$
quantum numbers whereas the third quantity is $U(1)_Y$ charge. As one can see from
Table~\ref{tab3} chiral supermultiplets in Eq.~(\ref{311}) have different $P$ and $P'$
parity assignments that result in different KK mode structures. These parity
assignments are such that the orbifold projection accomplishes doublet--triplet splitting,
in the sense that only one doublet superfield in Eq.~(\ref{311}) has zero mode
while the KK expansions of another doublet, triplet and antitriplet superfields do
not possess massless modes. Thus only $H_u$, $\overline{H}_u$, $H_d$ and $\overline{H}_d$
may survive to low--energies.
The $4D$ superfields $N^c_H$, $\overline{N}_H^c$, $S$ and $\overline{S}$ can stem from
the $5D$ SM singlet superfields that carry $U(1)_{\psi}$ and $U(1)_{\chi}$ charges
\begin{equation}
\Phi_{S}= \Phi_{\overline{S}}= \displaystyle\left(1,\,\dfrac{4}{\sqrt{24}},\,0\right),\qquad
\Phi_{N^c_H}= \Phi_{\overline{N}_H^c}= \displaystyle\left(1,\,\dfrac{1}{\sqrt{24}},\,-\dfrac{5}{\sqrt{40}}\right)\,.
\label{32}
\end{equation}
According to Eq.~(\ref{28}) only either $\phi_i$ and $\psi_i$ or $\phi^c_i$ and $\psi^c_i$
can have massless modes. Different parity assignments of $\Phi_{S}$ and $\Phi_{\overline{S}}$
as well as $\Phi_{N^c_H}$ and $\Phi_{\overline{N}_H^c}$ allow to project out different
components of these superfields so that only $4D$ superfields $N^c_H$, $\overline{N}_H^c$,
$S$ and $\overline{S}$ may be light (see Table~\ref{tab3}).
\begin{table}[ht]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$5D$ fields &$SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$&$Z_2\times Z'_2$&
Mass\\
& quantum numbers & parity &\\
\hline
& $(3,1,-1/3,-2,2)+(\bar{3},1,1/3,2,-2)$& $(+,-)$ & $(2n+1)/R$\\
$\Phi_{H_u}+\Phi_{\overline{H}_u}$ & $(1,2,1/2,-2,2)+(1,2,-1/2,2,-2)$ & $(+,+)$ & $2n/R$\\
& $(\bar{3},1,1/3,2,-2)+(3,1,-1/3,-2,2)$& $(-,+)$ & $(2n+1)/R$\\
& $(1,2,-1/2,2,-2)+(1,2,1/2,-2,2)$ & $(-,-)$ & $(2n+2)/R$\\
\hline
& $(3,1,-1/3,2,2)+(\bar{3},1,1/3,-2,-2)$& $(-,+)$ & $(2n+1)/R$\\
$\Phi_{H_d}+\Phi_{\overline{H}_d}$ & $(1,2,1/2,2,2)+(1,2,-1/2,-2,-2)$ & $(-,-)$ & $(2n+2)/R$\\
& $(\bar{3},1,1/3,-2,-2)+(3,1,-1/3,2,2)$& $(+,-)$ & $(2n+1)/R$\\
& $(1,2,-1/2,-2,-2)+(1,2,1/2,2,2)$ & $(+,+)$ & $2n/R$\\
\hline
$\Phi_{S}+\Phi_{\overline{S}}$ & $(1,1,0,4,0)+(1,1,0,-4,0)$ & $(+,+)$ & $2n/R$\\
& $(1,1,0,-4,0)+(1,1,0,4,0)$ & $(-,-)$ & $(2n+2)/R$\\
\hline
$\Phi_{N^c_H}+\Phi_{\overline{N}_H^c}$& $(1,1,0,1,-5)+(1,1,0,-1,5)$ & $(+,+)$ & $2n/R$\\
& $(1,1,0,-1,5)+(1,1,0,1,-5)$ & $(-,-)$ & $(2n+2)/R$\\
\hline
& $(3,1,-1/3,-1,-3)+(\bar{3},1,1/3,1,3)$& $(-,+)$ & $(2n+1)/R$\\
$\Phi_{L_4}+\Phi_{\overline{L}_4}$ & $(1,2,1/2,-1,-3)+(1,2,-1/2,1,3)$ & $(-,-)$ & $(2n+2)/R$\\
& $(\bar{3},1,1/3,1,3)+(3,1,-1/3,-1,-3)$& $(+,-)$ & $(2n+1)/R$\\
& $(1,2,-1/2,1,3)+(1,2,1/2,-1,-3)$ & $(+,+)$ & $2n/R$\\
\hline
& $(3,1,-1/3,-1,-3)+(\bar{3},1,1/3,1,3)$& $(-,-)$ & $(2n+2)/R$\\
$\Phi_{d^c_4}+\Phi_{\overline{d^c}_4}$& $(1,2,1/2,-1,-3)+(1,2,-1/2,1,3)$ & $(-,+)$ & $(2n+1)/R$\\
& $(\bar{3},1,1/3,1,3)+(3,1,-1/3,-1,-3)$& $(+,+)$ & $2n/R$\\
& $(1,2,-1/2,1,3)+(1,2,1/2,-1,-3)$ & $(+,-)$ & $(2n+1)/R$\\
\hline
\end{tabular}
\caption{Parity assignments and KK masses of fields in the $4D$ chiral supermultiplets resulting from
the $5D$ bulk supermultiplets $\Phi_{H_u}$, $\Phi_{\overline{H}_u}$, $\Phi_{H_d}$, $\Phi_{\overline{H}_d}$
$\Phi_{S}$, $\Phi_{\overline{S}}$, $\Phi_{N^c_H}$, $\Phi_{\overline{N}_H^c}$, $\Phi_{L_4}$, $\Phi_{\overline{L}_4}$
$\Phi_{d^c_4}$ and $\Phi_{\overline{d^c}_4}$.}
\label{tab3}
\end{table}
Finally, the particle spectrum below the scale $M_X$ should be supplemented by either
$L_4$ and $\overline{L}_4$ or $d^c_4$ and $\overline{d^c}_4$ (but not both) to allow
the lightest exotic quarks to decay. These $4D$ $N=1$ chiral superfields can come from
either $\Phi_{L_4}$ and $\Phi_{\overline{L}_4}$ or $\Phi_{d^c_4}$ and $\Phi_{\overline{d^c}_4}$
which are $5D$ $SU(5)$ bulk supermultiplets with quantum numbers
\begin{equation}
\Phi_{L_4}=\Phi_{\overline{L}_4}=\Phi_{d^c_4}=\Phi_{\overline{d^c}_4}=
\displaystyle\left(5,\,-\dfrac{1}{\sqrt{24}},\,-\dfrac{3}{\sqrt{40}}\right)\,.
\label{33}
\end{equation}
Again parity assignments guarantee that only two $4D$ doublet superfields $L_4$ and
$\overline{L}_4$ from $\Phi_{L_4}$ and $\Phi_{\overline{L}_4}$ can survive to low--energies
whereas the other $SU(2)_W$ doublet, color triplet and antitriplet partners do not have zero modes.
Using the freedom to flip the overall action of the $P'$ parity on the $SU(5)$ multiplets
by a sign relative to $\Phi_{L_4}+\Phi_{\overline{L}_4}$ one can get the KK spectrum
in which only triplet or antitriplet components of $SU(5)$ fundamental supermultiplets
possess massless modes. From Table~\ref{tab3} one can see that this freedom is used in the
case $\Phi_{d^c_4}$ and $\Phi_{\overline{d^c}_4}$ supermultiplets. Due to the different
structure of the KK spectrum only $4D$ triplet or antitriplet superfields, $\overline{d^c}_4$
and $d^c_4$, from $\Phi_{\overline{d^c}_4}$ and $\Phi_{d^c_4}$ are allowed to be light.
Since the three families of $27_i$ representations of $E_6$ are located on the $O$ brane,
where the $SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$ gauge symmetry remains intact,
the Yukawa interactions of quarks and leptons are necessarily $SU(5)$ symmetric. In general
the $SU(5)$ invariance yields the prediction for the first and second generation fermion
mass ratios $m_s/m_d=m_{\mu}/m_{e}$, which is in conflict with the data. In $4D$ GUTs
acceptable mass relations can be obtained using higher dimensional operators and relatively
large representations which acquire VEVs breaking $SU(5)$ or $SO(10)$
\cite{Altarelli:2000fu}, \cite{Ellis:1979fg}. In the case of the simplest 5D orbifold GUTs
there are no $SU(5)$ breaking VEVs. Nevertheless in this case one can introduce two additional
$5D$ bulk supermultiplets with quantum numbers given by Eq.~(\ref{33}) that
transform under $Z_2$ and $Z'_2$ as either $\Phi_{L_4}$ and $\Phi_{\overline{L}_4}$
or $\Phi_{d^c_4}$ and $\Phi_{\overline{d^c}_4}$. Furthermore we assume that
these bulk supermultiplets are odd under $\tilde{Z}^{H}_2$ symmetry which is defined
on the $O$ brane. Hence the zero modes of these extra $5D$ supermultiplets,
which are either weak doublets ($L_5$ and $\overline{L}_5$) or $SU(3)_C$ triplet and
antitriplet ($\overline{d^c}_5$ and $d^c_5$), can mix with quark or lepton superfields
from $27_i$ spoiling the $SU(5)$ relations between the down type quark and charged
lepton masses. Indeed, suppose that zero modes are weak doublet superfields $L_5$ and
$\overline{L}_5$. Then $\overline{L}_5$ can get combined with the superposition of
lepton doublet superfields from $27_i$ so that the resulting vectorlike states gain
masses slightly below $M_X$. The remaining three families of lepton doublets,
that survive to low energies, are superpositions of the corresponding components
from $27_i$ and $L_5$ while three generations of down type quarks stem from
$27_i$ completely. As a consequence the $SU(5)$ relations between the down type quark
and charged lepton masses may get spoiled entirely if the Yukawa couplings of $L_5$
to Higgs doublet $H_d$ are relatively large ($\sim 0.01-0.1$).
\begin{table}[ht]
\centering
\begin{tabular}{|l|l|l|l|l|}
\hline
$5D$ fields &$SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$&$Z_2\times Z'_2$&
Mass\\
& quantum numbers & parity &\\
\hline
& $(\bar{3},1,-2/3,1,-1)+(3,1,2/3,-1,1)$& $(+,+)$ & $2n/R$\\
& $(3,2,1/6,1,-1)+(\bar{3},2,-1/6,-1,1)$& $(+,-)$ & $(2n+1)/R$\\
$\Phi_{e^c}+\Phi_{\overline{e^c}}$ & $(1,1,1,1,-1)+(1,1,-1,-1,1)$ & $(+,+)$ & $2n/R$\\
& $(3,1,2/3,-1,1)+(\bar{3},1,-2/3,1,-1)$& $(-,-)$ & $(2n+2)/R$\\
& $(\bar{3},2,-1/6,-1,1)+(3,2,1/6,1,-1)$& $(-,+)$ & $(2n+1)/R$\\
& $(1,1,-1,-1,1)+(1,1,1,1,-1)$ & $(-,-)$ & $(2n+2)/R$\\
\hline
\end{tabular}
\caption{The $(Z_2,Z'_2)$ transformation properties and KK masses of $4D$ chiral supermultiplets
that stem from $SU(5)$ bulk supermultiplets $\Phi_{e^c}$ and $\Phi_{\overline{e^c}}$.}
\label{tab4}
\end{table}
Although the discussed specific realization of the mechanism which allows to obtain
the realistic pattern of fermion masses is the simplest one it is worth to consider
another very attractive possibility. Instead of two additional $5D$ $SU(5)$ fundamental
supermultiplets one can include two larger representations of $SU(5)$ that decompose
under $SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$ as follows:
\begin{equation}
\Phi_{e^c}=\Phi_{\overline{e^c}}=\displaystyle\left(10,\,\dfrac{1}{\sqrt{24}},\,-\dfrac{1}{\sqrt{40}}\right)\,.
\label{34}
\end{equation}
As before we assume that $\Phi_{e^c}$ and $\Phi_{\overline{e^c}}$ supermultiplets
are odd under $\tilde{Z}^{H}_2$ symmetry. Due to $P$ and $P'$ parity assignments each
$SU(5)$ bulk decuplet is divided into six pieces associated with different $N=1$ chiral
supermultiplets:
\begin{equation}
10=(\bar{3},1,-2/3)+(3,2,1/6)+(1,1,1)+(3,1,2/3)+(\bar{3},2,-1/6)+(1,1,-1)\,,
\label{35}
\end{equation}
where quantities in brackets are $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ quantum numbers.
The $Z_2$ and $Z'_2$ parity assignments and mass spectrum for all components of
the $5D$ decuplets are given in Table~\ref{tab4}. These parity assignments
guarantee that only two $4D$ $SU(2)_W$ singlet superfields ($e^c_5$ and $\overline{e^c}_5$)
as well as $4D$ triplet and antitriplet supermultiplets ($u^c_5$ and $\overline{u^c}_5$)
from $\Phi_{e^c}$ and $\Phi_{\overline{e^c}}$ can survive below scale $M_X\sim 1/R$.
Again $\overline{e^c}_5$ and $\overline{u^c}_5$ can get combined with the superposition
of the appropriate components of $27_i$ forming vectorlike states which may have
masses slightly below $M_X$. At the same time $e^c_5$ can mix with the corresponding
components of $27_i$ spoiling the $SU(5)$ relations between the masses of the
down type quarks and charged leptons. It is worth noting that together bulk supermultiplets
(\ref{31}), (\ref{32}), (\ref{33}) and (\ref{34}) form two complete $27$ representations
of $E_6$. This simplifies the structure of bulk supermultiplets making the considered
$5D$ orbifold GUT model more elegant.
For the consistency of the considered model it is crucial that all anomalies get
cancelled. In $5D$ theories no bulk anomalies exist. Nevertheless orbifold
compactification may lead to anomalies at orbifold fixpoints
\cite{ArkaniHamed:2001is}--\cite{Asaka:2002my}.
At the fixed point brane anomaly reduces to the anomaly of the unbroken subgroup of
the original group, i.e. $SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$ on the $O$ brane
and $SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\chi}\times U(1)_{\psi}$
on the $O'$ brane. It was also shown that the sum of the contributions to the
$4D$ anomalies at the fixpoint equals to the sum of the contributions of
the zero modes localized at the corresponding brane
\cite{ArkaniHamed:2001is}--\cite{Asaka:2002my}.
In this context it is worth to emphasize that the contributions of three families
of $27_i$ representations of $E_6$, which reside on the $O$ brane, to the anomalies
associated with this fixpoint get cancelled automatically. Moreover from
Tables \ref{tab3} and \ref{tab4} one can see that the $P$ and $P'$ parity assignments
are chosen so that the zero modes of the bulk fields localized at the $O$ and
$O'$ branes always form pairs of $N=1$ supermultiplets with opposite quantum numbers.
Such choice of parity assignments guarantees that the contributions of zero modes
of the bulk superfields to the brane anomalies are cancelled as well.
Another important issue for any GUT model is proton stability which was
discussed in the context of $5D$ orbifold GUT models in \cite{Hall:2001pg},
\cite{5d-susy-ogut-proton}--\cite{5d-susy-ogut-proton-unif}. In orbifold GUT models
the dimension five operators, which are caused by an exchange of the color triplet
Higgsino multiplets and give rise to proton decay in ordinary GUTs, do not get induced.
Indeed, in the considered class of models colored Higgsinos acquire mass via the KK mode
expansion of operators $\psi_i \partial_{5} \psi_i^c$ that leads to the Dirac mass terms
of the form $\psi^{(2n+1)}_i \psi_i^{c(2n+1)}$. Since $\psi_i^{c(2n+1)}$ do not couple
directly to the quarks (squarks) and sleptons (leptons) the dimension five operators
are not generated. It turns out that the absence of tree-level amplitudes caused by
the colored Higgsino exchange which result in proton decay is deeply entangled with
the orbifold construction and continuous global $U(1)_R$ symmetry that $5D$ bulk
Lagrangian possesses \cite{Hall:2001pg}. Although the dimension five operators discussed
above do not get induced within orbifold GUT models one must also suppress the brane
interactions $[QQQL]_F$ and $[u^c u^c d^c e^c]_F$ that may be already present on the $O$
brane as non--renormalizable interactions. Such operators can give a substantial contribution
to the proton decay rate if the fundamental scale of gravity is close to the GUT scale.
In the $5D$ orbifold GUT model considered here these dangerous operators are forbidden
by $U(1)_{\chi}$ and $U(1)_{\psi}$ gauge symmetries. Nevertheless proton decay is
mediated by dimension six operators induced by the leptoquark gauge bosons
\cite{Ellis:1979hy}.
Finally, one should mention that in the $5D$ orbifold GUT models gauge couplings of
the $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ interactions do not exactly unify at the scale
$M_X\sim 1/R$ where $SU(5)$ gauge symmetry gets broken. The reason for this is that
the symmetry of the model on the GUT--breaking brane $O'$ remains limited to the
SM gauge group. In particular, on this brane there are brane--localized $4D$ kinetic
terms for the SM gauge fields with $SU(5)$--violating coefficients $1/g_{O'i}^2$.
The part of the $5D$ effective SUSY Lagrangian that contains kinetic terms for the
SM gauge fields can be written as follows
\begin{equation}
\mathcal{L}_{eff}=\int d^2\theta
\biggl(\frac{1}{g_5^2}+\frac{1}{2g_{O}^2}\biggl\{\delta(y)+\delta(y-\pi R)\biggr\}\biggr)
\mbox{Tr}\,\mathcal{W}^{\alpha} \mathcal{W}_{\alpha}\qquad\qquad
\label{36}
\end{equation}
$$
\qquad\qquad + \sum_i \int d^2\theta
\frac{1}{2g_{O'i}^2}\biggl\{\delta(y-\frac{\pi}{2}R)+\delta(y+\frac{\pi}{2}R)\biggr\}
\mbox{Tr}\,\mathcal{W}^{\alpha}_i \mathcal{W}^i_{\alpha}+\mbox{h.c.},
$$
where $\mathcal{W}^i_{\alpha}$ $(i=1,2,3)$ are the supersymmetric gauge field
strengths of the $U(1)_Y$, $SU(2)_W$ and $SU(3)_C$ gauge interactions on the $O'$ brane,
and $\mathcal{W}_{\alpha}$ is the $SU(5)$ gauge field strength on the $O$ brane and
in the bulk \footnote{Note the $O'$ brane contribution vanish for $\mathcal{W}_{\alpha}$
associated with the leptoquark gauge bosons which are odd under $Z'_2$.}. Integrating
over $y$ one obtains zero--mode $4D$ SM gauge couplings at the scale $M_X\sim 1/R$
\begin{equation}
\frac{1}{g^2_i(M_X)}=\frac{2\pi R}{g_5^2}+\frac{1}{g_{O}^2}+\frac{1}{g_{O'i}^2}\,.
\label{37}
\end{equation}
Since $SU(5)$--violating coefficients $1/g_{O'i}^2$ may differ from each other
substantially the $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ gauge couplings $g^2_i(M_X)$
are not identical. However if in the $5D$ model the bulk and brane gauge couplings
have almost equal strength then after integrating out $y$ the zero--mode gauge couplings
are dominated by the bulk contributions because of the spread of the wavefunction of the
zero--mode gauge bosons. In other words the $SU(5)$--violating brane kinetic terms
are dominated by the bulk contributions when the linear extent of the $5$th dimension
is sufficiently large. Because the bulk contributions to the gauge couplings
(\ref{37}) are necessarily $SU(5)$ symmetric, a $4D$ observer sees an approximate
unification of the SM gauge couplings. The gauge coupling unification within
$5D$ orbifold GUT models was discussed in
\cite{5d-susy-ogut-proton-unif}--\cite{5d-susy-ogut-unif}.
As one can see from Eqs.~(\ref{36})--(\ref{37}) the discrepancy between $g^2_i(M_X)$
is determined by the $SU(5)$--violating gauge kinetic terms on the $O'$ brane.
This discrepancy is small when $g^2_i(M_X)$ are relatively small whereas $g_{O'i}^2$
are large ($g_{O'i}^2 \sim 4\pi $). On the other hand one can expect that the
relative contribution of the $SU(5)$--violating brane corrections to $g^2_i(M_X)$
becomes more sizable in the case when the $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ gauge
couplings are large at the scale $M_X$.
\subsection{$E_6$ orbifold GUT model in six dimensions}
Having discussed in detail the simplest 5D orbifold GUT model, that may lead at low
energies to the gauge group and field content of the $E_6$ inspired SUSY model specified
in section 2, we next study $E_6$ gauge theory in $6D$ with $N=1$ supersymmetry.
We consider the compactification on a torus $T^2$ with two fixed radii $R_5$ and $R_6$
so that two extra dimensions $y (=x_5)$ and $z (=x_6)$ are compact, i.e.
$y\in (-\pi R_5, \pi R_5]$ and $z\in (-\pi R_6, \pi R_6]$. The physical region
associated with the compactification on the orbifold $T^2/Z_2$ is a pillow with the
four fixed points of the $Z_2$ transformations ($y\to -y$, $z\to -z$) as corners.
The orbifold $T^2/Z_2$ has the following fixpoints $(0,0)$, $(\pi R_5,0)$, $(0,\pi R_6)$
and $(\pi R_5,\pi R_6)$.
Here we discuss $E_6$ gauge theory in $6D$ compactified on the orbifold
$T^2/(Z_2 \times Z^{I}_2 \times Z^{II}_2)$. The $Z_2$, $Z^{I}_2$ and $Z^{II}_2$
symmetries are reflections. The $Z_2$ transformations are defined as before,
i.e. $y\to -y$, $z\to -z$. The $Z^{I}_2$ reflection symmetry transformations
act as $y'\to -y'$, $z\to -z$ with $y' = y - \pi R_5/2$. The reflection $Z^{II}_2$
corresponds to $y\to -y$, $z'\to -z'$ where $z' = z - \pi R_6/2$. The
$Z^{I}_2$ and $Z^{II}_2$ reflection symmetries introduce additional fixed
points. As in the case of $5D$ orbifold GUT models extra reflection symmetries
lead to the reduction of the physical region which is again limited by the
appropriate fixed points. The $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ reflection symmetries
allow to work with the theory obtained by truncating to the physically irreducible
space in which $y\in [0, \pi R_5/2]$ and $z\in [0, \pi R_6/2]$ with the four
$4D$ walls (branes) located at its corners.
Again, we assume that the considered orbifold GUT model contains a set of $E_6$
bulk supermultiplets and another set of $N=1$ superfields which are confined on
one of the branes. The set of superfields that propagate in the bulk
$M^4\times T^2/(Z_2 \times Z^{I}_2 \times Z^{II}_2)$ includes $E_6$ gauge
supermultiplet and a few 27--plets. As before all quark and lepton superfields
are expected to be confined on one brane.
The $E_6$ gauge supermultiplet that exist in the bulk must involve vector bosons
$A_{M}$ ($M=0,1,2,3,5,6$) and $6D$ Weyl fermions (gauginos) which are composed of
two $4D$ Weyl fermions, $\lambda$ and $\lambda'$. These fields can be conveniently
grouped into vector and chiral multiplets of the $N=1$ supersymmetry in $4D$, i.e.
\begin{equation}
V=(A_{\mu}, \lambda)\,,\qquad\qquad \Sigma=\biggl((A_5+i A_6)/\sqrt{2},\lambda'\biggr)\,,
\label{38}
\end{equation}
where $V$, $A_{M}$, $\lambda$ and $\lambda'$ are matrices in the adjoint
representation of $E_6$. Two $N=1$ supermultiplets (\ref{38}) form $N=2$ vector
supermultiplet in $4D$. The bulk $27'$ supermultiplets also include $6D$ Weyl
fermion states (that involve two $4D$ Weyl fermions, $\psi_i$ and $\psi^c_i$)
together with two complex scalars $\phi_i$ and $\phi^c_i$. The fields
$\psi_i, \psi^c_i, \phi_i$ and $\phi^c_i$ compose $4D$ $N=2$ hypermultiplet
containing two $4D$ $N=1$ chiral superfields: $\hat{\Phi}_i=(\phi_i,\,\psi_i)$
and its conjugate $\hat{\Phi}^c_i = (\phi^c_i,\,\psi^c_i)$ with opposite
quantum numbers. Thus each bulk $27'$ supermultiplet involves two $4D$ $N=1$
supermultiplets $27'$ and $\overline{27'}$.
To ensure the consistency of the construction the Lagrangian of the considered
orbifold GUT model has to be invariant under $Z_2$, $Z^{I}_2$ and $Z^{II}_2$
symmetries.
As in the case of 5D orbifold GUT models each reflection symmetry, $Z_2$,
$Z^{I}_2$ and $Z^{II}_2$, has its own orbifold parity, $P$, $P_{I}$ and $P_{II}$.
The components $\hat{\Phi}$ and $\hat{\Phi}^c$ of the bulk $27'$ supermultiplet
$\Phi$ transform under $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ as follows
\begin{equation}
\begin{array}{ll}
\hat{\Phi}(x, -y, -z) = P \hat{\Phi}(x, y, z)\,,&\qquad
\hat{\Phi}^c(x, -y, -z) = -P \hat{\Phi}^c (x, y, z)\,,\\
\hat{\Phi}(x, -y', -z) = P_{I} \hat{\Phi}(x, y', z)\,,&\qquad
\hat{\Phi}^c(x, -y', -z) = -P_{I} \hat{\Phi}^c (x, y', z)\,,\\
\hat{\Phi}(x, -y, -z') = P_{II} \hat{\Phi}(x, y, z')\,,&\qquad
\hat{\Phi}^c(x, -y, -z') = -P_{II} \hat{\Phi}^c (x, y, z')\,,
\end{array}
\label{39}
\end{equation}
where $P$, $P_{I}$ and $P_{II}$ are diagonal matrices with eigenvalues $\pm 1$
that act on each component of the fundamental representation of $E_6$.
It is convenient to specify the matrix representation of the orbifold parity
assignments in terms of the $E_6$ weights $\alpha_{j}$ and gauge shifts,
$\Delta$, $\Delta_{I}$ and $\Delta_{II}$, associated with $Z_2$, $Z^{I}_2$ and
$Z^{II}_2$. The diagonal elements of the matrices $P$, $P_{I}$ and $P_{II}$
can be presented in the following form \cite{Braam:2010sy}
\begin{equation}
\begin{array}{c}
(P)_{jj}=\sigma\exp\{2\pi i \Delta \alpha_j\}\,,\qquad\qquad
(P_{I})_{jj}=\sigma_{I}\exp\{2\pi i \Delta_{I} \alpha_j\}\,,\\
(P_{II})_{jj}=\sigma_{II}\exp\{2\pi i \Delta_{II} \alpha_j\}\,,
\end{array}
\label{40}
\end{equation}
where $\sigma$, $\sigma_{I}$ and $\sigma_{II}$ are parities of the
bulk $27'$ supermultiplet, i.e. $\sigma, \sigma_{I}, \sigma_{II} \in \{+,-\}$.
The particle assignments of the weights in the fundamental representation
of $E_6$ are well known (see, for example \cite{Braam:2010sy}). Here we choose
the following gauge shifts
\begin{equation}
\begin{array}{c}
\Delta=\biggl(\dfrac{1}{2},\,\dfrac{1}{2},\,0,\,\dfrac{1}{2},\,\dfrac{1}{2},\,0\biggr)\,,\qquad
\Delta_{I}=\biggl(\dfrac{1}{2},\,\dfrac{1}{2},\,\dfrac{1}{2},\,\dfrac{1}{2},\,\dfrac{1}{2},\,0\biggr)\,,\\
\Delta_{II}=\biggl(\dfrac{1}{2},\,\dfrac{1}{2},\,0,\,0,\,\dfrac{1}{2},\,0\biggr)\,,
\end{array}
\label{41}
\end{equation}
that correspond to the orbifold parity assignments shown in Table~\ref{tab5}.
\begin{table}[ht]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
& $Q$ & $u^c$ & $e^c$ & $L$ & $d^c$ & $N^c$ & $S$ & $H^u$ & $D$ & $H^d$ & $\overline{D}$ \\
\hline
$Z_2$ & $-$ & $-$ & $-$ & $-$ & $-$ & $-$ & $+$ & $+$ & $+$ & $+$ & $+$ \\
\hline
$Z_2^{I}$ & $-$ & $+$ & $+$ & $-$ & $+$ & $+$ & $+$ & $-$ & $+$ & $-$ & $+$ \\
\hline
$Z_2^{II}$& $-$ & $-$ & $-$ & $+$ & $+$ & $+$ & $-$ & $+$ & $+$ & $-$ & $-$ \\
\hline
\end{tabular}
\caption{Orbifold parity assignments in the bulk $27'$ supermultiplet
with $\sigma=\sigma_{I}=\sigma_{II}=+1$.}
\label{tab5}
\end{table}
The components $V$ and $\Sigma$ of the $E_6$ gauge supermultiplet
transform under $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ as follows
\begin{equation}
\begin{array}{ll}
V(x, -y, -z) = P V(x, y, z) P^{-1}\,,&\qquad
\Sigma(x, -y, -z) = -P \Sigma (x, y, z) P^{-1}\,,\\
V(x, -y', -z) = P_{I} V(x, y', z) P^{-1}_{I}\,,&\qquad
\Sigma(x, -y', -z) = -P_{I} \Sigma (x, y', z) P^{-1}_{I}\,,\\
V(x, -y, -z') = P_{II} V(x, y, z') P^{-1}_{II}\,,&\qquad
\Sigma(x, -y, -z') = -P_{II} \Sigma (x, y, z') P^{-1}_{II}\,,
\end{array}
\label{42}
\end{equation}
where $V(x, y, z)=V^{A}(x, y, z) T^{A}$ and $\Sigma(x, y, z)=\Sigma^A(x, y, z) T^A$
while $T^A$ is the set of generators of the $E_6$ group. The boundary conditions
given by Eqs.~(\ref{39}) and (\ref{42}) break $4D$ $N=2$ supersymmetry because
different components of the $N=2$ supermultiplets transform differently
under $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ reflection symmetries. Moreover since
$P$, $P_{I}$ and $P_{II}$ are not unit matrices the $E_6$ gauge symmetry also
gets broken by these parity assignments.
The $P$ parity assignment indicates that on the $O$ brane at $y=z=0$ associated
with the $Z_2$ reflection symmetry the $E_6$ gauge group is broken down to
$SO(10)\times U(1)_{\psi}$ subgroup. Indeed, according to Table~\ref{tab5} the
$SO(10)$ representations that compose bulk $27'$ supermultiplet ($27\to 16+10+1$)
transform differently under $Z_2$ symmetry, i.e. $16\to -16$, $10\to 10$ and
$1\to 1$. Since the considered symmetry breaking mechanism preserves the rank
of the group the unbroken subgroup at the fixed point $O$ should be
$SO(10)\times U(1)_{\psi}$.
On the brane $O_{I}$ located at the fixed point $y=\pi R_5/2$, $z=0$
and associated with the $Z_2^{I}$ symmetry the $E_6$ gauge symmetry is broken
to $SU(6)\times SU(2)_W$. Again this follows from the $P_{I}$ parity assignment
in the bulk $27'$ supermultiplet. The fundamental representation of $E_6$
decomposes under the $SU(6)\times SU(2)_W$ as follows:
$$
27\to (\overline{15},\, 1) + (6,\,2)\,,
$$
where the first and second quantities in brackets are the $SU(6)$ and
$SU(2)_W$ representations respectively. The multiplet $(6,\,2)$ is formed
by all $SU(2)_W$ doublets which are contained in $27$--plet. From Table~\ref{tab5}
one can see that all $SU(2)_W$ doublet components of the $27'$ supermultiplet
transform differently under the $Z^{I}_2$ reflection symmetry as compared with
other components of this supermultiplet which form $(\overline{15},\, 1)$.
The $E_6$ gauge symmetry is also broken on the brane $O_{II}$ placed at the
fixed point $y=0$, $z=\pi R_6/2$ of the $Z_2^{II}$ symmetry transformations.
The $P_{II}$ parity assignment is such that $16$ components of the $27'$
are odd whereas $10+1$ components are even or viceversa. This implies that
$E_6$ group gets broken down to its $SO(10)'\times U(1)'$ subgroup.
It is worth to emphasize here that $SO(10)$ and $SO(10)'$ are not
the same $SO(10)$ subgroups of $E_6$. In particular, from Table~\ref{tab5}
one can see that the 16-plets of $SO(10)$ and $SO(10)'$ are formed by
different components of the fundamental representation of $E_6$. The
$U(1)_{\psi}$ and $U(1)'$ charge assignments should be also different.
In addition to the three branes mentioned above there is a fourth brane
located at the corner $O_{III}=(\pi R_5/2, \pi R_6/2)$ of the physically
irreducible space. The $Z^{III}_2$ reflection symmetry associated with
this brane is obtained by combining the three symmetries $Z_2$, $Z^{I}_2$
and $Z^{II}_2$ defined above. As a consequence the corresponding parity
assignment $P_{III}=P\, P_{I}\, P_{II}$. Combining three parity assignments
$P$, $P_{I}$ and $P_{II}$ it is easy to see that on the brane $O_{III}$
the unbroken subgroup is $SO(10)''\times \tilde{U}(1)$.
The unbroken gauge group of the effective $4D$ theory is given by the
intersection of the $E_6$ subgroups at the fixed points. Since $P$ and
$P_{II}$ commute with $SU(5)$ the intersection of the $E_6$ subgroups
$SO(10)\times U(1)_{\psi}$ and $SO(10)'\times U(1)'$ is
$SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$. The intersection of
$SU(6)\times SU(2)_W$ and $SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$
gives the SM gauge group with two additional $U(1)$ factors, $U(1)_{\psi}$
and $U(1)_{\chi}$.
The mode expansion for the $6D$ bulk fields $\phi(x,y,z)$ with any
combinations of parities reads \cite{Asaka:2001eh}:
\begin{eqnarray}
\phi_{+++}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{2^{\delta_{n,0}\delta_{m,0}}\pi \sqrt{R_5 R_6}}
\phi^{(2n,2m)}_{+++}(x)\cos\biggl(\frac{2ny}{R_5}+\frac{2mz}{R_6}\biggr)\,,
\label{43}
\end{eqnarray}
\begin{eqnarray}
\phi_{+-+}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{\pi \sqrt{R_5 R_6}}
\phi^{(2n+1,2m)}_{+-+}(x)\cos\biggl(\frac{(2n+1)y}{R_5}+\frac{2mz}{R_6}\biggr)\,,
\label{44}
\end{eqnarray}
\begin{eqnarray}
\phi_{++-}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{\pi \sqrt{R_5 R_6}}
\phi^{(2n,2m+1)}_{++-}(x)\cos\biggl(\frac{2ny}{R_5}+\frac{(2m+1)z}{R_6}\biggr)\,,
\label{45}
\end{eqnarray}
\begin{eqnarray}
\phi_{+--}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{\pi \sqrt{R_5 R_6}}
\phi^{(2n+1,2m+1)}_{+--}(x)\cos\biggl(\frac{(2n+1)y}{R_5}+\frac{(2m+1)z}{R_6}\biggr)\,,
\label{46}
\end{eqnarray}
\begin{eqnarray}
\phi_{-++}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{\pi \sqrt{R_5 R_6}}
\phi^{(2n+1,2m+1)}_{-++}(x)\sin\biggl(\frac{(2n+1)y}{R_5}+\frac{(2m+1)z}{R_6}\biggr)\,,
\label{47}
\end{eqnarray}
\begin{eqnarray}
\phi_{--+}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{\pi \sqrt{R_5 R_6}}
\phi^{(2n,2m+1)}_{--+}(x)\sin\biggl(\frac{2ny}{R_5}+\frac{(2m+1)z}{R_6}\biggr)\,,
\label{48}
\end{eqnarray}
\begin{eqnarray}
\phi_{-+-}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{\pi \sqrt{R_5 R_6}}
\phi^{(2n+1,2m)}_{--+}(x)\sin\biggl(\frac{(2n+1)y}{R_5}+\frac{2mz}{R_6}\biggr)\,,
\label{49}
\end{eqnarray}
\begin{eqnarray}
\phi_{---}(x,y,z) = \sum_{n,m}^{\infty} \frac{1}{\pi \sqrt{R_5 R_6}}
\phi^{(2n,2m)}_{---}(x)\sin\biggl(\frac{2ny}{R_5}+\frac{2mz}{R_6}\biggr)\,,
\label{50}
\end{eqnarray}
where $n$ and $m$ are non--negative integers. As follows from Eqs.~(\ref{43})--(\ref{50})
each bosonic and fermionic KK mode $\phi^{(k,\ell)}(x)$ is characterized by two integer
numbers and from the $4D$ perspective acquires mass
$\sqrt{\left(\dfrac{k}{R_5}\right)^2+\left(\dfrac{\ell}{R_5}\right)^2}$
upon compactification. Only fields for which all parities are positive
have zero modes, i.e. modes with $k=0$ and $\ell=0$. Such modes form
$4D$ $N=1$ massless vector multiplet of the unbroken
$SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$ subgroup
of $E_6$. The corresponding $6D$ bulk fields are non--vanishing on all branes.
All other KK modes of the bulk gauge fields combine to massive states.
In particular, one linear combination of $A^{a(k,\ell)}_{5}(x)$ and
$A^{a(k,\ell)}_{6}(x)$ play the role of the Nambu--Goldstone boson, i.e.
it is swallowed by $A^{a(k,\ell)}_{\mu}(x)$ leading to the formation of
the $4D$ vector boson state with mass
$\sqrt{\left(\dfrac{k}{R_5}\right)^2+\left(\dfrac{\ell}{R_5}\right)^2}$.
Thus the mass generation of the vector boson states is analogous to the
Higgs mechanism. The orthogonal superposition of $A^{a(k,\ell)}_{5}(x)$ and
$A^{a(k,\ell)}_{6}(x)$ compose a scalar state with the same mass.
The KK gaugino modes $\lambda^{a(k,\ell)}(x)$ and $\lambda^{'a(k,\ell)}(x)$
form $4D$ fermion state which is degenerate with the corresponding
vector and scalar states.
As before we assume that all incomplete $E_6$ supermultiplets in the E$_6$SSM,
which are even under the $\tilde{Z}^{H}_2$ symmetry, stem from the $6D$ bulk
superfields. Hereafter we also require that the three complete families of
$27_i$ representations of $E_6$ are located on the $O$ brane where $E_6$ gauge
group is broken down to $SO(10)\times U(1)_{\psi}$. The $4D$ superfields $H_u$
and $\overline{H}_u$ can originate from the bulk $27'$--plets $\Phi^{\prime}_{H_u}$
and $\Phi^{\prime}_{\overline{H}_u}$ that decompose as follows
\begin{equation}
\Phi^{\prime}_{H_u} = \displaystyle\left(27,\,+,\,-,\,+\right),\qquad
\Phi^{\prime}_{\overline{H}_u}= \displaystyle\left(27,\,-,\,+,\,-\right)\,,
\label{51}
\end{equation}
where first, second, third and fourth quantities in brackets are the $E_6$
representation as well as $\sigma$, $\sigma_{I}$ and $\sigma_{II}$
associated with this representation respectively. The parities of these bulk
$27'$--plets are chosen so that $H_u$ and $\overline{H}_u$ components of the
$N=1$ chiral superfields $\hat{\Phi}^{\prime}_{H_u}$ and
$\hat{\Phi}^{\prime c}_{\overline{H}_u}$ have positive parities with respect
to $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ reflection symmetries (see Table~\ref{tab5}).
In this context it is essential to keep in mind that the invariance of the
$6D$ action requires that the parities of the $4D$ chiral supermultiplets
$\hat{\Phi}^{\prime}_{\overline{H}_u}$ and $\hat{\Phi}^{\prime c}_{\overline{H}_u}$
are opposite. Since the parities of $H_u$ and $\overline{H}_u$ are positive
the KK expansions of the bulk $27'$--plets $\Phi^{\prime}_{H_u}$ and
$\Phi^{\prime}_{\overline{H}_u}$ contain zero modes that form $N=1$ chiral
superfields with quantum numbers of $H_u$ and $\overline{H}_u$.
The $SU(2)_W$ doublet chiral superfields $H_u$ and $\overline{H}_u$ are not
the only supermultiplets from $\Phi^{\prime}_{H_u}$ and $\Phi^{\prime}_{\overline{H}_u}$
that may survive below the scale $M_X\sim 1/R$. Indeed, the parity assignments
in Eq.~(\ref{51}) indicate that the $\overline{u}^{c}$ and $\overline{e}^{c}$
components of the $\hat{\Phi}^{\prime c}_{H_u}$ as well as $u^c$ and $e^c$
components of the $\hat{\Phi}^{\prime}_{\overline{H}_u}$ also have positive
parities with respect to $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ symmetries.
It means that the KK mode structures of the bulk supermultiplets
$\Phi^{\prime}_{H_u}$ and $\Phi^{\prime}_{\overline{H}_u}$ involve zero
modes that correspond to $N=1$ chiral superfields $u^c$, $e^c$, $\overline{u}^{c}$
and $\overline{e}^{c}$. Because the $E_6$ gauge symmetry is broken down to
the $SO(10)\times U(1)_{\psi}$ subgroup on the $O$ brane the zero modes
that come from the same bulk $27'$--plet but belong to different $SO(10)$
representations are not required to have the same transformation properties
under the custodial $\tilde{Z}^{H}_2$ symmetry. This permits us to assume that
$4D$ chiral superfields $u^c$, $e^c$, $\overline{u}^{c}$ and $\overline{e}^{c}$
are odd under the $\tilde{Z}^{H}_2$ symmetry. Then these supermultiplets
are expected to mix with the appropriate components from other $27$--plets
forming vectorlike states with masses slightly below $M_X$ and spoiling the
$SO(10)$ relations between the Yukawa couplings of quarks and leptons to $H_u$
and $H_d$ as it is discussed in the previous subsection.
The $4D$ superfields $H_d$ and $\overline{H}_d$ can originate from another
pair of bulk $27'$--plets
\begin{equation}
\Phi^{\prime}_{H_d} = \displaystyle\left(27,\,+,\,-,\,-\right),\qquad
\Phi^{\prime}_{\overline{H}_d}= \displaystyle\left(27,\,-,\,+,\,+\right)\,.
\label{52}
\end{equation}
Using the orbifold parity assignments presented in Table~\ref{tab5} it is
easy to check that all parities of $H_d$ and $\overline{H}_d$ components of
the $N=1$ superfields $\hat{\Phi}^{\prime}_{H_d}$ and
$\hat{\Phi}^{\prime c}_{\overline{H}_d}$ are positive so that the
KK expansions of $6D$ superfields $\Phi^{\prime}_{H_u}$ and
$\Phi^{\prime}_{\overline{H}_u}$ contain the appropriate zero modes.
On the other hand one can also find that $\overline{d}^{c}$ and $\overline{N}^{c}$
components of the $\hat{\Phi}^{\prime c}_{H_d}$ as well as $d^c$ and $N^c$
components of the $\hat{\Phi}^{\prime}_{\overline{H}_d}$ also have positive
parities with respect to $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ reflection
symmetries. Therefore the particle content below the scale $M_X$ includes
bosonic and fermionic states from $N=1$ chiral supermultiplets $d^c$, $N^c$,
$\overline{d}^{c}$ and $\overline{N}^{c}$ as well. The scalar components
of the $4D$ superfields $N^c$ and $\overline{N}^{c}$ can be used to break
$U(1)_{\psi}$ and $U(1)_{\chi}$ down to $U(1)_N\times Z_2^M$.
Because of this the supermultiplets $d^c$, $N^c$, $\overline{d}^{c}$ and
$\overline{N}^{c}$ are expected to be even under the $\tilde{Z}^{H}_2$
symmetry and therefore can not mix with the components of $27_i$ localised
on the $O$ brane. The large VEVs of $N^c$ and $\overline{N}^{c}$ ($\lesssim M_X$)
can give rise to the masses of the bosonic and fermionic components of
$N^c$ and $\overline{N}^{c}$ as well as $d^c$ and $\overline{d}^{c}$ which
are just slightly below $M_X$.
In order to achieve the appropriate breakdown of the $SU(2)_W\times U(1)_Y\times U(1)_{N}$
gauge symmetry at low energies the particle spectrum below the scale $M_X$ should be
supplemented by the $4D$ chiral superfields $S$ and $\overline{S}$ which are even under
the $\tilde{Z}^{H}_2$ symmetry. The corresponding zero modes can come from the pair of
bulk $27'$--plets
\begin{equation}
\Phi^{\prime}_{S} = \displaystyle\left(27,\,+,\,+,\,-\right),\qquad
\Phi^{\prime}_{\overline{S}}= \displaystyle\left(27,\,-,\,-,\,+\right)\,.
\label{53}
\end{equation}
The $S$ and $\overline{S}$ components of the $N=1$ superfields $\hat{\Phi}^{\prime}_{S}$
and $\hat{\Phi}^{\prime c}_{\overline{S}}$ have positive orbifold parities.
The $\overline{D}$ component of $\hat{\Phi}^{\prime}_{S}$ and the companion component
from the $\hat{\Phi}^{\prime c}_{\overline{S}}$ superfield have also positive parities
with respect to $Z_2$, $Z^{I}_2$ and $Z^{II}_2$ symmetries. It is convenient to assume
that the states associated with these exotic quark supermultiplets are odd under
the $\tilde{Z}^{H}_2$ symmetry so that the corresponding zero modes can mix with the
appropriate components of the $27$--plets localised on the $O$ brane leading to the
formation of the vectorlike states with masses slightly below $M_X$ and spoiling the
$SO(10)$ relations between the Yukawa couplings of $S$ to the inert Higgs and exotic
quark states. In addition to the components of $\hat{\Phi}^{\prime}_{S}$ and
$\hat{\Phi}^{\prime c}_{\overline{S}}$ mentioned above the orbifold parities of
$\overline{L}$ and $L$ components of $\hat{\Phi}^{\prime c}_{S}$ and
$\hat{\Phi}^{\prime}_{\overline{S}}$ are positive. If the zero modes
associated with these components survive to low energies and the corresponding
$N=1$ supermultiplets are even under the $\tilde{Z}^{H}_2$ symmetry then
the Yukawa couplings of these superfields to $Q_i$ and $\overline{D}_k$
allow the lightest exotic quarks to decay like in the case of Scenario A.
The discussion above indicate that the simplest $6D$ orbifold GUT model
based on the $E_6$ gauge group, which may lead at low energies to the gauge
group and field content of the Scenario A
specified in section 2, include six bulk $27'$--plets. The consistency of
this orbifold GUT model requires the absence of anomalies. In the 6D orbifold
models there are two types of anomalies: $4D$ anomalies \cite{Adler:1969gk}
intrinsic to the fixed points and bulk anomalies \cite{Asaka:2002my},
\cite{vonGersdorff:2006nt}--\cite{E6-anomaly-2}
which are induced by box diagrams with four gauge currents. For the $6D$
orbifold GUT model to be consistent it is necessary that both the fixed point
and the bulk anomalies must cancel. The contributions of the
anomalous box diagrams with four gauge currents to the $6D$ bulk anomalies
are determined by the trace of four generators of gauge group. This
trace contains nonfactorizable part and part which can be reduced to
the product of traces of two generators. The nonfactorizable part is
associated with the irreducible gauge anomaly while the factorized
contribution corresponds to what is known as reducible anomaly. The reducible
anomalies can be canceled by the Green--Schwarz mechanism \cite{Green:1984sg}.
For the consistency the chiral field content of the $6D$ orbifold model must
lead to the cancellation of the irreducible anomalies which is normally highly
restrictive requirement \cite{Hebecker:2001jb}. However $6D$ orbifold GUT
models based on the $E_6$ gauge group do not have irreducible bulk anomaly
\cite{vonGersdorff:2006nt}--\cite{E6-anomaly-2}. Moreover using the results
obtained in \cite{E6-anomaly-2} one can show that the reducible gauge
anomaly gets cancelled if the field content of the $6D$ orbifold model
involves six bulk $27'$--plets. The $4D$ anomalies at the fixpoints get
also cancelled within the $6D$ orbifold GUT model discussed above. Indeed,
the contributions of $27_i$ supermultiplets, that reside on the $O$ brane,
to the anomalies vanish. Since the orbifold parity assignments are such
that the KK modes of the bulk $27'$ superfields localized at the fixpoints
always form pairs of $N=1$ supermultiplets with opposite quantum numbers
the contributions of the bulk $27'$--plets to the $4D$ fixed point anomalies
are cancelled automatically as well.
Phenomenological viability of the $5D$ and $6D$ orbifold GUT models considered
in this section requires the adequate suppression of the baryon and lepton
number violating operators which can be induced at the scale $M_X$ giving rise
to proton decay. As it was mentioned before the dimension five operators,
that lead to the proton decay, are forbidden by the gauge symmetry in these
models. However baryon and lepton number violating operators, which are mediated
by the exchange of the leptoquark gauge bosons, are enhanced compared to the
usual $4D$ case due to the presence of KK towers of such states. The proton
decay rate in the $6D$ orbifold GUT models based on the $SO(10)$ gauge group
was studied in \cite{Buchmuller:2004eg} where it was
shown that in order to satisfy the experimental
lower limit on the proton lifetime the scale $M_X$ should be larger than
$9\cdot 10^{15}\,\mbox{GeV}$. This restriction on the scale $M_X$ can be
used in the case of the $E_6$ inspired SUSY models as well. However the analysis
of the RG flow of the gauge couplings, which we are going to consider next,
indicates that the value of $g^2_i(M_X)$ in these models are 3-5 times larger
than in the MSSM. This implies that the lower bound on the scale $M_X$
in the considered $E_6$ inspired models is expected to be
$1.5-2\cdot 10^{16}\,\mbox{GeV}$. It is worth noting here again that
the simplest $5D$ and $6D$ orbifold GUT models discussed in this section
do not lead to the exact gauge coupling unification at the scale $M_X$
due to the brane contributions to the gauge couplings. The relative
contribution of these brane corrections is expected to become more
sizable with increasing $g_i^2(M_X)$ as it was discussed before.
The gauge coupling unification in the $6D$ orbifold GUT models was
considered in \cite{6d-susy-ogut-unif}.
\section{RG flow of gauge couplings in the E$_6$SSM}
In this section we discuss the RG flow of the SM gauge couplings $g_i(t)$ above
the EW scale. The running of these couplings between $M_{X}$ and $M_Z$ is described
by a system of renormalisation group equations (RGEs). To simplify our analysis
we assume that $U(1)_{\psi}\times U(1)_{\chi}$ gauge symmetry is broken
down to $U(1)_{N}\times Z_{2}^{M}$ near the scale $M_X$. This permits us to
restrict our consideration to the analysis of the RG flow of four diagonal gauge
couplings $g_3(t)$, $g_2(t)$, $g_1(t)$ and $g'_1(t)$ which correspond to $SU(3)_C$,
$SU(2)_W$, $U(1)_Y$ and $U(1)_N$ gauge interactions respectively. Besides the
evolution of these gauge couplings is affected by a kinetic term mixing. The mixing
effect can be concealed in the interaction between the $U(1)_{N}$ gauge field and
matter fields that can be parametrized in terms of off--diagonal gauge coupling
$g_{11}$ (see \cite{Langacker:1998tc}, \cite{King:2005jy}, \cite{9}). In this
framework the RG equations can be written as follows:
\begin{equation}
\displaystyle\frac{d G}{d t}=G\times B\,,\qquad\qquad
\frac{d g_2}{dt}=\displaystyle\frac{\beta_2 g_2^3}{(4\pi)^2}\,,\qquad\qquad
\frac{d g_3}{dt}=\frac{\beta_3 g_3^3}{(4\pi)^2}\,,
\label{54}
\end{equation}
where $t=\ln\left(q/M_Z\right)$, $q$ is a renormalisation scale while $B$ and $G$
are $2\times 2$ matrices
\begin{equation}
G=\left(
\begin{array}{cc}
g_1 & g_{11}\\[2mm]
0 & g'_1
\end{array}
\right)\,,\qquad
B=\displaystyle\frac{1}{(4\pi)^2}
\left(
\begin{array}{cc}
\beta_1 g_1^2 & 2g_1g'_1\beta_{11}+2g_1g_{11}\beta_1\\[2mm]
0 & g^{'2}_1\beta'_1+2g'_1 g_{11}\beta_{11}+g_{11}^2\beta_1
\end{array}
\right)\,.
\label{55}
\end{equation}
In Eqs.~(\ref{54})--(\ref{55}) $\beta_i$ and $\beta_{11}$ are beta functions.
Here we examine the RG flow of gauge couplings in the two--loop approximation.
In general the two--loop diagonal $\beta_i$ and off--diagonal $\beta_{11}$ beta
functions may be presented as a sum of one--loop and two--loop contributions.
However the previous analysis performed in \cite{King:2007uj} revealed that an
off--diagonal gauge coupling $g_{11}$ being set to zero at the scale $M_X$ remains
very small at any other scale below $M_X$. Since it seems to be rather natural
to assume that just after the breakdown of the $E_6$ symmetry there is no mixing
in the gauge kinetic part of the Lagrangian between the field strengths associated
with the $U(1)_Y$ and $U(1)_{N}$ gauge interactions $g_{11}$ tends to be substantially
smaller than the diagonal gauge couplings. Because of this we can neglect two--loop
corrections to the off--diagonal beta function $\beta_{11}$. In the case of scenario A
the one--loop off--diagonal beta function is given by $\beta_{11}=-\displaystyle\frac{\sqrt{6}}{5}$
while in the scenario B $\beta_{11}=\displaystyle\frac{3\sqrt{6}}{10}$.
In the scenario A the two--loop diagonal beta functions $\beta_i$ are given by:
\begin{equation}
\begin{array}{rcl}
\beta_3&=&-9+3N_g+\displaystyle\frac{1}{16\pi^2}\Biggl[g_3^2(-54+34 N_g)+3 N_g\,g_2^2+ N_g\, g_1^2\\[3mm]
&&+N_g\,g_1^{'2}-4h_t^2-4h_b^2-2\Sigma_{\kappa}\Biggr]\,,\\[3mm]
\beta_2&=&-5+3N_g+\displaystyle\frac{1}{16\pi^2}\Biggl[8N_g g_3^2+(-17+21 N_g)g_2^2+ \left(\displaystyle\frac{3}{5}+N_g\right) g_1^2\\[3mm]
&&+\left(\displaystyle\frac{2}{5}+N_g\right) g_1^{'2}
-6 h_t^2-6 h_b^2-2h_{\tau}^2-2\Sigma_{\lambda}\Biggr]\,,\\[3mm]
\beta_1&=&\displaystyle\frac{3}{5}+3N_g+\displaystyle\frac{1}{16\pi^2}\Biggl[8N_g g_3^2+\left(\displaystyle\frac{9}{5}+3N_g\right)g_2^2+
\left(\displaystyle\frac{9}{25}+3 N_g\right) g_1^2\\[3mm]
&&+\left(\displaystyle\frac{6}{25}+N_g\right) g_1^{'2}-\displaystyle\frac{26}{5} h_t^2-\displaystyle\frac{14}{5}h_b^2-
\displaystyle\frac{18}{5}h_{\tau}^2-\displaystyle\frac{6}{5}\Sigma_{\lambda}-\displaystyle\frac{4}{5}\Sigma_{\kappa}\Biggr]\,,\\[3mm]
\beta'_1&=&\displaystyle\frac{2}{5}+3N_g+\displaystyle\frac{5}{4}n+
\displaystyle\frac{1}{16\pi^2}\Biggl[8N_g g_3^2+\left(\displaystyle\frac{6}{5}+3N_g\right)g_2^2+
\left(\displaystyle\frac{6}{25}+ N_g\right) g_1^2\\[3mm]
&&+\left(\displaystyle\frac{4}{25}+3N_g+\displaystyle\frac{25}{8}n \right) g_1^{'2}-
\displaystyle\frac{9}{5} h_t^2-\displaystyle\frac{21}{5}h_b^2-\displaystyle\frac{7}{5}h_{\tau}^2-
\displaystyle\frac{19}{5}\Sigma_{\lambda}-\displaystyle\frac{57}{10}\Sigma_{\kappa}\Biggr]\,,\\[3mm]
\Sigma_{\lambda}&=&\lambda_1^2+\lambda_2^2+\lambda^2\,,\qquad\qquad\qquad
\Sigma_{\kappa}=\kappa_1^2+\kappa_2^2+\kappa_3^2\,,
\end{array}
\label{56}
\end{equation}
where $N_g$ is a number of generations forming complete $E_6$ fundamental representations that
the considered model involves at low energies, i.e. $N_g=3$, whereas $n$ is a number of $S$ and
$\overline{S}$ supermultiplets from $27'_S$ and $\overline{27'}_S$ that survive to low energies
(i.e. $n=0$ or $1$). Here we assume that the structure of the Yukawa interactions appearing in the
superpotential (\ref{13}) is relatively simple, i.e. $\lambda_{\alpha\beta}=\lambda_{\alpha}\delta_{\alpha\beta}$,
and $\kappa_{ij}=\kappa_i\delta_{ij}$ while $\tilde{f}_{\alpha\beta}$, $f_{\alpha\beta}$, $g^D_{ij}$
and $h^E_{i\alpha}$ are small and can therefore be ignored ($i,\,j=1,\,2,\,3$ and $\alpha,\,\beta=1,\,2$).
We have also neglected all Yukawa couplings that may be associated with the presence of extra $S$
and $\overline{S}$ supermultiplets at low energies. In Eqs.~(\ref{56}) $h_t$, $h_b$ and $h_{\tau}$
are top quark, $b$-quark and $\tau$--lepton Yukawa couplings respectively. In the limit of $n=0$
the RG equations (\ref{56}) coincide with the ones presented in \cite{King:2007uj}.
In the scenario B the two--loop diagonal beta functions $\beta_i$ can be written
in the following form:
$$
\begin{array}{rcl}
\beta_3&=&-8+3N_g+\displaystyle\frac{1}{16\pi^2}\Biggl[g_3^2\left(-\displaystyle\frac{128}{3}+34 N_g\right)+3 N_g\,g_2^2
+ \left(N_g+\displaystyle\frac{4}{15}\right)\,g_1^2\\[3mm]
&& + \left(N_g+\displaystyle\frac{2}{5}\right)\,g_1^{'2}-4h_t^2-4h_b^2-2\Sigma_{\kappa}\Biggr]\,,
\end{array}
$$
\begin{equation}
\begin{array}{rcl}
\beta_2&=&-4+3N_g+\displaystyle\frac{1}{16\pi^2}\Biggl[8N_g g_3^2+(-10+21 N_g)g_2^2+ \left(\displaystyle\frac{6}{5}+N_g\right) g_1^2\\[3mm]
&&+\left(\displaystyle\frac{13}{10}+N_g\right) g_1^{'2}
-6 h_t^2-6 h_b^2-2h_{\tau}^2-2\tilde{\Sigma}_{\lambda}\Biggr]\,,\\[3mm]
\beta_1&=&\displaystyle\frac{8}{5}+3N_g+\displaystyle\frac{1}{16\pi^2}\Biggl[\left(8N_g + \displaystyle\frac{32}{15}\right)g_3^2+
\left(\displaystyle\frac{18}{5}+3N_g\right)g_2^2+
\left(\displaystyle\frac{62}{75}+3 N_g\right) g_1^2\\[3mm]
&&+\left(\displaystyle\frac{47}{50}+N_g\right) g_1^{'2}-\displaystyle\frac{26}{5} h_t^2-\displaystyle\frac{14}{5}h_b^2-
\displaystyle\frac{18}{5}h_{\tau}^2-\displaystyle\frac{6}{5}\tilde{\Sigma}_{\lambda}-\displaystyle\frac{4}{5}\Sigma_{\kappa}\Biggr]\,,\\[3mm]
\beta'_1&=&\displaystyle\frac{19}{10}+3N_g+\displaystyle\frac{5}{4}n+
\displaystyle\frac{1}{16\pi^2}\Biggl[\left(8N_g+\displaystyle\frac{16}{5}\right)g_3^2+\left(\displaystyle\frac{39}{10}+3N_g\right)g_2^2\\[4mm]
&&+\left(\displaystyle\frac{47}{50}+ N_g\right) g_1^2
+\left(\displaystyle\frac{121}{100}+3N_g+\displaystyle\frac{25}{8}n \right) g_1^{'2}\\[3mm]
&&-\displaystyle\frac{9}{5} h_t^2-\displaystyle\frac{21}{5}h_b^2-\displaystyle\frac{7}{5}h_{\tau}^2-
\displaystyle\frac{19}{5}\tilde{\Sigma}_{\lambda}-\displaystyle\frac{57}{10}\Sigma_{\kappa}\Biggr]\,,
\end{array}
\label{57}
\end{equation}
where $\tilde{\Sigma}_{\lambda}=\lambda_1^2+\lambda_2^2+\lambda_3^2+\lambda^2$.
As before we assume relatively simple structure of the Yukawa interactions in the
superpotential (\ref{18}), i.e. $\lambda_{ij}=\lambda_{i}\delta_{ij}$,
$\kappa_{ij}=\kappa_i\delta_{ij}$, and ignore $\tilde{f}_{\alpha i}$, $f_{\alpha i}$,
$g^{q}_{ij}$, $h^D_{ij}$ as well as all Yukawa couplings of extra $S$ and
$\overline{S}$ supermultiplets.
As one can see from Eqs.~(\ref{56})--(\ref{57}) $N_g=3$ is the critical value for
the one--loop beta function of the strong interactions in the case of scenario A.
Indeed, in the one--loop approximation the $SU(3)_C$ gauge coupling is equal to zero
in this case. In the scenario B the one--loop contribution to $\beta_3$ remains rather
small ($b_3=1$). Because of this any reliable analysis of the RG flow of
gauge couplings requires the inclusion of two--loop corrections to the diagonal
beta functions.
One can obtain an approximate solution of the two--loop RGEs presented above
(see \cite{Chankowski:1995dm}). At high energies this solution for the SM gauge
couplings can be written as
\begin{equation}
\displaystyle\frac{1}{\alpha_i(t)}=\frac{1}{\alpha_i(M_Z)}-\displaystyle\frac{b_i}{2\pi} t-\frac{C_i}{12\pi}-\Theta_i(t)
+\displaystyle\frac{b_i-b_i^{SM}}{2\pi}\ln\frac{T_i}{M_Z}\,,
\label{58}
\end{equation}
where $\alpha_i(t)=\displaystyle\frac{g_i^2(t)}{(4\pi)}$, $b_i$ and $b_i^{SM}$ are the coefficients
of the one--loop beta functions in the E$_6$SSM and SM respectively, the third term in the
right--hand side of Eq.~(\ref{58}) is the $\overline{MS}\to\overline{DR}$ conversion factor
with $C_1=0$, $C_2=2$, $C_3=3$ \cite{MS-DR}, while
\begin{equation}
\Theta_i(t)=\displaystyle\frac{1}{2\pi}\int_0^t (\beta_i-b_i)d\tau\,,\qquad\qquad
T_i=\prod_{k=1}^N\biggl(m_k\biggr)^{\displaystyle\frac{\Delta b^k_i}{b_i-b_i^{SM}}}\,.
\label{59}
\end{equation}
In Eq.~(\ref{59}) $m_k$ and $\Delta b_i^k $ are masses and one--loop contributions to
the beta functions due to new particles appearing in the E$_6$SSM. For the calculation of
$\Theta_i(t)$ the solutions of the one--loop RGEs are normally used.
In Eqs.~(\ref{58})--(\ref{59}) only leading one--loop threshold effects are
taken into account.
Using the approximate solution of the two--loop RGEs in Eqs.~(\ref{58})--(\ref{59})
one can establish the relationships between the values of the gauge couplings at
low energies and GUT scale. Then by using the expressions describing the RG flow
of $\alpha_1(t)$ and $\alpha_2(t)$ it is rather easy to find the scale $M_X$ where
$\alpha_1(M_X)=\alpha_2(M_X)=\alpha_0$ and the value of the overall gauge coupling
$\alpha_0$ at this scale. Substituting $M_X$ and $\alpha_0$ into the solution of
the RGE for the strong gauge coupling one finds the value of $\alpha_3(M_Z)$ for
which exact gauge coupling unification occurs (see \cite{Carena:1993ag}):
\begin{equation}
\begin{array}{c}
\displaystyle\frac{1}{\alpha_3(M_Z)}=\frac{1}{b_1-b_2}\biggl[\displaystyle\frac{b_1-b_3}{\alpha_2(M_Z)}-
\displaystyle\frac{b_2-b_3}{\alpha_1(M_Z)}\biggr]-\frac{1}{28\pi}+\Theta_s+\frac{19}{28\pi}\ln\frac{T_{S}}{M_Z}\,,\\[4mm]
\Theta_s=\biggl(\displaystyle\frac{b_2-b_3}{b_1-b_2}\Theta_1-\frac{b_1-b_3}{b_1-b_2}\Theta_2+\Theta_3\biggr)\,,
\qquad \Theta_i=\Theta_i(M_X)\,.
\end{array}
\label{60}
\end{equation}
The combined threshold scale $T_{S}$, that appears in Eq.~(\ref{60}), can be expressed
in terms of the effective threshold scales $T_1$, $T_2$ and $T_3$.
The expression for $T_{S}$ is model--dependent. In the scenario A $T_{S}$ is given by
$$
\begin{array}{rcl}
T_{S}&=&\displaystyle\frac{T_2^{172/19}}{T_1^{55/19} T_3^{98/19}}\,,\\[0mm]
T_1&=&\tilde{M}_1^{5/11} \mu_{L}^{4/55} m_{L}^{2/55}
\Biggl(\prod_{i=1,2,3}m_{\tilde{D}_i}^{4/165}\mu_{D_i}^{8/165}\Biggr)
\Biggl(\prod_{\alpha=1,2}m_{H_{\alpha}}^{2/55}\mu_{\tilde{H}_{\alpha}}^{4/55}\Biggr)\,,\\[0mm]
\end{array}
$$
\vspace{-2mm}
\begin{eqnarray}
T_2&=&\tilde{M}_2^{25/43} \mu_{L}^{4/43} m_{L}^{2/43}
\Biggl(\prod_{\alpha=1,2} m_{H_{\alpha}}^{2/43}\mu_{\tilde{H}_{\alpha}}^{4/43}\Biggr)\,,\qquad\qquad\qquad\qquad\qquad\quad\nonumber\\[0mm]
T_3&=&\tilde{M}_3^{4/7}\Biggl(\prod_{i=1,2,3}m_{\tilde{D}_i}^{1/21}\mu_{D_i}^{2/21}\Biggr)\,,
\label{61}
\end{eqnarray}
where $\mu_{D_i}$ and $m_{\tilde{D}_i}$ are the masses of exotic quarks and their superpartners,
$m_{H_{\alpha}}$ and $\mu_{\tilde{H}_{\alpha}}$ are the masses of Inert Higgs and Inert Higgsino
fields, $m_{L}$ and $\mu_{L}$ are the masses of the scalar and fermion components of $L_4$ and
$\overline{L}_4$ while $\tilde{M}_1$, $\tilde{M}_2$ and $\tilde{M}_3$ are the effective threshold
scales in the MSSM
\begin{eqnarray}
\tilde{M}_1&=& \mu^{4/25} m_{A}^{1/25}
\Biggl(\prod_{i=1,2,3} m_{\tilde{Q}_i}^{1/75} m_{\tilde{d}_i}^{2/75} m_{\tilde{u}_i}^{8/75}
m_{\tilde{L}_i}^{1/25} m_{\tilde{e}_i}^{2/25}\Biggr)\,,\nonumber \\[0mm]
\tilde{M}_2&=& M_{\tilde{W}}^{8/25} \mu^{4/25} m_A^{1/25}
\Biggl(\prod_{i=1,2,3} m_{\tilde{Q}_i}^{3/25} m_{\tilde{L}_i}^{1/25}\Biggr)\,,\nonumber\\[0mm]
\tilde{M}_3&=& M_{\tilde{g}}^{1/2}
\Biggl(\prod_{i=1,2,3} m_{\tilde{Q}_i}^{1/12} m_{\tilde{u}_i}^{1/24} m_{\tilde{d}_i}^{1/24}\Biggr)\,.
\label{62}
\end{eqnarray}
In Eqs.~(\ref{62}) $M_{\tilde{g}}$ and $M_{\tilde{W}}$ are masses of gluinos and winos
(superpartners of $SU(2)_W$ gauge bosons), $\mu$ and $m_A$ are effective $\mu$--term and
masses of heavy Higgs states respectively; $m_{\tilde{u}_i}$, $m_{\tilde{d}_i}$ and
$m_{\tilde{Q}_i}$ are the masses of the right--handed and left--handed squarks and
$m_{\tilde{L}_i}$ and $m_{\tilde{e}_i}$ are the masses of the left--handed and right--handed
sleptons.
In the case of scenario B we find
\begin{eqnarray}
\tilde{T}_{S}&=&\displaystyle\frac{\tilde{T}_2^{196/19}}{\tilde{T}_1^{65/19} \tilde{T}_3^{112/19}}\,,\nonumber \\[0mm]
\tilde{T}_1&=&\tilde{M}_1^{5/13} \mu_{d_4}^{8/195} m_{d_4}^{4/195}
\mu_{H_u}^{4/65} m_{H_u}^{2/65} \mu_{H_d}^{4/65} m_{H_d}^{2/65}
\Biggl(\prod_{i=1,2,3}m_{\tilde{D}_i}^{4/195}\mu_{D_i}^{8/195}\Biggr)
\Biggl(\prod_{\alpha=1,2}m_{H_{\alpha}}^{2/65}\mu_{\tilde{H}_{\alpha}}^{4/65}\Biggr)\,,\nonumber\\[0mm]
\tilde{T}_2&=&\tilde{M}_2^{25/49} \mu_{H_u}^{4/49} m_{H_u}^{2/49} \mu_{H_d}^{4/49} m_{H_d}^{2/49}
\Biggl(\prod_{\alpha=1,2} m_{H_{\alpha}}^{2/49}\mu_{\tilde{H}_{\alpha}}^{4/49}\Biggr)\,,\nonumber\\[0mm]
\tilde{T}_3&=&\tilde{M}_3^{1/2} \mu_{d_4}^{1/12} m_{d_4}^{1/24}
\Biggl(\prod_{i=1,2,3} m_{\tilde{D}_i}^{1/24} \mu_{D_i}^{1/12}\Biggr)\,,
\label{63}
\end{eqnarray}
where $\mu_{d_4}$, $\mu_{H_u}$ and $\mu_{H_d}$ are the masses of the fermionic components
of $d^c_4$ and $\overline{d^c}_4$, $H^u_{i}$ and $\overline{H}_u$ as well as $H^d_{i}$ and $\overline{H}_d$,
that form vector-like states at low energies, whereas $m_{d_4}$, $m_{H_u}$ and $m_{H_d}$ are the
masses of the scalar components of the corresponding supermultiplets.
In general the effective threshold scales derived above can be quite different.
Since our purpose is to establish the range of the values of $T_S$ and $\tilde{T}_{S}$
that leads to the unification of gauge couplings we shall set these effective
threshold scales equal to each other. Then from Eqs.~(\ref{61}) and (\ref{63}) it
follows that $T_1=T_2=T_3=T_S$ and $\tilde{T}_1=\tilde{T}_2=\tilde{T}_3=\tilde{T}_S$.
The results of our numerical studies of the two--loop RG flow of gauge couplings in the
case of scenarios A and B are summarized in Figs.~\ref{essmfig1} and \ref{essmfig2}
respectively. We use the two--loop SM beta functions to describe the running of gauge
couplings between $M_Z$ and $T_1=T_2=T_3=T_S$ (or $\tilde{T}_1=\tilde{T}_2=\tilde{T}_3=\tilde{T}_S$),
then we apply the two--loop RGEs of the E$_6$SSM to compute the flow of $g_i(t)$ from
$T_S$ (or $\tilde{T}_S$) to $M_X$ which is equal to $3\cdot 10^{16}\,\mbox{GeV}$
in the case of the E$_6$SSM. The low energy values of $g'_1$ and $g_{11}$ are chosen
so that all four diagonal gauge couplings are approximately equal near the GUT scale
and $g_{11}=0$ at this scale. For the calculation of the evolution of Yukawa couplings
a set of one--loop RGEs is used. The corresponding one--loop RG equations are specified
in \cite{King:2005jy}.
\begin{figure}
\begin{center}
\hspace*{-11cm}{$\alpha_i(t)$}\\[1mm]
\includegraphics[height=70mm,keepaspectratio=true]{gc-scen-a1.eps}\\
\hspace*{0cm}{$2\log[q/M_Z]$}\\[1mm]
\hspace*{0cm}{\bf (a)}\\[3mm]
\hspace*{-11cm}{$\alpha_i(t)$}\\[1mm]
\includegraphics[height=70mm,keepaspectratio=true]{gc-scen-a11.eps}\\
\hspace*{0cm}{$2\log[q/M_Z]$}\\[1mm]
\hspace*{0cm}{\bf (b) }\\
\vspace{-3mm}
\caption{Two--loop RG flow of gauge couplings in the Scenario A:
{\it (a)} RG flow of $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ couplings
from $M_Z$ to $M_X$ for $T_S=400\,\mbox{GeV}$ and $n_S=1$;
{\it (b)} running of SM gauge couplings in the vicinity of $M_X$
for $T_S=400\,\mbox{GeV}$ and $n_S=1$. Thick, dashed and solid lines
correspond to the running of $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ couplings
respectively. We used $\tan\beta=10$, $\alpha_s(M_Z)=0.118$,
$\alpha(M_Z)=1/127.9$, $\sin^2\theta_W=0.231$ and $\kappa_1(T_S)=\kappa_2(T_S)
=\kappa_3(T_S)=\lambda_1(T_S)=\lambda_2(T_S)=\lambda_3(T_S)=g^{'}_1(T_S)$.
The dotted lines represent the uncertainty in $\alpha_i(t)$ caused by
the variation of the strong gauge coupling from 0.116 to 0.120 at the
EW scale.}
\end{center}
\label{essmfig1}
\end{figure}
In Fig.~\ref{essmfig1} we fix the effective threshold scale to be equal to $400\,\mbox{GeV}$.
In Fig.~1a we plot the running of the gauge couplings from $M_Z$ to $M_X$ assuming
that the low energy matter content involves three 27-plets of $E_6$ as well as $L_4$,
$\overline{L}_4$, $S$ and $\overline{S}$ supermultiplets. Fig.~1b shows a blow--up
of the crucial region in the vicinity of the GUT scale. Dotted lines show the interval
of variations of gauge couplings caused by $1\,\sigma$ deviations of $\alpha_3(M_Z)$
around its average value, i.e. $\alpha_3(M_Z)\simeq 0.118\pm 0.002$. The results
of the numerical analysis presented in Fig.~\ref{essmfig1} demonstrate that in the
scenario A almost exact unification of the SM gauge couplings can be achieved for
$\alpha_3(M_Z)=0.118$ and $\tilde{T}_S=400\,\mbox{GeV}$. With increasing (decreasing)
the effective threshold scale the value of $\alpha_3(M_Z)$, at which exact gauge coupling
unification takes place, becomes lower (greater). Thus in this case the gauge coupling
unification can be achieved for any phenomenologically reasonable value of $\alpha_3(M_Z)$,
consistent with the central measured low energy value, unlike in the MSSM
where it is rather problematic to get the exact unification of gauge couplings
\cite{Chankowski:1995dm}, \cite{gc-unif-mssm-2}--\cite{gc-unif-mssm-1}.
Indeed, it is well known that in order to achieve gauge coupling
unification in the MSSM with $\alpha_s(M_Z)\simeq 0.118$, the combined threshold
scale, which is given by \cite{Chankowski:1995dm}, \cite{Carena:1993ag},
\cite{gc-unif-mssm-1}--\cite{Langacker:1992rq}
\begin{equation}
\tilde{M}_{S}=\displaystyle\frac{\tilde{M}_2^{100/19}}{\tilde{M}_1^{25/19} \tilde{M}_3^{56/19}}
\simeq \mu/6\,,
\label{64}
\end{equation}
must be around $\tilde{M}_S\approx 1\,\mbox{TeV}$. However the correct pattern of EW symmetry
breaking requires $\mu$ to lie within the $1-2\,\mbox{TeV}$ range which implies
$\tilde{M}_S<200-300\,\mbox{GeV}$, so that, ignoring the effects of high energy threshold
corrections, the exact gauge coupling unification in the MSSM requires significantly
higher values of $\alpha_3(M_Z)$, well above the experimentally measured central value
\cite{Chankowski:1995dm}, \cite{Carena:1993ag}, \cite{gc-unif-mssm-1}--\cite{gc-unif-mssm-3}.
It was argued that it is possible to get the unification of gauge couplings
in the minimal SUSY model for $\alpha_3(M_Z)\simeq 0.123$ \cite{gc-unif-mssm-4}.
On the other hand in the case of scenario A the combined threshold scale $T_{S}$
can be substantially larger than in the MSSM. This can be seen directly from the
explicit expression for $T_S$. Combining Eqs.~(\ref{61}) we find
\begin{eqnarray}
T_{S}&=&\tilde{M}_{S}\cdot
\Biggl(\displaystyle\frac{\mu_{L}^{12/19} m_{L}^{6/19}}{\mu_{D_3}^{12/19} m_{\tilde{D}_3}^{6/19}}\Biggr)
\Biggl(\prod_{\alpha=1,2}
\displaystyle\frac{m_{H_{\alpha}}^{6/19}\mu_{\tilde{H}_{\alpha}}^{12/19}}{m_{\tilde{D}_{\alpha}}^{6/19}\mu_{D_{\alpha}}^{12/19}}
\Biggr)\,.
\label{65}
\end{eqnarray}
From Eq.~(\ref{65}) it is obvious that $T_{S}$ is determined by the masses of the scalar
and fermion components of $L_4$ and $\overline{L}_4$. The term $\mu_L L_4\overline{L}_4$
in the superpotential (\ref{13}) is not involved in the process of EW symmetry breaking.
As a consequence the parameter $\mu_L$ remains arbitrary\footnote{When $\mu_L$ is considerably
larger than the SUSY breaking scale $m_L\simeq \mu_L$.}. In particular, since the corresponding
mass term is not suppressed by the $E_6$ symmetry the components of the doublet superfields
$L_4$ and $\overline{L}_4$ may be much heavier than the masses of all exotic states resulting in
the large combined threshold scale $T_{S}$ that lies in a few hundred GeV range even
when scale $\tilde{M}_S$ is relatively low. The large range of variation of $T_{S}$ allows
to achieve the exact unification of gauge couplings in the scenario A for any value of
$\alpha_3(M_Z)$ which is in agreement with current data.
It is worth noting here that, in principle, one could naively expect that large two--loop
corrections to the diagonal beta functions would spoil the unification of the SM gauge couplings
entirely in the considered case. Indeed, in the scenario A these corrections affect the RG
flow of gauge couplings much more strongly than in the case of the MSSM because at any intermediate
scale the values of the gauge couplings in the E$_6$SSM are substantially larger as compared to
the ones in the MSSM. Nevertheless the results of our analysis discussed above are not as surprising
as they may first appear. The analysis of the RG flow of the SM gauge couplings performed in
\cite{King:2007uj} revealed that the two--loop corrections to $\alpha_i(M_X)$ are a few times
bigger in the E$_6$SSM than in the MSSM. At the same time due to the remarkable cancellation
of different two--loop corrections the absolute value of $\Theta_s$ is more than three times
smaller in the E$_6$SSM as compared with the MSSM. This cancellation is caused by the structure
of the two--loop corrections to the diagonal beta functions in the considered model.
As a result, the prediction for the value of $\alpha_3(M_Z)$ at which exact gauge coupling
unification takes place is considerably lower in the E$_6$SSM than in the MSSM.
The only difference between the E$_6$SSM scenario, which was studied in \cite{King:2007uj},
and scenario A discussed above is in the possible presence of extra $S$ and $\overline{S}$
supermultiplets at low energies. From Eqs.~(\ref{56}) it follows that these supermultiplets
do not contribute to the diagonal beta functions of the SM gauge couplings. Our analysis of
the RG flow of $g_i(t)$ reveals that the evolution of the SM gauge couplings does not change
much when the low energy particle spectrum is supplemented by the bosonic and fermionic
components that originate from the extra $S$ and $\overline{S}$ chiral superfields. This
explains why our results are so similar to those previously obtained in \cite{King:2007uj}.
It is also worthwhile to point out that at high energies the uncertainty in $\alpha_3(t)$
caused by the variations of $\alpha_3(M_Z)$ is much bigger in the E$_6$SSM than in the MSSM.
This is because in the E$_6$SSM the strong gauge coupling grows slightly with increasing
renormalisation scale whereas in the MSSM it decreases at high energies. This implies that
the uncertainty in the high energy value of $\alpha_3(t)$ in the E$_6$SSM is approximately equal
to the low energy uncertainty in $\alpha_3(t)$ while in the MSSM the interval of variations of
$\alpha_3(t)$ near the scale $M_X$ shrinks drastically. The relatively large uncertainty in
$\alpha_3(M_X)$ in the E$_6$SSM, compared to the MSSM, allows one to achieve exact unification
of gauge couplings for values of $\alpha_3(M_Z)$ which are within one standard deviation of its
measured central value.
The RG flow of the SM gauge couplings changes substantially in the case of scenario B
as can be seen from Figs.~\ref{essmfig2}. As before we assume that the effective threshold
scales are equal, i.e. $\tilde{T}_1=\tilde{T}_2=\tilde{T}_3=\tilde{T}_S$. Our numerical
analysis reveals that the evolution of $\alpha_i(t)$ depends very strongly on $\tilde{T}_S$.
When $\tilde{T}_S\lesssim 1\,\mbox{TeV}$ the gauge couplings become rather large
near the GUT scale, i.e. $\alpha_i(M_X) \sim 1$, where as before we set
$M_X\simeq 3\cdot 10^{16}\,\mbox{GeV}$. For so large values of $\alpha_i(t)$
the perturbation theory method becomes inapplicable.
Therefore in our analysis we consider the range
of scales $\tilde{T}_S$ which are much higher than $1\,\mbox{TeV}$. In Figs.~\ref{essmfig2}
we set the threshold scale $\tilde{T}_S$ to be equal to $3\,\mbox{TeV}$. As one
can see from these figures for $\tilde{T}_S=3\,\mbox{TeV}$ the values of $\alpha_i(M_X)$
are about $0.2$ that still allows us to use the perturbation theory up to
the scale $M_X$.
\begin{figure}
\begin{center}
\hspace*{-11cm}{$\alpha_i(t)$}\\[1mm]
\includegraphics[height=70mm,keepaspectratio=true]{gc-scenb-3tev.eps}\\
\hspace*{0cm}{$2\log[q/M_Z]$}\\[1mm]
\hspace*{0cm}{\bf (a)}\\[3mm]
\hspace*{-11cm}{$\alpha_i(t)$}\\[1mm]
\includegraphics[height=70mm,keepaspectratio=true]{gc-scenb1-3tev.eps}\\
\hspace*{0cm}{$2\log[q/M_Z]$}\\[1mm]
\hspace*{0cm}{\bf (b) }\\
\vspace{-3mm}
\caption{Two--loop RG flow of gauge couplings in the Scenario B:
{\it (a)} evolution of $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ couplings
from the EW scale to the GUT scale for $\tilde{T}_S=3\,\mbox{TeV}$ and $n_S=0$;
{\it (b)} running of SM gauge couplings near the scale $M_X$ for
$\tilde{T}_S=3\,\mbox{TeV}$ and $n_S=0$. The parameters and notations
are the same as in Fig.~\ref{essmfig1}.
}
\end{center}
\label{essmfig2}
\end{figure}
The effective threshold scale that we consider in our analysis $\tilde{T}_S$
is in the multi TeV range. At first glance, it is not clear if so large values of
$\tilde{T}_i$ and $\tilde{T}_S$ can be obtained for a reasonable set of parameters.
In particular, to satisfy naturalness requirements the third generation sfermions
as well as neutralino and chargino states which are superparners of the SM gauge
bosons and Higgs fields are expected to have masses below $1\,\mbox{TeV}$. Because
of this in the MSSM naturalness arguments constrain the combined threshold scale
$\tilde{M}_{S}$ to be lower than $200-300\,\mbox{GeV}$ as it was mentioned above.
In the case of scenario B the analytical expression for the threshold scale
$\tilde{T}_{S}$ can be obtained by combining Eqs.~(\ref{63}) that gives
\begin{eqnarray}
\tilde{T}_{S}&=&\tilde{M}_{S}\cdot
\Biggl(\displaystyle\frac{\mu_{H_u}^{12/19} m_{H_u}^{6/19} \mu_{H_d}^{12/19} m_{H_d}^{6/19}}
{\mu_{d_4}^{12/19} m_{d_4}^{6/19} \mu_{D_3}^{12/19} m_{\tilde{D}_3}^{6/19}}\Biggr)
\Biggl(\prod_{\alpha=1,2}
\displaystyle\frac{m_{H_{\alpha}}^{6/19}\mu_{\tilde{H}_{\alpha}}^{12/19}}{m_{\tilde{D}_{\alpha}}^{6/19}\mu_{D_{\alpha}}^{12/19}}
\Biggr)\,.
\label{66}
\end{eqnarray}
Eq.~(\ref{66}) indicates that the combined threshold scale $\tilde{T}_{S}$ tends to be
very large if, for example, $\mu_{H_u}\simeq m_{H_u}\simeq \mu_{H_d}\simeq m_{H_d}$
are considerably larger than the masses of the scalar and fermion components of $d^c_4$ and
$\overline{d^c}_4$ as well as the masses of all exotic states. In this case $\tilde{T}_{S}$
can be as large as $10\,\mbox{TeV}$ even when $\tilde{M}_{S}$ lies in a few hundred GeV range
and $\mu_{H_u}\simeq m_{H_u}\simeq \mu_{H_d}\simeq m_{H_d}\lesssim 10\,\mbox{TeV}$.
This can be achieved if the components of $d^c_4$ and $\overline{d^c}_4$ and
some of the exotic quark and squark states have masses below $1\,\mbox{TeV}$.
The effective threshold scales $\tilde{T}_1$, $\tilde{T}_2$ and $\tilde{T}_3$ can be
also as large as a few $\mbox{TeV}$ if the scalar superpartners of the first and second
generation fermions and some of the exotic states have masses above $10\,\mbox{TeV}$.
Naturalness does not require these states to be light and, in fact, allowing them to be
heavy ameliorates SUSY flavor and CP problems. As a consequence the several TeV threshold
scales $\tilde{T}_1$, $\tilde{T}_2$, $\tilde{T}_3$ and $\tilde{T}_S$ can naturally emerge
in the scenario B.
In Fig.~2a we show the running of the SM gauge couplings from the EW scale to high
energies. We assume that in this case the low energy matter content includes three
27-plets of $E_6$ as well as $d^c_4$, $\overline{d^c}_4$, $H_u$, $\overline{H}_u$
$H_d$ and $\overline{H}_d$ supermultiplets. Fig.~2b shows the same RG flow of the
SM gauge couplings but just around the scale where the values of $\alpha_i(t)$ become
rather close. Again dotted lines in Figs.~2a and 2b represent the changes of the
evolution of the SM gauge couplings induced by the variations of $\alpha_3(M_Z)$ within
$1\,\sigma$ around its average value.
From Figs.~2a and 2b one can see that the interval of variations of $\alpha_3(t)$
enlarges with increasing renormalisation scale. The growth of the uncertainty in
the high energy value of $\alpha_3(t)$ is caused by the raise of this coupling itself.
As follows from Figs.~\ref{essmfig1} and \ref{essmfig2} in the scenario B the SM gauge
couplings grow faster with increasing renormalisation scale than in the case of
scenario A. This happens because the one--loop beta functions of these couplings are
larger in the scenario B as compared to the ones in the scenario A. As a consequence
the interval of variations of $\alpha_3(t)$ at high energies is also a bit bigger in
the former than in the latter. However as one can see from Figs.~2a and 2b this does
not facilitate the gauge coupling unification in scenario B. In fact, these figures
demonstrate that large two--loop corrections spoil the unification of gauge couplings
in this case. Indeed, in the one--loop approximation Eq.~(\ref{60}) leads to the same
prediction for $\alpha_3(M_Z)$ in the scenarios A and B because extra matter in these
scenarios form complete $SU(5)$ representations which contribute equally to the
one--loop beta functions of the $SU(3)_C$, $SU(2)_W$ and $U(1)_Y$ interactions so
that the differences of the coefficients of the one--loop beta functions $b_i-b_j$
remain intact. At the same time the contributions of two--loop corrections to
$\alpha_i(M_X)$ ($\Theta_i$) and $\alpha_3(M_Z)$ ($\Theta_s$) are different in these
cases. Our numerical analysis reveals that for $\tilde{T}_S\simeq 3\,\mbox{TeV}$ the
exact gauge coupling unification can be achieved in the scenario B only if the value
of $\alpha_3(M_Z)$ is around $0.112$. For higher scale $T_S$ the exact unification of
$\alpha_i(t)$ requires even smaller values of $\alpha_3(M_Z)$ which are disfavoured
by the recent fit to experimental data. The lower scales $T_S\lesssim 3\,\mbox{TeV}$
lead to the larger values of $\alpha_i(M_X)$ making questionable the validity of our
calculations.
As before extra $S$ and $\overline{S}$ superfields, that may survive to
low energies, do not contribute to the diagonal beta functions of the SM gauge couplings
and, therefore, do not change much the RG flow of $\alpha_i(t)$. As a result the value
of $\alpha_3(M_Z)$ at which exact gauge coupling unification takes place does not
change much as well after the inclusion of the bosonic and fermionic components of
these supermultiplets. Thus it seems to be rather difficult to reconcile the
unification of gauge couplings with present data in the Scenario B. Nevertheless
the values of $\alpha_i(M_X)$ are not so much different from each other. From Fig.~2b
it follows that the relative discrepancy of $\alpha_i(M_X)$ is about 10\% . This brings
us back to the orbifold GUT framework which was discussed in the previous section.
As it has been already mentioned orbifold GUTs do not imply the exact gauge
coupling unification near the scale $M_X$, which is associated with the size of
compact extra dimensions, due to the brane contributions to the gauge couplings
(see Eq.~(\ref{37})). Since one can expect that these brane corrections
become more sizable when $\alpha_i(M_X)$ are large, the relative discrepancy
of 10\% between $\alpha_i(M_X)$ should not be probably considered as a big problem
in the case of scenario B.
\section{Phenomenological implications}
We now consider cosmological implications and collider signatures of the $E_6$
inspired SUSY models discussed above. The phenomenological implications of these
models are determined by the structure of the particle spectrum that can vary
substantially depending on the choice of the parameters. For example, the masses
of the $Z'$ boson, exotic quarks, Inert Higgsinos and Inert singlinos are set by
the VEVs of the Higgs fields. In this section we primarily focus on the simplest
case when only $H_u$, $H_d$ and $S$ acquire non--zero VEVs breaking
$SU(2)_W\times U(1)_Y\times U(1)_{N}$ symmetry to $U(1)_{em}$ associated with
electromagnetism. Assuming that $f_{\alpha\beta}$ and $\tilde{f}_{\alpha\beta}$
are sufficiently small the masses of the exotic quarks, Inert Higgsino states and
$Z'$ boson are given by
\begin{equation}
\mu_{D_i}=\dfrac{\kappa_i}{\sqrt{2}}\,s\,, \qquad\qquad
\mu_{H_{\alpha}}=\dfrac{\lambda_{\alpha}}{\sqrt{2}}\,s\,, \qquad\qquad
M_{Z'}\simeq g^{'}_1 \tilde{Q}_S s\,,
\label{67}
\end{equation}
where $s$ is a VEV of the field $S$, i.e. $\langle S \rangle=s/\sqrt{2}$.
Here without loss of generality we set $\kappa_{ij}=\kappa_i\delta_{ij}$
and $\lambda_{\alpha\beta}=\lambda_{\alpha}\delta_{\alpha\beta}$.
Since $\mu_{D_i}$, $\mu_{H_{\alpha}}$ and $M_{Z'}$ are determined by $s$,
that remains a free parameter, the $Z'$ boson mass and the masses of exotic
quarks and Inert Higgsinos cannot be predicted. Because recent measurements
from the LHC experiments exclude $E_6$ inspired $Z'$ with masses lower than
$2-2.15\,\mbox{TeV}$ \cite{Chatrchyan:2012it} the singlet field $S$
must acquire a large VEV ($s\gtrsim 5.5-6\,\mbox{TeV}$) to induce sufficiently
large $M_{Z'}$. The couplings $\kappa_i$ should be also large enough to ensure
that the exotic fermions are sufficiently heavy to avoiding conflict with direct
particle searches at present and former accelerators. However the exotic
fermions (quarks and Inert Higgsinos) can be relatively light in the E$_6$SSM.
This happens, for example, when the Yukawa couplings of the exotic particles
have hierarchical structure similar to the one observed in the ordinary
quark and lepton sectors. Then $Z'$ mass lie beyond $10\,\mbox{TeV}$ and
the only manifestation of the considered models may be the presence of
light exotic quark and/or Inert Higgsino states in the particle spectrum.
Since the qualitative pattern of the particle spectrum and associated collider
signatures are so sensitive to the parameter choice it is worth to discuss
first the robust predictions that the considered models have.
It is well known that SUSY models predict that the mass of the lightest Higgs
particle is limited from above. The E$_6$SSM is not an exception.
In the simplest case when only $H_u$, $H_d$ and $S$ develop the VEVs, so that
$\langle H_d\rangle =\displaystyle\frac{v_1}{\sqrt{2}}$,\,$\langle H_u\rangle
=\displaystyle\frac{v_2}{\sqrt{2}}$ and $\langle S\rangle =\displaystyle\frac{s}{\sqrt{2}}$,
the Higgs sector involves ten degrees of freedom. However four of them are
massless Goldstone modes which are swallowed by the $W^{\pm}$, $Z$ and $Z'$
gauge bosons that gain non-zero masses. If CP--invariance is preserved
the other degrees of freedom form two charged, one CP--odd and three CP-even
Higgs states. When the SUSY breaking scale is considerably larger than
the EW scale, the mass matrix of the CP-even Higgs sector has a hierarchical
structure and can be diagonalised using the perturbation
theory \cite{Nevzorov:2001um}-\cite{Nevzorov:2004ge}. In this case the mass
of one CP--even Higgs particle is always very close to the $Z'$ boson mass
$M_{Z'}$. The masses of another CP--even, the CP--odd and the charged Higgs
states are almost degenerate. When $\lambda\gtrsim g'_1$, the qualitative
pattern of the Higgs spectrum is rather similar to the one which arises
in the PQ symmetric NMSSM \cite{Nevzorov:2004ge}-\cite{Miller:2005qua}.
In the considered limit the heaviest CP--even, CP--odd and charged states
are almost degenerate and lie beyond the $\mbox{TeV}$ range \cite{King:2005jy}.
Finally, like in the MSSM and NMSSM, one of the CP--even Higgs bosons is
always light irrespective of the SUSY breaking scale. However, in contrast
with the MSSM, the lightest Higgs boson in the E$_6$SSM can be heavier
than $110-120\,\mbox{GeV}$ even at tree level. In the two--loop approximation
the lightest Higgs boson mass does not exceed $150-155\,\mbox{GeV}$ \cite{King:2005jy}.
\subsection{Dark matter}
The structure of the Yukawa interactions in the E$_6$SSM leads to another
important prediction. Using the method proposed in \cite{Hesselbach:2007te}
one can argue that there are theoretical upper bounds on the masses of the
lightest and second lightest inert neutralino states \cite{Hall:2010ix}.
To simplify the analysis we assume that the fermion components of
the supermultiplets $\overline{S}$, $\overline{H}_u$ and $\overline{H}_d$,
which may survive below the scale $M_X$, get combined with the corresponding
superpositions of the fermion components of the superfields $S_i$, $H^u_i$
and $H^d_i$ resulting in a set of heavy vectorlike states. Furthermore
we also assume that these vectorlike states completely decouple so that
the particle spectrum below the TeV scale contains only two generations of
inert Higgsinos ($\tilde{H}^u_{\alpha}$ and $\tilde{H}^d_{\alpha}$) and
two generations of inert singlinos $\tilde{S}_{\alpha}$. The Yukawa
interactions of these superfields are described by the superpotential
\begin{eqnarray}
W_{IH}=\lambda_{\alpha\beta} S (H^d_{\alpha} H^u_{\beta})+
f_{\alpha\beta} S_{\alpha} (H_d H^u_{\beta})+
\tilde{f}_{\alpha\beta} S_{\alpha} (H^d_{\beta} H_u)\,,
\label{essm2}
\end{eqnarray}
where $\alpha,\beta=1,2$\,.
Thus below the TeV scale the inert neutralino states are linear superposition
of the inert singlino states ($\tilde{S}_1$, $\tilde{S}_2$) and neutral components
of inert Higgsinos ($\tilde{H}^{d0}_1$, $\tilde{H}^{d0}_2$, $\tilde{H}^{u0}_1$,
$\tilde{H}^{u0}_2$). The charged components of the inert Higgsinos
$(\tilde{H}^{u+}_2,\,\tilde{H}^{u+}_1,\,\tilde{H}^{d-}_2,\,\tilde{H}^{d-}_1)$,
form inert chargino sector. In order to avoid the LEP lower limit on the masses
of inert charginos the couplings $\lambda_{\alpha\beta}$ and $s$ must be chosen
so that all inert chargino states are heavier than $100\,\mbox{GeV}$.
In addition, the requirement of the validity of perturbation theory up to
the GUT scale constrains the allowed range of Yukawa couplings $\lambda_{\alpha\beta}$,
$f_{\alpha\beta}$ and $\tilde{f}_{\alpha\beta}$. The restrictions specified
above set very stringent limits on the masses of two lightest inert neutralinos.
The analysis performed in \cite{Hall:2010ix} indicates that the lightest and
second lightest inert neutralinos ($\tilde{H}^0_{1}$ and $\tilde{H}^0_{2}$)
are typically lighter than $60-65\,\mbox{GeV}$. These neutralinos are predominantly
inert singlinos so that they can have rather small couplings to the $Z$--boson.
Therefore any possible signal which these neutralinos could give rise to at LEP
would be extremely suppressed. On the other hand the couplings of $\chi^0_{1}$ and
$\chi^0_{2}$ to the lightest CP--even Higgs boson $h_1$ are proportional to the
$\mbox{mass}/\sqrt{v_1^2+v_2^2}$ in the leading approximation \cite{Hall:2010ix}.
As a consequence the couplings of two lightest inert neutralino to the lightest Higgs
state are always large if the corresponding states have appreciable masses.
The discussion above indicates that the lightest and second lightest inert
neutralinos tend to be the lightest states which are odd under the $Z_{2}^{E}$ symmetry.
It is worth to remind here that in the considered $E_6$ inspired SUSY models
$U(1)_{\psi}\times U(1)_{\chi}$ gauge symmetry is broken down to $U(1)_{N}\times Z_{2}^{M}$
where $Z_{2}^{M}=(-1)^{3(B-L)}$ is the so--called matter parity which is a discrete
subgroup of $U(1)_{\psi}$ and $U(1)_{\chi}$. Since the low--energy effective
Lagrangian is invariant under both $Z_{2}^{M}$ and $\tilde{Z}^{H}_2$ symmetries and
$\tilde{Z}^{H}_2 = Z_{2}^{M}\times Z_{2}^{E}$ (see Table~\ref{tab1}), the $Z_{2}^{E}$
symmetry is also conserved. This means that the lightest exotic state, which is odd
under the $Z_{2}^{E}$ symmetry, is absolutely stable and contributes to the relic
density of dark matter.
Because the lightest inert neutralino is also the lightest
$R$--parity odd state either the lightest $R$--parity even exotic state or
the lightest $R$--parity odd state with $Z_{2}^{E}=+1$ must be absolutely
stable. When $f_{\alpha\beta}$ and $\tilde{f}_{\alpha\beta}$ are large enough
($f_{\alpha\beta}\sim \tilde{f}_{\alpha\beta}\sim 0.5$) the large mixing in
the inert Higgs sector may lead to the lightest CP--even (or CP--odd) inert Higgs
state with mass of the order of the EW scale. The corresponding exotic state
is $R$--parity even neutral particle. If it is substantially lighter than the
lightest ordinary neutralino state $\chi_1^0$ and the decay of $\chi_1^0$ into
the lightest inert neutralino and the lightest inert Higgs scalar (pseudoscalar)
is kinematically allowed then this lightest inert Higgs scalar (pseudoscalar) is
absolutely stable and may result in considerable contribution to the relic dark
matter density.
Although the possibility mentioned above looks very attractive a substantial
fine-tuning is normally required to make the lightest inert Higgs scalar (pseudoscalar)
lighter than $\chi_1^0$. Most commonly $\chi_1^0$ is considerably lighter than
the lightest inert Higgs scalar (pseudoscalar) so that the lightest CP--even
(CP--odd) inert Higgs state can decay into $\chi_1^0$ and the lightest inert
neutralino state. In other words, in the considered $E_6$ inspired SUSY models
the lightest $R$--parity odd state with
$Z_{2}^{E}=+1$, i.e. $\chi_1^0$, tend to be substantially lighter than
the $R$--parity even exotic states. As a result the lightest neutralino
state $\chi_1^0$ is a natural candidate for a cold component of dark matter
in these models.
In the neutralino sector of the E$_6$SSM there are two extra neutralinos besides
the four MSSM ones. One of them is an extra gaugino $\tilde{B}'$ coming from the
$Z'$ vector supermultiplet. The other one is an additional singlino $\tilde{S}$
which is a fermion component of the SM singlet superfield $S$. Extra neutralinos form
two eigenstates $(\tilde{B}'\pm\tilde{S})/\sqrt{2}$ with masses around $M_{Z'}$\,
\cite{King:2005jy}. Since LHC experiments set very stringent lower bound on the mass
of the $Z'$ boson extra neutralino eigenstates tend to be the heaviest ones
and decouple. The mixing between these heavy neutralino states and other gauginos and
Higgsinos is very small. Therefore the lightest neutralino states in the E$_6$SSM,
that determine the composition of $\chi_1^0$ and as a consequence its contribution
to the relic dark matter density, become almost indistinguishable from the ones
in the MSSM. This means that in the E$_6$SSM, like in the MSSM, the lightest neutralino
$\chi_1^0$ can give a substantial contribution to the relic density which is in
agreement with the measured abundance of cold dark matter
$\Omega_{\mathrm{CDM}}h^2 = 0.1099 \pm 0.0062$ \cite{cdm}.
In the E$_6$SSM the lightest inert neutralino can account for all or some of the
observed cold dark matter relic density if $\chi_1^0$ has mass close to half the
$Z$ mass. In this case the lightest inert neutralino states annihilate mainly through
an $s$--channel $Z$--boson, via its Inert Higgsino doublet components which couple
to the $Z$--boson \cite{Hall:2010ix}, \cite{Hall:2009aj}. When $|m_{\tilde{H}^0_{1}}|\ll M_Z$
the lightest inert neutralino states are almost inert singlinos and the couplings of
$\tilde{H}^0_1$ to gauge bosons, Higgs states, quarks (squarks) and leptons (sleptons)
are quite small leading to a relatively small annihilation cross section for
$\tilde{H}^0_1\tilde{H}^0_1\to \mbox{SM particles}$. Since the dark matter
number density is inversely proportional to the annihilation cross section
at the freeze-out temperature the lightest inert neutralino state with mass
$|m_{\tilde{H}^0_{1,2}}|\ll M_Z$ gives rise to a relic density which is typically
much larger than its measured value\footnote{When $f_{\alpha\beta},\,
\tilde{f}_{\alpha\beta}\to 0$ the masses of $\tilde{H}^0_1$ and $\tilde{H}^0_2$
tend to zero and inert singlino states essentially decouple from the rest of the
spectrum. In this limit the lightest non-decoupled Inert neutralino may be rather
stable and can play the role of dark matter \cite{Hall:2011zq}. The presence of very
light neutral fermions in the particle spectrum might have interesting implications
for the neutrino physics (see, for example \cite{Frere:1996gb}).}.
Because the scenarios with $|m_{\tilde{H}^0_{1,2}}|\sim M_Z/2$ imply that the couplings
of $\tilde{H}^0_1$ and $\tilde{H}^0_2$ to the lightest Higgs boson are much larger than
the $b$--quark Yukawa coupling the lightest Higgs state decays more than 95\% of the
time into $\tilde{H}^0_1$ and $\tilde{H}^0_2$ in these cases while the total branching
ratio into SM particles varies from 2\% to 4\% \cite{Hall:2010ix}. At the same time
the LHC production cross section of the lightest Higgs state in the considered $E_6$
inspired SUSY models is almost the same as in the MSSM. Therefore the evidence for the
Higgs boson recently presented by ATLAS \cite{:2012gk} and CMS \cite{:2012gu} indicates
that the corresponding scenarios are basically ruled out.
In this context one should point out another class of scenarios that might have interesting
cosmological implications. Let us consider a limit when
$f_{\alpha\beta}\sim \tilde{f}_{\alpha\beta}\sim 10^{-5}$. So small values of the
Yukawa couplings $f_{\alpha\beta}$ and $\tilde{f}_{\alpha\beta}$ result in extremely light
inert neutralino states $\tilde{H}^0_1$ and $\tilde{H}^0_2$ which are basically inert singlinos.
These states have masses about $1\,\mbox{eV}$.
Since $\tilde{H}^0_1$ and $\tilde{H}^0_2$ are so light and absolutely stable they form hot
dark matter in the Universe\footnote{In the context of $E_6$ inspired SUSY models warm dark
matter was recently discussed in \cite{King:2012wg}.}.
These inert neutralinos have
negligible couplings to $Z$ boson and would not have been observed at earlier collider
experiments. These states also do not change the branching ratios of the $Z$ boson and
Higgs decays. Moreover if $Z'$ boson is sufficiently heavy the presence of such light
Inert neutralinos does not affect Big Bang Nucleosynthesis \cite{Hall:2011zq}.
When the masses of $\tilde{H}^0_1$ and $\tilde{H}^0_2$ are about $1\,\mbox{eV}$
these states give only a very minor contribution to the dark matter density while
the lightest neutralino may account for all or some of the observed dark matter density.
In this case one can expect that the lifetime of the next-to-lightest exotic state
(for example, inert chargino) is given by
\begin{equation}
\tau_{NLES}\sim \frac{8\pi^2}{f^2 M_{NLES}}\,,
\label{73}
\end{equation}
where $f_{\alpha\beta}\sim \tilde{f}_{\alpha\beta}\sim f$ and
$M_{NLES}$ is the mass of the next-to-lightest exotic state.
Assuming that $M_{NLES}\sim 1\,\mbox{TeV}$ we get $\tau_{NLES}\sim 10^{-15}\,s$.
With increasing $f_{\alpha\beta}$ and $\tilde{f}_{\alpha\beta}$ the masses of the
lightest inert neutralino states grow and their contribution to the relic density of
dark matter becomes larger. This may lead to some interesting cosmological implications.
The detailed study of these implications is beyond the scope of this paper and will
be considered elsewhere.
\subsection{LHC signatures}
We can now turn to the possible collider signatures of the $E_6$ inspired SUSY models
with exact custodial $\tilde{Z}^{H}_2$ symmetry. The presence of $Z'$ boson and exotic
multiplets of matter in the particle spectrum is a very peculiar feature that may permit
to distinguish the considered $E_6$ inspired SUSY models from the MSSM or NMSSM.
Although the masses of the $Z'$ boson and exotic states cannot be predicted there are
serious reasons to believe that the corresponding particles should be relatively light.
Indeed, in the simplest scenario the VEVs of $H_u$, $H_d$ and $S$ are determined by the
corresponding soft scalar masses. Since naturalness arguments favor SUSY models with
$O(1\,\mbox{TeV})$ soft SUSY breaking terms the VEV $s$ is expected to be of the order
of $1-10\,\mbox{TeV}$. On the other hand the requirement of the validity of perturbation
theory up to the GUT scale sets stringent upper bounds on the low--energy values of the
Yukawa couplings $\kappa_i$ and $\lambda_{\alpha}$ whereas the gauge coupling unification
implies that $g^{'}_1(q)\simeq g_1(q)$. As a consequence the $Z'$ boson and exotic states
are expected to have masses below $10\,\mbox{TeV}$.
Collider experiments and precision EW tests set stringent limits on the mass of the $Z'$
boson and $Z-Z'$ mixing. The direct searches at the Fermilab Tevatron
$(p\overline{p}\to Z'\to l^{+}l^{-})$ exclude $Z'$, which is associated with $U(1)_N$,
with mass below $892\,\mbox{GeV}$ \cite{Accomando:2010fz}
\footnote{Slightly weaker lower bound on the mass of the $Z_N'$ boson was obtained
in \cite{Erler:2011ud}.}. Recently ATLAS and CMS experiments ruled out $E_6$ inspired $Z'$
with masses lower than $2-2.15\,\mbox{TeV}$ \cite{Chatrchyan:2012it}.
The analysis performed in \cite{ZprimeE6} revealed that $Z'$ boson in the $E_6$ inspired
models can be discovered at the LHC if its mass is less than $4-4.5\,\mbox{TeV}$.
The determination of its couplings should be possible if $M_{Z'}\lesssim 2-2.5\,\mbox{TeV}$
\cite{Dittmar:2003ir}. The precision EW tests bound the $Z-Z'$ mixing angle to be around
$[-1.5,\,0.7]\times 10^{-3}$ \cite{Erler:2009jh}. Possible $Z'$ decay channels in $E_6$
inspired supersymmetric models were studied in \cite{Accomando:2010fz}, \cite{Gherghetta:1996yr}.
The potential influence of gauge kinetic mixing on $Z'$ production at the 7 TEV LHC was
considered in \cite{Rizzo:2012rf}.
The production of a TeV scale exotic states will also provide spectacular LHC signals.
Several experiments at LEP, HERA, Tevatron and LHC have searched for colored objects
that decay into either a pair of quarks or quark and lepton. But most searches focus on
exotic color states, i.e leptoquarks or diquarks, have integer--spin. So they are either
scalars or vectors. These colored objects can be coupled directly to either
a pair of quarks or to quark and lepton. Moreover it is usually assumed that leptoquarks and
diquarks have appreciable couplings to the quarks and leptons of the first generation.
The most stringent constraints on the the masses of leptoquarks come from the nonobservation
of these exotic color states at the ATLAS and CMS experiments. Recently ATLAS collaboration ruled
out first and second generation scalar leptoquarks (i.e. leptoquarks that couple to the first and
second generation fermions respectively) with masses below $600-700\,\mbox{GeV}$ \cite{Aad:2011ch}.
The CMS collaboration excluded first and second generation scalar leptoquarks which are lighter
than $640-840\,\mbox{GeV}$ \cite{:2012dn}. The experimental lower bounds on the masses of dijet
resonances (in particular, diquarks) tend to be considerably higher (see, for example, \cite{90}).
However the LHC lower bounds on the masses of exotic quarks mentioned above are not directly
applicable in the case of the $E_6$ inspired SUSY models considered here.
Since $Z_{2}^{E}$ symmetry is conserved every interaction vertex contains an even number of
exotic states. As a consequence each exotic particle must eventually decay into a final state
that contains at least one lightest Inert neutralino (or an odd number of the lightest Inert
neutralinos). Since stable lightest Inert neutralinos cannot be detected directly each exotic
state should result in the missing energy and transverse momentum in the final state.
The $Z_{2}^{E}$ symmetry conservation also implies that in collider experiments exotic
particles can only be created in pairs.
In this context let us consider the production and sequential decays of the lightest exotic
quarks at the LHC first. Because $D$ and $\overline{D}$ states are odd under the $Z_{2}^{E}$
symmetry they can be only pair produced via strong interactions. In the scenario A the lifetime
and decay modes of the lightest exotic quarks are determined by the operators
$g^D_{ij} (Q_i L_4) \overline{D}_j$ and $h^E_{i\alpha} e^c_{i} (H^d_{\alpha} L_4)$ in the
superpotential (\ref{13}). These operators ensure that the lightest exotic quarks decay into
$$
D\to u_i (d_i) + \ell (\nu) + E^{\rm miss}_{T} + X\,,
$$
where $\ell$ is either electron or muon. Here $X$ may contain extra
charged leptons that can originate from the decays of intermediate
states (like Inert chargino or Inert neutralino). Since lightest exotic
quarks are pair produced these states may lead to a substantial enhancement
of the cross section $pp\to jj\ell^{+}\ell^{-}+E^{\rm miss}_{T}+X$ if they
are relatively light. In the scenario B the decays of the lightest exotic quarks
are induced by the operators $g^{q}_{ij}\overline{D}_i d^c_4 u^c_j$ and $h^D_{ij} d^c_4 (H^d_{i} Q_j)$.
As a consequence the lightest diquarks decay into
$$
D\to u^c_i + d^c_j + E^{\rm miss}_{T}+X\,,
$$
where $X$ again can contain charged leptons that may come from the decays
of intermediate states. In this case the presence of light $D$-fermions in the particle spectrum
could result in an appreciable enhancement of the cross section $pp\to jjjj+E^{\rm miss}_{T}+X$.
In general exotic squarks are expected to be substantially heavier than the exotic quarks
because their masses are determined by the soft SUSY breaking terms. Nevertheless the exotic
squark associated with the heavy exotic quark maybe relatively light. Indeed, as in the case
of the superpartners of the top quark in the MSSM, the large mass of the heaviest exotic quark
in the E$_6$SSM gives rise to the large mixing in the corresponding exotic squark sector
that may result in the large mass splitting between the appropriate mass eigenstates. As a
consequence the lightest exotic squark can have mass in TeV range. Moreover, in principle,
the lightest exotic squark can be even lighter than the lightest exotic quark. If this is
a case then the decays of the lightest exotic squark are induced by the same operators
which give rise to the decays of the lightest exotic quarks when all exotic squarks are heavy.
Therefore the decay patterns of the lightest exotic color states are rather similar in both
cases. In other words when exotic squark is the lightest exotic color state in the particle
spectrum it decays into either
$$
\tilde{D}\to
u_i (d_i) + \ell (\nu) + E^{\rm miss}_{T} + X\,,
$$
if exotic squark is a scalar leptoquark or
$$
\tilde{D}\to u^c_i + d^c_j + E^{\rm miss}_{T} + X\,,
$$
if it is a scalar diquark. Due to the $Z_{2}^{E}$ symmetry conservation $E^{\rm miss}_{T}$
should always contain contribution associated with the lightest exotic particle. However
since the lightest exotic squark is $R$--parity even state whereas the lightest Inert neutralino
is $R$--parity odd particle the final state in the decay of $\tilde{D}$ should also involve
the lightest neutralino to ensure that $R$--parity is conserved. Again, $X$ may contain
charged leptons that can stem from the decays of intermediate states. Because the $Z_{2}^{E}$
symmetry conservation implies that the lightest exotic squarks can be only pair produced in
the considered case the presence of light $\tilde{D}$ is expected to lead to an appreciable
enhancement of the cross section of either $pp\to jj\ell^{+}\ell^{-}+E^{\rm miss}_{T}+X$ if
$\tilde{D}$ is scalar leptoquark or $pp\to jjjj+E^{\rm miss}_{T}+X$ if $\tilde{D}$ is scalar
diquark.
Thus one can see that in both scenarios when the lightest exotic color state is either
$D$-fermion or $\tilde{D}$-scalar the collider signatures associated with these new states
are rather similar. Moreover since the decays of the lightest exotic color particles lead
to the missing energy and transverse momentum in the final state it might be rather
problematic to distinguish the corresponding signatures from the ones which are associated
with the MSSM. For example, the pair production of gluinos at the LHC should also result
in the enhancement of the cross section of $pp\to jjjj+E^{\rm miss}_{T}+X$. In this context
the presence of additional charged leptons in $X$ can play an important role leading to
characteristic signatures such as $\ell^{+}\ell^{-}$ pairs together with large missing energy
in the final state. The situation also becomes a bit more promising if one assumes
that the Yukawa couplings of the exotic particles have hierarchical structure similar
to the one observed in the ordinary quark and lepton sectors. In this case all states
which are odd under the $Z^{E}_2$ symmetry couple to the third generation fermions and
sfermions mainly\footnote{This possibility was discussed at length in
\cite{King:2005jy}--\cite{Accomando:2006ga}, \cite{8}.}. As a consequence the
presence of the relatively light exotic color states should give rise to the enhancement
of the cross section of either $pp\to t\bar{t}\ell^{+}\ell^{-}+E^{\rm miss}_{T}+X$ or
$pp\to t\bar{t}b\bar{b}+E^{\rm miss}_{T}+X$.
Here it is worthwhile to point out that the collider signatues associated with the light
scalar leptoquarks or diquarks in the considered $E_6$ inspired SUSY models are very
different from the commonly established ones which have been thoroughly studied.
For instance, it is expected that scalar diquarks may be produced singly at the LHC
and decay into quark--quark without missing energy in the final state. The scalar
leptoquarks can be only pair produced at the LHC but it is commonly assumed that
these states decay into quark--lepton without missing energy as well. On the other
hand in the $E_6$ inspired SUSY models considered here the $Z_{2}^{E}$ symmetry
conservation necessarily leads to the missing energy and transverse momentum
in the corresponding final state.
The presence of relatively light exotic quark and squark can substantially modify
the collider signatures associated with the production and decay of gluinos
\footnote{Novel gluino decays in the $E_6$ inspired models were recently considered
in \cite{Belyaev:2012si}}.
Indeed, if all squarks except the lightest exotic squark are rather heavy and
the decay of the gluino into exotic quark and squark are kinematically allowed
then the gluino pair production at the LHC results in $D\bar{D}\tilde{D}\tilde{\bar{D}}$
in the corresponding final state. The sequential decays of exotic quarks and squarks
give rise to the enhancement of either $pp\to 4\,\ell+4\, j + E^{\rm miss}_{T}+X$
if exotic color states are leptoquarks or $pp\to 8\,j + E^{\rm miss}_{T}+X$
if exotic color states are diquarks, modulo of course effects of QCD
radiation and jet merging. The modification of the gluino collider
signatures discussed above might be possible only if there are non--zero
flavor-off-diagonal couplings $\theta^g_{ij}$ of gluino to $D_i$ and $\tilde{D}_j$
($i\ne j$). This is a necessary condition because the lightest exotic squark is
normally associated with the heaviest exotic quark. Rough estimates indicate
that the corresponding modification of the gluino collider signatures can occur
even when the gluino flavour-off-diagonal couplings $\theta^g_{ij}$ are relatively
small, i.e. $\theta^g_{ij}\gtrsim 0.01$.
If gluino is heavier than the lightest exotic color state, but is substantially lighter
than the second lightest exotic color state than the branching ratios of the nonstandard
gluino decays mentioned above are suppressed. In this case the second lightest exotic
color state can decay mostly into the lightest exotic color state and gluino if the
corresponding decay channel is kinematically allowed. This happens when the lightest exotic
color state is exotic $D$-fermion while the second lightest exotic color state is
$\tilde{D}$-scalar or vice versa.
Other possible manifestations of the $E_6$ inspired SUSY models considered
here are related to the presence of vectorlike states $d^c_4$ and $\overline{d^c}_4$
as well as $L_4$ and $\overline{L}_4$. In the case of scenario B the fermionic components
of the supermultiplets $d^c_4$ and $\overline{d^c}_4$ can have mass below the TeV scale.
One of the superpartners of this vectorlike quark state may be also relatively light
due to the mixing in the corresponding squark sector. If these quark and/or squark
states are light they can be pair produced at the LHC via strong interactions. Since
the superfields $d^c_4$ and $\overline{d^c}_4$ are odd under the $Z^{E}_2$
symmetry the decays of the corresponding quarks ($d_4$) and squarks ($\tilde{d}_4$) must
always lead to the missing energy in the final state. In the limit when the lightest
exotic color states include $d_4$ and/or $\tilde{d}_4$ whereas all other exotic states
and sparticles are much heavier, the operators $h^D_{ij} d^c_4 (H^d_{i} Q_j)$ give
rise to the following decay modes of $d_4$ and $\tilde{d}_4$
$$
d_4 \to q_i + E^{\rm miss}_{T} + X\,,\qquad\qquad\qquad
\tilde{d}_4 \to d_i + E^{\rm miss}_{T} + X\,,
$$
where $q_i$ can be either up-type or down-type quark while $X$ may contain charged leptons
which can appear as a result of the decays of intermediate states. As in the case of
exotic squark the final state in the decay of $d_4$ should contain the lightest neutralino
and the lightest Inert neutralino to ensure the conservation of $R$--parity and $Z^{E}_2$
symmetry. Again due to the $Z_{2}^{E}$ symmetry conservation $d_4$ and $\tilde{d}_4$ can be only
pair produced at the LHC resulting in an enhancement of $pp\to jj+E^{\rm miss}_{T}+X$.
If $d_4$ and $\tilde{d}_4$ couple predominantly to the third generation fermions and sfermions
then the pair production of these quarks/squarks should lead to the presence of two heavy
quarks in the final state. As before these collider signatures do not permit to distinguish
easily the considered $E_6$ inspired SUSY models from other supersymmetric models.
For example, squark pair production at the LHC can also lead to two jets and missing energy
in the final state. Again, the presence of additional charged leptons in $X$ can lead to
the signatures that may help to distinguish the considered $E_6$ inspired SUSY models from
the simplest SUSY extensions of the SM.
In the case of scenario A the fermionic components of the supermultiplets $L_4$ and
$\overline{L}_4$ as well as one of the superpartners of this vectorlike state may have masses
below the TeV scale. If all other exotic states and sparticles are rather heavy the
corresponding bosonic ($\tilde{L}_4$) and fermionic ($L_4$) states can be produced at the
LHC via weak interactions only. Because of this their production cross section is relatively
small. In the considered limit the decays of $L_4$ and/or $\tilde{L}_4$ are induced by
the operators $h^E_{i\alpha} e^c_{i} (H^d_{\alpha} L_4)$. As a consequence the decays
of $L_4$ and/or $\tilde{L}_4$ always lead to either $\tau$--lepton or electron/muon
as well as missing energy in the final state. In the case of $\tilde{L}_4$ decays the
missing energy in the final state can be associated with only one lightest Inert neutralino
whereas the final state of the $L_4$ decays must contain at least one lightest Inert neutralino
and one lightest ordinary neutralino to ensure the conservation of $R$--parity and $Z^{E}_2$
symmetry. More efficiently $L_4$ and/or $\tilde{L}_4$ can be produced through the decays of the
lightest exotic color states (i.e. $D$ and/or $\tilde{D}$) if these states are relatively light
and the corresponding decay channels are kinematically allowed.
The Inert Higgs bosons and/or Inert neutralino and chargino states, which are predominantly Inert
Higgsinos, can be also light or heavy depending on their free parameters. Indeed, as follows from
Eq.~(\ref{67}) the lightest Inert Higgsinos may be light if the corresponding Yukawa coupling
$\lambda_{\alpha}$ is rather small. On the other hand if at least one coupling $\lambda_{\alpha}$
is large it can induce a large mixing in the Inert Higgs sector that may lead to relatively
light Inert Higgs boson states. Since Inert Higgs and Higgsino states do not couple to quarks
directly at the LHC the corresponding states can be produced in pairs via off--shell $W$ and
$Z$--bosons. Therefore their production cross section remains relatively small even when these
states have masses below the TeV scale. The lightest Inert Higgs and Higgsino states are
expected to decay via virtual lightest Higgs, $Z$ and $W$ exchange. The conservation of $R$--parity
and $Z^{E}_2$ symmetry implies that the final state in the decay of Inert Higgsino involves at
least one lightest Inert neutralino while the final state in the decay of Inert Higgs state
should contain at least one lightest ordinary neutralino and one lightest Inert neutralino.
As it was mentioned in the beginning of this subsection in the simplest scenario, when
only $H_u$, $H_d$ and $S$ acquire VEVs at low energies, there are serious reasons to believe
that the $Z'$ boson and all exotic states from three complete $27_i$ representations of $E_6$
have masses below $10\,\mbox{TeV}$. However the situation may change dramatically when
$\tilde{Z}^H_2$ even superfield $\overline{S}$ survive to low energies. In order to demonstrate
this, let us consider a simple toy model, where $U(1)_N$ gauge symmetry is broken by VEVs of
a pair of SM singlet superfields $S$ and $\overline{S}$. Assuming that the superpotential of the
considered model involves bilinear term $\mu_S\, S \overline{S}$ the part of the tree--level scalar
potential, which depends on the scalar components of the superfields $S$ and $\overline{S}$ only,
can be written as
\begin{equation}
V_S = (m^2_S+\mu_S^2) |S|^2 + (m^2_{\overline{S}}+\mu_S^2) |\overline{S}|^2
+ (B_S \mu_S S \overline{S} + h.c.)
+\displaystyle\frac{Q_S^2 g^{'2}_1}{2}\left(|S|^2-|\overline{S}|^2\right)^2\,,
\label{74}
\end{equation}
where $m_S^2$, $m^2_{\overline{S}}$ and $B_S$ are soft SUSY breaking parameters and $Q_S$ is
a $U(1)_N$ charge of the SM singlet superfields $S$. The last term in Eq.~(\ref{74}), which
is the $U(1)_N$ D--term contribution to the scalar potential, forces the minimum of the
corresponding potential to be along the $D$--flat direction
$\langle S \rangle = \langle \overline{S} \rangle$. Indeed, in the limit
$\langle S \rangle = \langle \overline{S} \rangle$ the quartic terms in the potential (\ref{74})
vanish. In the considered case the scalar potential (\ref{74}) remains positive definite only if
$(m^2_S + m^2_{\overline{S}}+ 2 \mu_S^2 - 2|B_S \mu_S|)>0$. Otherwise physical vacuum becomes
unstable, i.e. $\langle S \rangle = \langle \overline{S} \rangle \to \infty$.
The scalar potential can be easily stabilized the if bilinear term $\mu_S\, S \overline{S}$
in the superpotential is replaced by
\begin{equation}
W_S = \lambda_0 \tilde{\phi} S \overline{S} + f(\tilde{\phi})\,,
\label{75}
\end{equation}
where $\tilde{\phi}$ is $\tilde{Z}^H_2$ even superfield that does not participate in the
$SU(3)_C\times SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$ gauge interactions.
When $\lambda_0$ is small (i.e. $\lambda_0\ll 0.1$) the $U(1)_N$ D--term contribution to
the scalar potential still forces the minimum of the scalar potential to be along the
nearly $D$--flat direction if $m^2_S + m^2_{\overline{S}}<0$. This condition can be satisfied
because sufficiently large values of $\kappa_i$ affect the evolution of $m_S^2$ rather
strongly resulting in negative values of $m_S^2$ at low energies \cite{8}.
If $m^2_S + m^2_{\overline{S}}<0$ and $\lambda_0$ is small then the scalar components of
the superfields $\tilde{\phi}$, $S$ and $\overline{S}$ acquire very large VEVs, i.e.
\begin{equation}
\langle \tilde{\phi} \rangle \sim \langle S \rangle \simeq \langle \overline{S} \rangle \sim M_{SUSY}/\lambda_0\,,
\label{76}
\end{equation}
where $M_{SUSY}$ is a supersymmetry breaking scale.
If $\lambda_0\simeq 10^{-3}-10^{-4}$ the VEVs of the SM singlet superfields $S$ and $\overline{S}$
are of the order of $10^{3}-10^4\,\mbox{TeV}$ even when $M_{SUSY}\sim 1\,\mbox{TeV}$.
So large VEV of the superfield $S$ may give rise to the extremely heavy spectrum of exotic
particles and $Z'$. This can lead to the MSSM type of particle spectrum at the $\mbox{TeV}$
scale.
Nevertheless even in this case the broken $U(1)_N$ symmetry leaves its imprint on the
MSSM sfermion mass spectrum. Since $m^2_S\ne m^2_{\overline{S}}$ the VEVs of the SM singlet
superfields $S$ and $\overline{S}$ deviates from the $D$--flat direction
\begin{equation}
Q_S^2 g^{'2}_1 \left(\langle S \rangle^2 - \langle \overline{S} \rangle^2\right)\simeq m^2_{\overline{S}}-m^2_S\,.
\label{77}
\end{equation}
As a consequence all sfermions receive an additional contribution to the mass that come
from the $U(1)_N$ $D$--term quartic interactions in the scalar potential \cite{Kolda:1995iw}.
This contribution $\Delta_i$ is proportional to the $U(1)_N$ charge of the corresponding sfermion
$Q_i$, i.e.
\begin{equation}
\Delta_{i}=\dfrac{g^{'2}_1}{2}\biggl(Q_1 v_1^2 + Q_2 v_2^2 +
2 Q_S \left(\langle S \rangle^2 - \langle \overline{S} \rangle^2\right)\biggr)
Q_{i}=M_0^2\, \sqrt{40}\, Q_{i} \,,
\label{78}
\end{equation}
where $Q_1$ and $Q_2$ are the $U(1)_N$ charges of $H_d$ and $H_u$.
Thus for the superpartners of the first and second generation quarks and leptons one finds
$$
\begin{array}{rcl}
m_{\tilde{d}_{L\,i}}^2&\simeq &m_{Q_i}^2+\left(-\dfrac{1}{2}+\dfrac{1}{3}\sin^2\theta_W\right)M_Z^2\cos 2\beta + M_0^2\,,\\
m_{\tilde{u}_{L\,i}}^2&\simeq &m_{Q_i}^2+\left(\dfrac{1}{2}-\dfrac{2}{3}\sin^2\theta_W\right)M_Z^2\cos 2\beta + M_0^2\,,\\
m_{\tilde{u}_{R\,i}}^2&\simeq &m_{u^c_i}^2+\dfrac{2}{3} M_Z^2 \sin^2\theta_W \cos 2\beta + M_0^2\,,\\
m_{\tilde{d}_{R\,i}}^2&\simeq &m_{d^c_i}^2-\dfrac{1}{3} M_Z^2 \sin^2\theta_W \cos 2\beta + 2 M_0^2\,,\\[2mm]
m_{\tilde{e}_{L\,i}}^2&\simeq &m_{L_i}^2+\left(-\dfrac{1}{2}+\sin^2\theta_W\right)M_Z^2\cos 2\beta + 2 M_0^2\,,\\
m_{\tilde{\nu}_{i}}^2&\simeq &m_{L_i}^2+\dfrac{1}{2} M_Z^2\cos 2\beta + 2 M_0^2\,,\\
m_{\tilde{e}_{R\,i}}^2&\simeq &m_{e^c_i}^2- M_Z^2 \sin^2\theta_W \cos 2\beta + M_0^2\,.
\end{array}
$$
\section{Conclusions}
In this paper we have considered the $E_6$ inspired SUSY models in which a single discrete
$\tilde{Z}^{H}_2$ symmetry forbids the tree--level flavor--changing transitions and baryon
number violating operators. We assumed that the breakdown of $E_6$ symmetry or its subgroup
lead to the rank--6 SUSY models below the GUT scale $M_X$. These models are based on the
Standard Model (SM) gauge group together with extra $U(1)_{\psi}$ and $U(1)_{\chi}$ gauge
symmetries. We also allow three copies of $27_i$ representations of $E_6$ to survive below
the scale $M_X$ so that anomalies get canceled generation by generation. If extra exotic
states from $27_i$--plets survive to low energies they give rise to tree--level non--diagonal
flavor transitions and rapid proton decay. In order to suppress baryon number violating
operators one can impose $\tilde{Z}^{H}_2$ discrete symmetry. We assumed that all matter
superfields, that fill in complete $27_i$ representations of $E_6$, are odd under this
discrete symmetry. Thus $\tilde{Z}^{H}_2$ symmetry is defined analogously to the matter
parity $Z_{2}^{M}$ in the simplest $SU(5)$ SUSY GUTs, that lead to the low--energy spectrum
of the MSSM.
In addition to three complete fundamental representations of $E_6$ we further assumed the
presence of of $M_{l}$ and $\overline{M}_l$ supermultiplets from the incomplete $27'_l$ and
$\overline{27'}_l$ representation just below the GUT scale. Because multiplets $M_{l}$ and
$\overline{M}_l$ have opposite $U(1)_{Y}$, $U(1)_{\psi}$ and $U(1)_{\chi}$ charges their
contributions to the anomalies get cancelled identically.
As in the MSSM we allowed the set of multiplets $M_{l}$ to be used for the breakdown of
gauge symmetry and therefore assumed that all multiplets $M_{l}$ are even under $\tilde{Z}^{H}_2$
symmetry. In order to ensure that the $SU(2)_W\times U(1)_Y\times U(1)_{\psi}\times U(1)_{\chi}$
symmetry is broken down to $U(1)_{em}$ associated with the electromagnetism the set of multiplets
$M_{l}$ should involve $H_u$, $H_d$, $S$ and $N^c_H$.
We argued that $U(1)_{\psi}\times U(1)_{\chi}$ gauge symmetry can be broken by the VEVs
of $N^c_H$ and $\overline{N}_H^c$ down to $U(1)_{N}\times Z_{2}^{M}$ because matter parity
is a discrete subgroup of $U(1)_{\psi}$ and $U(1)_{\chi}$. Such breakdown of $U(1)_{\psi}$ and
$U(1)_{\chi}$ gauge symmetries guarantees that the exotic states which originate from $27_i$
representations of $E_6$ as well as ordinary quark and lepton states survive to low energies.
On the other hand the large VEVs of $N^c_H$ and $\overline{N}_H^c$ can induce the large Majorana
masses for right-handed neutrinos allowing them to be used for the see--saw mechanism.
For this reason we assumed that the $U(1)_{\psi}\times U(1)_{\chi}$ symmetry is broken down
to $U(1)_{N}\times Z_{2}^{M}$ just below the GUT scale.
The $\tilde{Z}^{H}_2$ symmetry allows the Yukawa interactions in the superpotential that originate
from $27'_l \times 27'_m \times 27'_n$ and $27'_l \times 27_i \times 27_k$. Since the set of
multiplets $M_{l}$ contains only one pair of doublets $H_d$ and $H_u$ the $\tilde{Z}^{H}_2$ symmetry
defined above forbids not only the most dangerous baryon and lepton number violating operators
but also unwanted FCNC processes at the tree level. Nevertheless if the set of $\tilde{Z}^{H}_2$
even supermultiplets $M_{l}$ involve only $H_u$, $H_d$, $S$ and $N^c_H$ then the lightest exotic
quarks are extremely long--lived particles because $\tilde{Z}^{H}_2$ symmetry forbids all Yukawa
interactions in the superpotential that allow the lightest exotic quarks to decay. Since models
with stable charged exotic particles are ruled out by different terrestrial experiments the set
of supermultiplets $M_{l}$ in the phenomenologically viable $E_6$ inspired SUSY models should be
supplemented by some components of $27$-plet that carry $SU(3)_C$ colour or lepton number.
In this work we required that extra matter beyond the MSSM fill in complete $SU(5)$ representations
because in this case the gauge coupling unification remains almost exact in the one--loop
approximation. As a consequence we restricted our consideration to two scenarios that result in
different collider signatures associated with the exotic quarks. In the scenario A the set of
$\tilde{Z}^{H}_2$ even supermultiplets $M_{l}$ involves lepton superfields $L_4$. To ensure the
unification of gauge couplings we assumed that $\overline{H}_u$ and $\overline{H}_d$ are odd under
the $\tilde{Z}^{H}_2$ symmetry whereas supermultiplet $\overline{L}_4$ is even. Then $\overline{H}_u$
and $\overline{H}_d$ from the $\overline{27'}_l$ get combined with the superposition of the
corresponding components from $27_i$ so that the resulting vectorlike states gain masses of order
of $M_X$. In contrast, $L_4$ and $\overline{L}_4$ should form vectorlike states at low energies
facilitating the decays of exotic quarks. The superfield $\overline{S}$ can be either odd or even
under the $\tilde{Z}^{H}_2$ symmetry. The bosonic and fermionic components of $\overline{S}$ may or may
not survive to low energies. In the scenario A the exotic quarks are leptoquarks.
Another scenario, that permits the lightest exotic quarks to decay within a reasonable time, implies
that the set of multiplets $M_{l}$ together with $H_u$, $H_d$, $S$ and $N^c_H$ contains extra $d^c_4$
supermultiplet. Because in this scenario B the $\tilde{Z}^{H}_2$ even supermultiplets $d^c_4$ and
$\overline{d^c}_4$ give rise to the decays of the lightest exotic color states they are expected
to form vectorlike states with the TeV scale masses. Then to ensure that the extra matter
beyond the MSSM fill in complete $SU(5)$ representations $\overline{H}_u$ and $\overline{H}_d$ should
survive to the TeV scale as well. Again we assumed that $\overline{H}_u$ and $\overline{H}_d$ are
odd under the $\tilde{Z}^{H}_2$ symmetry so that they can get combined with the superposition of the
corresponding components from $27_i$ forming vectorlike states at low energies. As in the case of
scenario A the superfield $\overline{S}$ can be either even or odd under the $\tilde{Z}^{H}_2$
symmetry and may or may not survive to the TeV scale. In the scenario B the exotic quarks manifest
themselves in the Yukawa interactions as superfields with baryon number $\left(\pm\dfrac{2}{3}\right)$.
The gauge group and field content of the $E_6$ inspired SUSY model discussed here can originate from the
5D and 6D orbifold GUT models in which the splitting of GUT multiplets can be naturally achieved.
In particular, we studied $SU(5)\times U(1)_{\chi}\times U(1)_{\psi}$ SUSY GUT model in 5D compactified on
the orbifold $S^1/(Z_2\times Z'_2)$. At low energies this model may lead to the scenarios A and B. We also
considered $E_6$ gauge theory in $6D$ compactified on the orbifold $T^2/(Z_2 \times Z^{I}_2 \times Z^{II}_2)$
that can lead to the scenario A at low energies. In these orbifold GUT models all anomalies get cancelled
and GUT relations between Yukawa couplings get spoiled. The adequate suppression of the operators, that
give rise to proton decay, can be also achieved if the GUT scale $M_X\sim 1/R$ is larger than
$1.5-2\cdot 10^{16}\,\mbox{GeV}$.
We examined the RG flow of gauge couplings from $M_Z$ to $M_X$ in the case of scenarios A and B using both
analytical and numerical techniques. We derived the corresponding two--loop RG equations and studied the
running of the gauge couplings with and without extra $S$ and $\overline{S}$ superfields at the TeV scale.
In the scenario A the gauge coupling unification can be achieved for any phenomenologically reasonable value
of $\alpha_3(M_Z)$ consistent with the central measured low energy value. This was already established in
the case of the SUSY model with extra $U(1)_N$ gauge symmetry and low energy matter content that involves
three 27-plets of $E_6$ as well as $L_4$ and $\overline{L}_4$ \cite{King:2007uj}. Our analysis here revealed
that the evolution of the SM gauge couplings does not change much when the low energy particle spectrum is
supplemented by the $S$ and $\overline{S}$ chiral superfields. Thus this is not so surprising that the
unification of the SM gauge couplings can be so easily achieved even in this case. In the scenario B
large two--loop corrections spoil the unification of gauge couplings. Indeed, in this case the exact gauge
coupling unification can be achieved only if $\alpha_3(M_Z)\lesssim 0.112$. As before the inclusion of
extra $S$ and $\overline{S}$ superfields does not change much the RG flow of $\alpha_i(t)$ and therefore
does not improve gauge coupling unification. However the relative discrepancy of $\alpha_i(M_X)$ is about 10\% .
At the same time orbifold GUT framework does not imply the exact gauge coupling unification near the scale
$M_X\sim 1/R$ because of the brane contributions to the gauge couplings. Therefore relative discrepancy of
10\% between $\alpha_i(M_X)$ should not be probably considered as a big problem.
Finally we also discussed the cosmological implications and collider signatures of the $E_6$ inspired
SUSY models discussed above. As it was mentioned the low--energy effective Lagrangian of these models is invariant
under both $Z_{2}^{M}$ and $\tilde{Z}^{H}_2$ symmetries. Since $\tilde{Z}^{H}_2 = Z_{2}^{M}\times Z_{2}^{E}$ the
$Z_{2}^{E}$ symmetry associated with exotic states is also conserved. As a result the lightest exotic
state, which is odd under the $Z_{2}^{E}$ symmetry, must be stable. In the scenarios A and B the lightest and
second lightest inert neutralinos tend to be the lightest exotic states in the particle spectrum. On the other
hand the $Z_{2}^{M}$ symmetry conservation implies that $R$--parity is conserved. Because the lightest inert
neutralino $\tilde{H}^0_1$ is also the lightest $R$--parity odd state either the lightest $R$--parity even exotic
state or the lightest $R$--parity odd state with $Z_{2}^{E}=+1$ must be absolutely stable. Most commonly the second
stable state is the lightest ordinary neutralino $\chi_1^0$ ($Z_{2}^{E}=+1$). Both stable states are natural
dark matter candidates in the considered $E_6$ inspired SUSY models.
When $|m_{\tilde{H}^0_{1}}|\ll M_Z$ the lightest inert neutralino is predominantly inert singlino and its couplings
to the gauge bosons, Higgs states, quarks and leptons are very small resulting in too small annihilation cross
section for $\tilde{H}^0_1\tilde{H}^0_1\to \mbox{SM particles}$. As a consequence the cold dark matter density is much
larger than its measured value. In principle, $\tilde{H}^0_1$ could account for all or some of the observed cold
dark matter density if it had mass close to half the $Z$ mass. In this case the lightest inert neutralino states
annihilate mainly through an $s$--channel $Z$--boson. However the usual SM-like Higgs boson decays more than 95\%
of the time into either $\tilde{H}^0_1$ or $\tilde{H}^0_2$ in these cases while the total branching ratio into
SM particles is suppressed. Because of this the corresponding scenarios are basically ruled out nowadays.
The simplest phenomenologically viable scenarios imply that the lightest and second lightest inert neutralinos are
extremely light. For example, these states can have masses about $1\,\mbox{eV}$. The lightest and second lightest
inert neutralinos with masses about $1\,\mbox{eV}$ form hot dark matter in the Universe but give only a very minor
contribution to the dark matter density while the lightest ordinary neutralino may account for all or some of
the observed dark matter density.
The presence of two types of dark matter is a very peculiar feature that affect the collider
signatures of the considered $E_6$ inspired SUSY models. The most spectacular LHC signals
associated with these models may come from the TeV scale exotic color states and $Z'$.
The production of the $Z'$ boson, that corresponds to the $U(1)_N$ gauge
symmetry, should lead to unmistakable signal $pp\to Z'\to l^{+}l^{-}$ at the LHC.
The $Z_{2}^{E}$ symmetry conservation implies that in collider experiments exotic particles
can only be created in pairs. Moreover each exotic particle has to decay into a final state
that contains at least one lightest inert neutralino resulting in the missing energy.
Because of this the lightest exotic color state, that can be either $D$-fermion or $\tilde{D}$-scalar,
decay into either $u_i (d_i) + \ell (\nu) + E^{\rm miss}_{T} + X$ if exotic
quark (squark) is leptoquark or $u^c_i + d^c_j + E^{\rm miss}_{T} + X$ if exotic quark
(squark) is diquark. The $Z_{2}^{E}$ symmetry conservation requires that $E^{\rm miss}_{T}$
should always contain contribution associated with the lightest inert neutralino. Since the
lightest exotic squark is $R$--parity even state while the lightest inert neutralino is $R$--parity
odd particle the final state in the decay of $\tilde{D}$ should also involve the lightest
ordinary neutralino to ensure $R$--parity conservation. Thus the pair production of the lightest
exotic color state is expected to lead to a substantial enhancement of the cross section of
either $pp\to jj\ell^{+}\ell^{-}+E^{\rm miss}_{T}+X$ or
$pp\to jjjj+E^{\rm miss}_{T}+X$. If the Yukawa couplings of the exotic particles
have hierarchical structure similar to the one observed in the ordinary quark and lepton
sectors then all states which are odd under the $Z^{E}_2$ symmetry couple to the third
generation fermions and sfermions mainly. As a result the TeV scale exotic color
states should give rise to the enhancement of the cross section of either
$pp\to t\bar{t}\ell^{+}\ell^{-}+E^{\rm miss}_{T}+X$ or $pp\to t\bar{t}b\bar{b}+E^{\rm miss}_{T}+X$.
Our consideration indicates that $\tilde{D}$-scalars in the considered $E_6$ inspired
SUSY models lead to rather unusual collider signatures. Indeed, it is commonly expected
that scalar diquarks decay into quark--quark without missing energy in the final state
while the scalar leptoquarks decay into quark--lepton without missing energy as well.
In the models considered here the $Z_{2}^{E}$ symmetry conservation necessarily
leads to the missing energy in the corresponding final states. In addition relatively
light exotic quark and squark can modify the collider signatures associated with gluinos
if the decay of the gluino into exotic quark and squark is kinematically allowed.
In this case gluino pair production at the LHC may result in $D\bar{D}\tilde{D}\tilde{\bar{D}}$
in the final state. The sequential decays of $D$-fermions and $\tilde{D}$-scalars give rise to the
enhancement of either $pp\to 4\,\ell+4\, j + E^{\rm miss}_{T}+X$ or $pp\to 8\,j + E^{\rm miss}_{T}+X$.
In the scenario B the fermionic components of the supermultiplets $d^c_4$ and $\overline{d^c}_4$
that form vectorlike quark state as well as their superpartner may have TeV scale masses.
Then these quark and/or squark states can be pair produced at the LHC via strong interactions
and decay into $q_i + E^{\rm miss}_{T} + X$ where $q_i$ can be either up-type or down-type quark.
This may lead to an enhancement of $pp\to jj+E^{\rm miss}_{T}+X$.
The discovery of $Z'$ and new exotic particles predicted by the $E_6$ inspired SUSY models considered
here will open a new era in elementary particle physics. This would not only represent a revolution
in particle physics, but would also point towards an underlying $E_6$ gauge structure at high energies.
\section*{Acknowledgements}
\vspace{-2mm}
R.N. thanks X.~Tata for sharing his valuable ideas in connection with this work.
R.N. acknowledges fruitful discussions with S.~F.~King, J.~Kumar, S.~Moretti, S.~Pakvasa
and T.~Rizzo. R.N. is also grateful to P.~Athron, J.~Bjorken, K.~R.~Dienes, J.~Hewett,
S.~Kraml, D.~J.~Miller, M.~M\"{u}hlleitner, M.~Sher, M.~A.~Shifman, L.~B.~Okun, B.~D.~Thomas,
D.~G.~Sutherland, A.~I.~Vainshtein, M.~I.~Vysotsky for valuable comments and remarks.
The work of R.N. was supported by the U.S. Department of Energy under Contract DE-FG02-04ER41291.
\newpage
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,190 |
María Teresa Antoñanzas Garro, née le , est une femme politique espagnole membre du Parti populaire.
Biographie
Vie privée
Elle est célibataire.
Profession
Carrière politique
Le , elle est élue sénatrice pour La Rioja au Sénat et réélue en 2016. Elle est première secrétaire de la commission générale des communautés autonomes.
Notes et références
Voir aussi
Article connexe
Sénateurs de la XIIe législature de l'Espagne
Lien externe
Fiche sur le site du Sénat (Espagne)
Sénateur espagnol de la XIe législature
Sénateur espagnol de la XIIe législature
Personnalité politique espagnole du XXIe siècle
Personnalité du Parti populaire (Espagne)
Femme politique espagnole
Naissance en octobre 1979 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,673 |
Antarctic Intermediate Water (AAIW) is a cold, relatively low salinity water mass found mostly at intermediate depths in the Southern Ocean. The AAIW is formed at the ocean surface in the Antarctic Convergence zone or more commonly called the Antarctic Polar Front zone. This convergence zone is normally located between 50°S and 60°S, hence this is where almost all of the AAIW is formed.
Properties
The AAIW is unique water mass in that it is a sinking water mass with a moderately low salinity, unlike most sinking water masses which have a relatively high salinity. This salinity minimum, unique to the AAIW, can be recognized throughout the Southern Ocean at depths ranging from 700 to 1200 meters. Typical temperature values for the AAIW are 3-7°C, and a salinity of 34.2-34.4 psu upon initial formation. Due to vertical mixing at intermediate depths in the Southern Ocean, the salinity slowly rises as it moves northward. Typical density of AAIW water is between 1026.82 kg/m3 and 1027.43 kg/m3. The thickness of the AAIW ranges greatly between where it forms and its most northern extent.
Formation
The formation of AAIW can be explained very simply through the Ekman transport process and the divergence and convergence of water masses. The winds over Antarctica are called the polar easterlies where winds blow from the east to the west. This creates a counter-clockwise surface current near the coast of Antarctica, called the Antarctic Coastal Current. Ekman transport causes the water to push towards the left of the surface motion in the Southern Hemisphere. Thus, this westward directed coastal current in Antarctica will push the water towards Antarctica.
At the same time there is a strong current north of the Antarctic Coastal Current, called the Antarctic Circumpolar Current (ACC) created by the strong westerlies in this region which flows clockwise around Antarctica. Again, Ekman transport will push this water to the left of the surface motion, meaning away from Antarctica. Because water just offshore of Antarctica is being pushed away and into Antarctica, it leads to the Antarctic Divergence region. Here, upwelling of North Atlantic Deep Water (NADW) takes place. NADW is cold and quite saline. Once the NADW is upwelled to the surface some of it diverges towards Antarctica, gets colder, and sinks back down as Antarctic Bottom Water.
The NADW water also diverges away from Antarctica when it is upwelled. This diverged water moves northward (equatorward), and at the same time persistent precipitation (location is near the polar lows ~60°S) along with an influx of melt water decreases the salinity of the original NADW. Because the salinity of the NADW has changed by so much and it has essentially lost all its unique characteristics to be NADW, this northward propagating surface water is now called Antarctic Surface Water (AASW). Also, the AASW movement northward has gained some heat from the atmosphere, thereby increasing the temperature slightly.
When this water reaches between 50°S and 60°S it encounters the Antarctic Convergence zone. At this point the Subantarctic waters, which are characterized as being much warmer than the Antarctic waters, are just north of the Antarctic Polar Front and the Antarctic waters are just south of the Antarctic Polar Front. This region is referred to as the Antarctic Convergence Zone/Antarctic Polar Front because of the sharp gradients in both temperature and salinity (esp. temperature) between the Antarctic waters and the Subantarctic waters. It is also a region of strong vertical mixing. It is important to note that this convergence zone does not occur simply because the Subantarctic water is flowing southward and the AASW is flowing northward, but due to Ekman convergence.
Once the northward propagating Antarctic Surface Water reaches the Antarctic Convergence zone it begins to sink because it is more dense than the Subantarctic water to its north, but less dense than the Antarctic water to its south. This water is then referred to as AAIW. The sinking AAIW becomes sandwiched between the Subantarctic water (above) which is much warmer, but more saline and the NADW (below) which is cold and quite salty.
For many years the aforementioned formation of AAIW was thought to be the only formation process. Recent studies have found that there exists some evidence that some Subantarctic mode water is able to penetrate through the Subantarctic front (frontal region separating the Polar frontal zone from the Subantarctic zone) and become the dominant source of AAIW, rather than the AASW. Because of the difficulty of getting observations in this very treacherous area, this research on Subantarctic mode water mixing theory is still being worked out, but a lot of evidence exists for its inclusion in the formation of AAIW. It is important to note that the biggest source of AAIW formation is just southwest of the southern tip of South America.
Areal extent and movement
The interesting characteristic of AAIW is how far it extends northward. The salinity minima associated with the AAIW can be seen in intermediate waters (~1000m) as far north as 20°N, with trace amounts as far as 60°N. It is by far the largest spreading intermediate water of all the ocean intermediate water masses. It continues northward until it encounters other intermediate water masses (e.g.AIW). The movement of the AAIW is predominantly northward due to the Ekman volume transport mostly directed in that way. When the AAIW is initially formed, the ACC is able to transport the AAIW into all ocean basins because the ACC flows clockwise around Antarctica with no land based boundaries.
References
External links
Process Study of AAIW
Polar Discovery at WHOI
Glossary of Physical Oceanography and Related Disciplines Antarctic Intermediate Water (AAIW)
Geography of the Southern Ocean
Water masses | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,275 |
Q: NSString encoding-name from WebDataSource? I have a WebDataSource and I want to get pure HTML out of it. How can I do it?
The way I did it so far is:
get it's webArchive, and then get WebArchive's data. But when I try to store it in a string, I don't know the encoding name? What should I use? I used several and it gives me nil. How can I find it from the WebDataSource what encoding its using?
Thanks!
A: I'm not sure WebDataSource is really what you want for this. If you want to get the HTML, you should be using NSURLRequest, which will pull the HTML in as NSData. You can then encode it into a string with NSUTF8StringEncoding.
Are you familiar with NSURLRequest?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,199 |
The energy released by the Indian Ocean tsunami of 2004 equalled 40,000 Hiroshima-sized atomic bombs, according to eminent earth scientist Harsh Gupta.
The massive undersea earthquake measuring 9.1 on Richter Scale, which triggered the worst-ever tsunami, caused 1,500 km long, 300 km wide and 20 meter vertical rupture.
Tsunami Early Warning Centre (ITEWC) here to mark the 10th anniversary of the disaster.
The tsunami claimed 2.38 lakh lives in 14 countries. As many as 10,749 people were killed in India and 5,640 went missing. More than 1.5 million people were displaced in the affected countries.
Gupta, former secretary of ministry of earth sciences, said it was a rare event which occurs once or twice a century.
The experts at the workshop recalled that the disaster was a rude shock as most people had not even heard of tsunami.
"We all watched on TV how the tragic events unfolded that Sunday," said another former secretary. P.S. Goel.
They said everyone including institutes which monitor earthquakes were caught off guard.
Gupta, who later played a key role in developing the tsunami early warning system, recalled the efforts taken in the aftermath of the horror.
He pointed out that not every earthquake triggers tsunami.
"Whenever an earthquake occurs through the ocean bottom and the rupture occurs in the sea bed creating a vertical displacement of the ocean bed and the entire column of water above gets displaced; this column gets oscillate and moves out and it is called tsunami wave," he said.
One of them was the Andaman-Nicobar-Sumatra island arc.
The scientist said the world had seen three great tsunamis: in 1755, 2004 and 2011.
The first major tsunami occurred on Nov 1, 1755, caused by the Lisbon earthquake. The most damaging tsunami ever in the Atlantic ocean claimed 90,000 lives.
As no major tsunami occurred till 2004, there was no awareness among people.
He pointed out that those walking on the beach in Chennai on the morning of Dec 26, 2004 rushed to pick up pebbles and shells when water receded in the ocean. In no time high waves lashed the coast, washing them away.
Gupta said the destruction was massive as the rule which prohibits any commercial activity in the 500 meter area along the coast was flouted.
He pointed out that 2011 earthquake and the resultant tsunami in Japan were a surprise to the seismologists around the world as the height of the tsunami far exceeded the estimated heights.
The disaster claimed about 20,000 human lives and also caused nuclear accidents. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,691 |
ACCEPTED
#### According to
Index Fungorum
#### Published in
Mycologia 29: 592 (1937)
#### Original name
Micromyces longispinosus Couch
### Remarks
null | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,496 |
Q: How to add query string to the pager(codeigniter I have a link for example:
http://mysite.com/5 / 5 - is page number
and here is all ok, but what if i have a search filter activated with the pagination:
http://mysite.com/5?filter=something
so the question is: how can i add the query string to the pager links, leaving the pager links itself as uri segments, pager links are uri segments and filter is a query string, is that possible?
A: In the config.php file there is section of code to allow you to enable query strings. It should look like this.
$config['allow_get_array'] = TRUE;
$config['enable_query_strings'] = FALSE;
$config['controller_trigger'] = 'c';
$config['function_trigger'] = 'm';
$config['directory_trigger'] = 'd'; // experimental not currently in use
What are these set too for you?
A: I would advice to add the searched key word to a cookie or a session
since codeigniter pagination uses url segment,
using
http://mysite.com/5?filter=something will destroy the segment because CI will think that the second parameter will be the limit and this will return false since your second uri segment is 5?filter=something
this is how i would do it
public function search()
{
//config pagination
if($this->session->usedata('search_keyword') == false)//no search has been made
{
if($this->input->post('search_keyword'))
{
$data_to_query = $this->input->post('search_keyword');
$this->session->set_userdata('search',$data_to_query);
// query for database when search_keyword is entered
$this->model->get($this->session->userdata('search')); // assuming this is your query model
}else{
// if session search_keyword is false then no
// search has been made do the normal query
}
}else{
// if search_keyword session is true then a search has been made
// use the data gathered on the session as a query parameter
}
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,098 |
We are looking at a new song by Iration called Danger feat. J Boog and Tyrone's Jacket. This song has a great feel and the groove is moving! It is also sporting a few killer features, I love verse 2! The main rhythm of the song is a F#m – E – F#m – C#m four chord progression. This two bar progression plays throughout the song. This rhythm part makes this a great easy Reggae song for a beginners to learn. And for the more advanced guitar players try mixing in the lead guitar while hoping back on the rhythm part without losing the beat.
The feel off this song is straight Irie! A heavy swing and guitar hits on the and of every beat. I have shown the tab with the most simple shapes available for these chords. They are basically the top strings off an E shape barre chord. You could play around with fragmented or inverted A chord hand shapes as well. Another tip would be to fret the whole bar chord even if your striking just the top three strings. This can be useful for A shapes if your just starting to learn your different chord shapes. If you can memorize songs by there progressions remembering them becomes a lot easier. The lead line can be clearly heard on the fifth measure of the intro.
I hear three main guitar parts on the recording. The main rhythm, which may even be played by a keyboard only, the lead guitar line, and a super reverb drenched part similar to the lead line playing some choice 16th note variations. I have tabbed the main lead line, listen for the reverb part I am describing. If you have any question or comments drop them below! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,552 |
Ep55: Cultivating Curling Computations w/ Gerry Geurts
June 29, 2020 Uncategorizedrocksacrossthepond
We welcome CurlingZone's Gerry Geurts to the show to discuss the impact COVID-19 will have on the upcoming season, changes to curling's ratings systems and ways competitive curling tours can help promote the growth of the sport. Jonathan also has an update for us on Scottish Curling's selection system for the World Curling Championships.
Why Curling Needs a World Amateur Championship
The Rift Between Professionals and Amateurs
Long time listeners of Rocks Across the Pond know that one of our concerns is with the grassroots of curling. The abrupt end to the 2019 – 2020 curling season forced many curling associations to change the selection process for their national teams. Russia recently announced it was replacing its national playoff with a coach selected team. Scottish Curling's AGM will focus on a vote to repeal its recent move to a coach selected team. Members of national curling associations are angry that curling's traditional playdown process is under threat. However national high performance programs and national coaches worry about Olympic qualification and national funding for curling.
A rift has opened up in curling between the professionals and the amateurs. The Olympics is causing this rift. For a handful of teams at the top the Olympics brings with it large pots of money. But for competitive club curlers the Olympics runs roughshod over the grassroots and traditions of the game.
Olympic funding is highly competitive. National Olympic organizations allocate funding entirely on results. British Curling, for example, gets £1.7 million ($2.1 USD) a year to run its high performance program. But UK Sport reviews this funding every cycle. Sports that do not win medals get their funding cut. Sports that fail to qualify for the Olympics get no funding at all. For the coaches, athletes, and administrators in high performance curling missing the playoffs at next year's world championships will cost them their jobs.
What the Professionals Want
Given that context, the decision makers in these organizations want to select teams that give them the best chance to qualify for the Olympics. National championships are entertaining to watch precisely because they provide the opportunities for Cinderella teams to pull off upsets. Two years ago the Sophie Jackson rink upset the Muirhead rink to winning the Scottish championship. The British Curling coaching staff were concerned that team Jackson was not ready to perform on the world stage. British Curling tried to to send team Muirhead instead. A big uproar ensued, but in the end team Jackson was sent to the Worlds as Team Scotland.
While for the grassroots of Scottish curling that was the end of the story — the integrity of the national championship and the playdown tradition was protected — for British Curling disaster ensued. The Jackson team finished 10th at the world championships. If that happens in 2021, team GB would have to enter the Olympic Qualification Event. A bad week there, and Britain has no women's team at the Olympics in Beijing.
It is worth noting that the Jackson result at worlds was entirely predictable. Heading into the event they were 65th on the World Curling Tour Order of Merit. The were also the 10th highest ranked team in the 13 team field. Therefore it was not a surprise when they finished the tournament in 10th place. While order of merit standings cannot guarantee success at a single event, they do measure a team's performance over the course of a whole season. Team Muirhead has been a fairly consistent top 10 OOM team for the last 8 years. They are normally good for a top six finish at a world championship. If my funding and job depended on results at a world championship, I know which team I would pick.
What the Amateurs Want
Of course the traditionalists make very good points too. The whole point of sports is that debates over who is the best are settled on the field of play. The whole idea of a team being selected in a boardroom by a shadowy committee smacks of unfairness. The great advantage of a national playdown process is that everyone knows beforehand what a team has to do to get to a world championship — win the national championship. When decisions are handed over to a committee the process becomes more murky and more subjective. Is it simply decided by OOM standings on a certain date? Do national high performance programs try to create a combine (like USA curling did a few years ago) and factor in things like personality and IQ tests, body fat measurements, and one rep maxes in the bench press? This may sound preposterous but many other professional sports do exactly this when making decisions to draft or sign players. So why not curling?
The other major objection to selected teams is that it takes away a clear pathway for aspiring competitive curlers. Under a playdown process it is clear what a team has to do. Go out and compete in the events that qualify you for the national championship and then win the national championship. In this new world of professional curling it isn't always clear how an athlete can become an Olympian. In Great Britain, British Curling puts out an application process every year to select the teams and then invites some athletes to a trial. I've spoken to several athletes who have been through the process — both those who were selected and those who were not — and the decision making process is not clear at all. It is more like applying for a job than competing in a sport.
The Olympics: A Double Edged Sword
So what to do? I'm firmly of the view that the Olympics are a double edged sword for curling's grassroots. They drive tremendous attention to the sport that can help grow curling club membership, but the money and the pressure to win now compels high performance directors to prioritize the interests of a handful of elite curlers without doing or giving anything in return to the grassroots. Furthermore the professionalization of the sport means that events like national championships are less and less necessary (or even desirable) for selecting Olympic and national teams. It is time to face reality: curling is splitting into a professional and an amateur tier.
How to divide the sport
Let the high performance directors select their national teams how they see fit. But grassroots curling organizations should not support the process at all. I have seen the fight play out in USA curling when I was a board member there. And we are watching it unfold in Scottish curling now. The grassroots curlers are upset because the championships they fund with their membership dues are being fundamentally altered by the high performance directors. The high performance directors are upset that grassroots curlers through their clubs and national governing bodies won't let them take the decisions they need to win medals. Let the high performance directors select the national teams how they see fit, but let the grassroots keep ownership of national playdowns.
What does this mean in practice? In the case of Scottish Curling, give over the power to select national teams for events leading to an Olympics entirely to the high performance coaches. The national championships should be run entirely by Scottish Curling. Team GB funded athletes should be prohibited from entering the Scottish championship.
A World Championship for the Rest of Us
Why would somebody want to enter the Scottish Curling championship then? To make the championship more appealing we need it to lead to a new world amateur championship. Canada and the U.S. already have national amateur championships, called national club championships. These have strict entry requirements that ensure grassroots curlers from the same club all enter. The standard of play is fairly high and it respects the traditional curling playdown process. All we — the grassroots curlers — need to do is link these national club championships together into a world club championship. If Canada, the U.S.A., and Scotland all get behind a world club championships, I'm certain club level curlers from around the world would be eager to send teams.
One of the reasons playdown numbers are down across the world is that many competitive curlers know that they have no chance against the new emerging class of professional curlers. To be a top 20 team on the Order of Merit Rankings a team has to play at least 10 tournaments a year. Professional teams spend at least $100,000 a year on entry fees and travel expenses. Professional curlers can practice at least 6 days a week for two plus hours, and commit to five plus days a week of strength and conditioning training. Curlers who have jobs in other fields simply do not have the time or money to do all that.
By creating a specific world championship for amateurs we give back to the grassroots the traditional playdown process. A team can enter at the club level and advance all the way to a world championship to compete against other amateurs. National governing bodies should offer an all expenses paid trip to a world amateur championship to the winners of a club playdowns. Many teams would sign up for such an event if it had a clear path to victory and a meaningful prize. We are losing talented competitive curlers because they don't have the time or money to compete against the pros. But if we created a playdown and a world championship for competitive amateur curlers they would have something to compete in, while the fully funded professionals chase Olympic glory and Grand Slams.
Ep54: Take This Hammer – Making Our Community More Inclusive
We are joined by Adriana Camarena—a writer, rule of law consultant, anti-police brutality activist and member of Mexico's national curling team—for a much-needed discussion on how curling can be more inclusive, the Black Lives Matter movement and the fight against racism in the United States.
Unsettlers.org
Justice for Luis Góngora Pat | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,050 |
Q: User Persistant React I have a login system for an app am creating, everything works fine but the problem is the users are logged out when the page refreshes any help??
PersistLogin component -which is supposed to handle the persistence.
Children here are the components wrapped by persistlogin in app below
import React, { useState } from 'react'
import { useEffect } from 'react'
import useAuth from '../hooks/useAuth'
import UseRefreshToken from '../hooks/UseRefreshToken'
import CircularProgress from '@mui/material/CircularProgress';
import { Box } from '@mui/material';
function Persistlogin({ children }) {
const [isLoading, setIsLoading] = useState(true)
const refresh = UseRefreshToken()
const { auth } = useAuth()
useEffect(() => {
const verifyRefreshToken = async () => {
try {
await refresh()
} catch (error) {
console.log(error)
}
finally {
setIsLoading(false)
}
}
!auth.accessToken ? verifyRefreshToken() : setIsLoading(false)
}, [])
useEffect(() => {
console.log(`isLoading:${isLoading}`)
console.log(`at:${JSON.stringify(auth?.accessToken)}`)
console.log(auth)
}, [isLoading])
return (
<>
{isLoading
?
<Box sx={{
display: 'flex',
flexDirection:'row',
width: '100%',
justifyContent: 'center',
alignItems:'center',
top: '20%',
}}>
<CircularProgress />
</Box>
: children
}
</>
)
}
export default Persistlogin
useRefresh Component (incase the current acccess token expires)
import axiosInstance from "../axios"
import useAuth from "./useAuth"
function UseRefreshToken() {
const {auth,setAuth}=useAuth()
const refresh=async ()=>{
const response=await axiosInstance.post(`bp_refresh_token_information/ ${auth.refreshToken}`,{
withCredentials:true,
})
console.log(response)
setAuth(prev=>{
console.log(JSON.stringify(prev))
console.log(response.data.accessToken)
return {...prev,acessToken:response.data.accessToken}
})
return response.data.accessToken
}
return refresh
}
export default UseRefreshToken
App Component
which will handle the routing and I have wrapped every component with persistLogin
<Routes>
<Route path="/">
<Route path="CompanyDetails" element={<CompanyDetails />} />
<Route path="register" element={<Register />} />
<Route path="passwordreset" element={<PasswordReset />} />
<Route path="Verification" element={<Verification />} />
<Route path="login" element={<Login />} />
<Route path="/" element={<Persistlogin><RequireAuth><Home /></RequireAuth></Persistlogin>} />
<Route path="employees">
<Route index element={<Persistlogin><RequireAuth><List title="Add New Employees" inputs={userInputs} url={url}/></RequireAuth></Persistlogin>} />
<Route path=":employeeId" element={<Persistlogin><RequireAuth><Single chartTitle='Employee Payment' tableTitle='Payment History' inputs={userInputs}/></RequireAuth></Persistlogin>} />
</Route>
<Route path="products">
<Route index element={<Persistlogin><RequireAuth><List title="Add New Products" productheader={productheader} url={url}/></RequireAuth></Persistlogin>} />
<Route path=":productId" element={<Persistlogin><RequireAuth><Single chartTitle='Number of Sales' tableTitle='Payment History' /></RequireAuth></Persistlogin>} />
</Route>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,715 |
ACCESS TO MENTORSHIP
Thursday, 08 November 2018 / Published in Uncategorized
By Lorraine Anyango
Foot football clinic
Ronaldo De Assis Moreira, also known Ronaldinho, made history for his technical skills and creativity due to his agility, pace and dribbling ability as well as his use of tricks, he rose to become FIFA world player of the year.
He demonstrated feints, overhead kicks, no look passes and accuracy from free kicks. He was considered the best player of his generation and one of greatest of all time.
Ronaldinho enjoyed a stellar 17-year career, playing 97 matches for brazil national team, scoring 33 goals and representing his county in two FIFA world cups.
This is the man that young foot ballers in Kisumu and its environs are set to glean from as he graces their matches at the Moi stadium in Kisumu.
He is scheduled to grace a grass root football match between Kisumu All starts and Bondo United during which he is expected to also mentor the young players.
"We shall be learning from the best, we are already building confidence and desire within ourselves that with Ronald wisdom we can scale the heights of world class football." Alfred Adu. Kisumu all Stars team manager said.
"For me it's a dream come true, I have been playing football passionately since child hood and people nicknamed me Gaucho, I lack words, I can't wait" He added.
Apart from gracing the exhibition match on Saturday November 10 th 2018 from 3 pm, Ronaldinho who is also the brand ambassador for Betika will open the firm's office in Kisumu.
Kisumu county Sports directorate is working in conjunction with Extreme Sports, a sports management company based in Nairobi in together with Gina Din communications and sports Betika towards the event.
"We will use this opportunity to strengthen the County's global linkages, and work on uplifting the career horizons of our talented young footballers." The Kisumu county minister for Tourism Sports culture and Information Archie Alai said.
Plans are under way to provide adequate security during the event that is likely to draw masses of football fans, the county commander is expected to deploy 50 uniformed officers to ensure safety at the venue.
This event comes ahead of the planned sports stake holder's forum in Kisumu scheduled for Friday, 16th November 2018 and geared towards discussions on sporting heritage, using it as a foundation and inspiration for establishing initiatives aiming at becoming the country's sports powerhouse.
Kisumu all-stars members comprising Kisumu all Stars were drawn from the sub counties after PALOS FC disbanded from the National super league.
The County government has since last year been sponsoring the team, providing an upkeep stipend to the players as well as equipping the team and organizing for matches under the National Super league.
Further in a move to encourage the sporting fraternity, Kisumu County appointed the legendary football player Peter Dawo as the head of Sports Directorate and held a meeting with Kisumu all stars to forge a way for sustainable development.
"The coming of Ronaldihno will go a long way in motivating the current and upcoming footballers in Kisumu county." Mr Dawo said noting the fact for about eight years, Kisumu county was not participating in any premier leagues leading to low quality football in the region.
Other football stars that have visited Kenya in recent years include ex Liverpool star Bruce Grobbelaar, Nigerian Yakubu Ayegbeni, Frenchmen Thierry Henry, Robert Pires and German World Cup winner Lothar Matthäus.
Radio as A Critical Driver of Gender Equality in Kisumu
COUNTY STAKEHOLDER BRIEFING ON SHISHA BAN STATUS
COUNTY GOVERNMENT OF KISUMU NEWSLETTER | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,503 |
Quentin Anthony Anderson is a Baton Rouge, Louisiana native who leads Groundswell's communications and public affairs efforts as it's Associate Director. He is a dynamic non-profit & communications professional with years of experience working with community-based organizations, and a rich background in social justice, political & community organizing, and policy advocacy. In 2015, Q led a statewide coalition of high school-aged young people in Illinois during their school discipline reform campaign, resulting in one of the nation's most comprehensive school discipline reform bill ever signed into law. Prior to that, he served as Regional Field Director for NextGen Climate, helping lead GOTV efforts in southwest Colorado during the 2014 congressional midterm elections. Quentin earned his JD and graduate degree from Louisiana State University, his BA in Political Science from Louisiana Tech University, and has previously worked for United Way, The Justice Alliance, and Obama for America.
Groundswell's Quentin Anthony Anderson reflects on his recent attendance and participation at the first annual Coalition for Community Solar Association Summit in Denver, CO this past July. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,813 |
PSMJ's A/E Capacity Budget Template - PSMJ Resources, Inc.
This one-of-a-kind, interactive budget tool allows you to use a simple spreadsheet to input data on the employees who work at your firm. From your unique information, PSMJ's Capacity Budget Template generates critical information you can use to plan better and budget quicker.
PSMJ has more than 40 years of experience helping firms plan for the year ahead. We know what factors affect net revenue, chargeability, and other critical factors that impact financial decision-making. All you need to do is fill out a simple spreadsheet.
You can do all this work manually—but it would take hours to calculate what this tool does instantaneously, reliably, and consistently. | {
"redpajama_set_name": "RedPajamaC4"
} | 647 |
This is from Facebook--A New Map of the Southeastern US.
Wow, what a weekend! (Scroll down to read what it's like to wonder if your house will be flooded).
First off, I'm fine. The house didn't flood.
Pray for safety: Rivers are all at flood stage or above, and as these waters travel they will continue to flood places and pose hazards. The water can contain three yucky "S"s-- 1. Sharp and swiftly moving things that are very dangerous like broken glass and alligators. 2. Snakes, many are poisonous and 3. Sewage and other bacterial nasties like drowned rats.
Please pray for families who are grieving over lost loved ones, including those of the men lost on the cargo ship. Here in SC the last death toll I saw was at 7. Just 1 is too many.
Pray for those who had to evacuate. It's very traumatic to escape with only the clothes on your back and what you can carry, to sleep with strangers in a shelter, to leave your pets and belongings behind not knowing what will still be there when you return. How will you rebuild?
Pray for our governor-- Nikki Haley, for relief workers, the Coast Guard, and others who are coordinating recovery. Pray for their safety, strength and clear thinking. Pray that they'll receive the cooperation needed.
Pray for the weather and that it won't be too windy. Our trees here grow tall and fast and without a deep root system. With the ground so soggy they might topple more easily and take out power lines. We really need our electricity to help dry things out quickly before mold starts growing. Pray for some sunny days and low humidity ahead!
Thanks so much, God bless you! Please add to this list as needed in the comments.
Continuing to pray for you, Ferree, and for those in your area! Sandy E. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,202 |
Copyright © 2017 by Liz Moody
Photography copyright © 2017 by Lauren Volo
All rights reserved.
Published in the United States by Clarkson Potter/Publishers, an imprint of the Crown Publishing Group, a division of Penguin Random House LLC, New York.
crownpublishing.com
clarksonpotter.com
CLARKSON POTTER is a trademark and POTTER with colophon is a registered trademark of Penguin Random House LLC.
Library of Congress Cataloging-in-Publication Data
Names: Moody, Liz, author.
Title: Glow pops : super-easy superfood recipes to help you look and feel your best / Liz Moody.
Description: First edition. | New York : Clarkson Potter, [2017]
Identifiers: LCCN 2016031171| ISBN 9780451496447 (hardcover : alk. paper) |
ISBN 9780451496454 (ebook)
Subjects: LCSH: Ice pops. | LCGFT: Cookbooks.
Classification: LCC TX796.I46 M66 2017 | DDC 641.86/2--dc23 LC record available at https://lccn.loc.gov/2016031171
ISBN 9780451496447
Ebook ISBN 9780451496454
Photography by Lauren Volo
Author Photo by Douglas Gorenstein
v4.1
prh
CONTENTS
INTRODUCTION
GETTING STARTED
Equipment
Ingredients
FRUITY
CREAMY
CHOCOLATEY
SAVORY
GREEN
APPENDIX
GLOW GLOSSARY
ACKNOWLEDGMENTS
INDEX
INTRODUCTION
I've always been a firm believer that having a healthy life doesn't have to be limiting or difficult. Every time a new health trend sweeps the country—whether it's expensive pressed juices or imported Asian jungle fruits or powdered superfoods with exotic names—I find myself equally excited and frustrated. Excited because I _love_ that the health conversation has become mainstream! No longer relegated to the subcultures, the desire to eat well and nourish our bodies has become common. Natural supermarkets are popping up all over the country, sales at McDonald's are at an all-time low, and people are finally asking, "What are all these man-made ingredients I can't pronounce doing in my food? Is this something I should be putting in my body?"
I'm frustrated, though, because the health world can feel exclusive and intimidating. My health philosophy relies on a simple rule: eat whole, real foods. That's it! You don't need a dehydrator or hours and hours of free time or a local farmers' market that's filled with heirloom apples and microgreens (although if you have any of those things, that's wonderful too!). All you need is a desire to nourish your body and an appetite for the delicious bounty that comes from our earth.
This is one of the many reasons I'm such a fan of Glow Pops, healthy, nourishing pops that fill you with energy and nutrients and make you absolutely glow. Once you nail a few simple tricks to ensure you get the right texture every time, you can have a ton of fun playing around with different flavor profiles. A Glow Pop is the opposite of a boring, flavorless salad—and it often has far more in the way of nutrients. Anything that gets people not only eating nutritious food but truly enjoying it—well, that's a real win.
I didn't always view food like this. When I was in college, I was obsessed with calorie counting, giving far more credence to the numbers on the back of my packaged foods than what was actually _in_ them. I was perpetually hungry, plagued with skin problems, and couldn't get rid of the excess fat I had around my belly, no matter how few calories I ate.
At the time, I was working as a newspaper columnist. For my column, I'd often find myself traveling around the world, going on adventures that I could share with my audience. Along the way, I began to notice how local cultures viewed food. Most approached both making and eating it with much more joy than I did. Instead of nuking a two-minute meal, people were soaking grains overnight, letting a tagine simmer for hours, and spending entire evenings at the table, laughing and chatting and sipping wine well after the food was finished. In many of these cultures, the people were also healthier than most of the population back home: strong and slender, with clear skin and long, healthy hair.
Intrigued, I dove deeper into the gastronomical world. I took cooking classes in Italy and Syria, and a short sommelier course in Paris. I even worked on a farm in the South of France, plowing fields and turning goat's milk into creamy, tangy cheese. At the same time, I began to note all the ways food was used as medicine.
When I returned to the United States, my approach to food had completely changed—and so had my body. My skin had cleared up and my body had effortlessly found its happy weight (absolutely no calorie counting required!). I felt vibrant and full of energy, excited about the possibilities food held, for both my taste buds and my health, like never before.
I started my blog, _Sprouted Routes,_ as a way to share this knowledge and all the easy, healthy, delectable recipes I've developed in the years since. _Sprouted Routes_ is all about easy paths to a healthy, beautiful life. Health doesn't have to be difficult. It doesn't have to be bland or ugly or more "crunchy" or less delicious. It can be glamorous, gorgeous, and bursting with flavor and life. And it can be _fun,_ which is where Glow Pops come in!
I've infused many of these recipes with the flavor profiles and ancient, natural remedies I picked up on my international adventures. Every pop in this book is made with whole, nutritious foods and is gluten-free, dairy-free, refined-sugar-free, and optionally vegan. There are pops for weight loss, pops to decrease your bloat, and pops to make your skin glow. There are antiviral, antifungal, and antibacterial pops as well as antioxidant-packed pops. There are pops designed to soothe specific conditions—when my summer allergies kick in, I make a batch of Turmeric Golden Milk pops weekly to keep the itchy eyes and sniffles at bay. If I know I'm going to be spending a lot of time outside, I rely on Cucumber Mint Mojito pops, which contain ingredients that help protect against skin cancer. And, of course, I keep a stash of Tomato Beet Bloody Mary pops in the freezer at all times—there's no better (or better tasting!) way to quickly get rid of a hangover.
While I love to eat pops as a healthy way to satisfy my sweet tooth in lieu of traditional desserts, many of the pops contain enough fiber, good fat, and protein to make incredibly healthy meals. Because you make pops in batches and then simply grab them, fully finished, from the freezer, they're ready in less time than it takes to whip up a smoothie or toast an English muffin. During the summer, my husband grabs a Peanut Butter & Jelly pop on his way to work most mornings.
And by the way—pops are simply fun! While adults love Glow Pops, often kids love them even more, and they're a great way to sneak more healthy foods into even the pickiest eater's diet. Because most of the recipes are quite easy to prepare, it's fun to bring kids into the kitchen from the beginning, letting them be involved in the actual preparation of the pops. My four-year-old niece is obsessed with the Cardamom Cinnamon Sweet Potato pop, while my girlfriend's seven-year-old son, who turns his nose up at all vegetables, is obsessed with the spinach-packed Mint Chocolate Chip green smoothie pop.
If you want to up the ante on the health benefits of your pops even more, be sure to check out the superfoods section (this page), where I tell you how to sneak in ingredients like vitamin C–rich camu camu, protein-packed hemp seeds, hormone-balancing maca, and many more. These pops are a great option if you have specific needs you're trying to address or simply want to take it to the next level!
So if you're already health conscious, I applaud you, and invite you to treat yourself to a Glow Pop. And if you're not—well, you've picked a great way to start your journey, and I'm so proud to be part of its beginning!
GETTING STARTED
Glow Pops are super simple to make, but there are a few things that will ensure you have perfect pops every single time. In this section, we'll discuss the few pieces of equipment you'll need, some notes on ingredients to make sure you're getting all the glow-inducing health benefits, and information on how to unmold and store your pops. Are you ready? Let's get glowing!
EQUIPMENT
Pops are one of the easiest things you can make in your kitchen, but there are still a few key pieces of equipment you need, from the molds you'll freeze your pops in to the blender that turns fresh, beautiful ingredients into a sweet, smooth pop mixture! Luckily, there's quite a bit of flexibility in both, and you'll likely find you have much of what you need in your kitchen already.
MOLDS
All pops begin with molds, also known as the things that give them their shape and make them, well, pops! There are plenty of different brands available on the market, and they come in shapes from swirls and stars to fish and rocket ships. You don't need to spend much to get a great one.
Two of my favorites, from Zoku and Norpro, are available for under $20 in home goods stores, big box retailers, specialty kitchen goods shops, and online. The Zoku Classic Pop Molds come with built-in, reusable plastic sticks that have drip guards, while the Norpro Ice Pop Maker utilizes wooden sticks and has the classic shape you might remember from childhood, with lines down the side. There are also silicone molds that make push-up pops, and some newer, quick-freezing models out there, for those of you who are impatient (I'm raising my hand!), although they cost a bit more. (Because of the fast freezing time, they'll also result in a slightly creamier pop texture.) You can get great results with almost any mold. I've made pops in over twenty different types of molds, and the results were almost identical across the board, so it really comes down to your preference. My one caveat: Always look for a mold that is BPA-free, which will almost always be touted on the box or in the description.
Because they're relatively inexpensive, I do recommend buying a mold if you're planning on making pops regularly. There are, however, a few good alternatives. Small paper or plastic cups (like the kind typically found in the bathroom or the dentist's office) make cute, fun-size pops, and plastic Champagne flutes make a surprisingly classic shape (just leave room at the top for the liquid to expand, and be careful when placing them in the freezer). Ice cube trays will make bite-size pops, which are fun for parties or kids. While most molds do come with sticks, if you're DIY-ing it, you can buy wooden sticks at most craft stores or online.
Note that the recipes in this book are all designed to make five or six 3-ounce pops, which is the standard size of most molds. If you're using a DIY mold, the recipes will likely yield a slightly different amount. If you have any extra unfrozen mixture left over, you can either drink it (sometimes I can't help but sneak sips anyway!) or store it in an airtight container in the fridge until the first batch of pops is frozen, and then use it to make a second batch.
BLENDER
My blender is one of my kitchen workhorses. If you can, I recommend investing in a high-power blender: it will give you creamier, smoother results, especially with things that can be a bit harder to blend, like dates or greens. I splurged on a Vitamix a few years ago, and my only regret is not doing it sooner. I use it every single day and love using it to make Glow Pops. If you're not quite there yet, no worries! There are wonderful blenders at every price point—both the NutriBullet Pro 900 and the Waring Pro PBB225 make wonderful pops and clock in at under $100. If you're using a less powerful blender, just let it run a little longer than recommended in the recipe, then turn off the blender and sneak in a spoon for a taste. If it's not quite smooth enough for you, give it an extra 30 seconds or a minute!
FOOD PROCESSOR
For the Fruity pops (this page), it also helps to have a food processor, although it's not completely necessary. Because fruits are so juicy, we're adding little to no additional liquid, and food processors are more adept at making all-fruit purees. You can find great food processors between $100 and $200 at kitchen stores and online. I personally love the 6-, 8-, and 11-cup KitchenAid and Cuisinart models, but really any type will work fine. You don't need anything fancy in a food processor—I strongly recommend ignoring all the crazy new features and getting a simple machine that's built to last. If you don't have a food processor, don't worry—where I suggest using one, just add water 1 tablespoon at a time until your blender whizzes the fruit right up.
UNMOLDING AND STORING YOUR POPS
To unmold your pops, simply run the molds under warm water for about 10 seconds, then pull the sticks to remove the pops. You can eat them immediately or, if you want to save them for later, lay them flat on a piece of parchment paper on a baking sheet or plate. Place in the freezer for 1 hour after unmolding, or until completely frozen again; after that, you can transfer to a plastic zip-top bag and store for up to 3 months. If you're using small paper cups to make your pops, simply store in the freezer until you're ready to eat, then peel off the paper and devour!
INGREDIENTS
While the basic form of pops is always the same—you mix or blend ingredients, then freeze them—one of my favorite things about them is their flexibility. They're nowhere near as finicky as traditional baked desserts, which require exact chemical reactions and a precise order to their process. Pops are almost impossible to truly mess up, and you can taste them as you go along and adjust accordingly. Feel free to use this book as a rough guide, and play around with the recipes to make them suit you! Let's go over some basic ingredients you'll see again and again.
SWEETENERS
The recipes here use five different sweeteners: maple syrup, coconut sugar, coconut syrup, honey, and dates. Unlike white or cane sugar, these are all still in their whole food form, and therefore haven't been stripped of their nutrients. They all have different advantages and nutritional properties. Maple syrup, coconut syrup, and honey are liquid and thus easy to dissolve at room temperature, for instance, while coconut syrup and coconut sugar have some of the lowest glycemic index numbers of any sweetener, making them a great choice for diabetics or others with blood sugar issues. Dates are a whole fruit and still contain all of their fiber, but may not blend as well in the smoother pops, so I prefer to use them only in green smoothie pops.
For each recipe, I've selected the sweetener I think works best for the pop, in terms of both texture and flavor. If you prefer something different, go ahead and substitute it! Maple syrup, coconut sugar, coconut syrup, and honey can all be substituted for each other in a 1:1 ratio, and each tablespoon of maple syrup, coconut sugar, coconut syrup, and honey is equal to 1 date. If you choose to use dates, you'll typically have the best results with the Medjool variety, which tend to be softer and more caramel-like in texture, and are, as a result, easier to blend with the rest of your ingredients (see this page). If you're using another type of date, be sure to soak them in boiling water for 10 to 20 minutes to soften them quite a bit and make them far more blendable; drain them well before using.
All these can be purchased online and at most grocery stores and health food stores. Coconut syrup is a bit rarer than coconut sugar, but it's super easy to make at home. Here's how:
COCONUT SYRUP
MAKES ½ CUP
•1 cup coconut sugar
•½ cup water
In a small saucepan, whisk together the coconut sugar and water, then bring to a boil over medium-high heat. Reduce the heat to medium-low and simmer, uncovered, for 20 minutes, or until thick and syrupy—no need to stir. Remove the pan from the heat and let the syrup cool completely before using. Store in an airtight container in the fridge for up to 2 weeks.
For optimal nutrition, I recommend buying raw honey, as it still contains all its enzymes and nutrients. As an added bonus, if you buy local raw honey, it'll act as a mild inoculant against allergies prevalent in your area!
If you're having a hard time finding reasonably priced Medjool dates, I recommend checking out Middle Eastern markets (a great place to buy affordable spices as well!).
Flavor preferences are individual, and what I consider perfectly sweet might be too sweet or not sweet enough for you, so be sure to taste your pop mix before pouring it into the molds, and adjust the sweetener as needed. Remember that the cold dulls our perception of sweetness, so you want your unfrozen pop mixture to taste just a little bit sweeter than how you want the final pops to taste.
MILKS
I prefer to use nondairy milks to make pops, for reasons of both health and texture. The trick to a good creamy pop is having milk that's less watery than typical dairy milk. Canned coconut milk, available at almost any grocery store, works beautifully in pops—it's thick enough to give a super-creamy, hearty texture. Coconut milk is definitely my favorite kind to use in any of these recipes. I've had good results with both light and full-fat. The latter is slightly creamier, but both will give great results. If you're wondering if all your pops will taste like coconut, the answer is no! The flavor of coconut milk actually is interpreted by our tongues as sweetness, so if anything, that's all it will add in flavor.
When cold, canned coconut milk will separate into a thick white top layer (sometimes called coconut cream) and a watery bottom layer. This is completely normal, so don't panic! To make the texture homogenous again, simply toss both layers into a blender and whiz for a few seconds, or heat it gently over low heat on the stove, stirring occasionally, then use as the recipe directs. It's always a good idea to shake the can before opening it, too.
As for almond, rice, hemp, and soy milks, the types that now fill supermarket aisles have too thin a consistency to make a great pop. You can use them if you like, but you might end up with an icier pop. There are a few ways to combat this, so be sure to check out A Note About Texture.
Homemade milks are incredibly easy to make—the process basically involves soaking, blending, and straining. I often will make a big batch on Sunday or Monday and leave it my fridge for smoothies, pops, and other recipes (curry, pancakes, cereal) throughout the week. See this page for the recipes. But if a recipe calls for another type of milk—let's say, almond—and you don't feel like making your own, you can _always_ substitute canned coconut milk in a 1:1 ratio.
COCONUT MILK
MAKES 2 CUPS
•1¼ cups water
•1 cup dried unsweetened flaked coconut
In a small pot, bring the water to a boil. Put the coconut in a bowl and pour the boiling water over it. Set aside to soak for 30 minutes. Transfer the mixture to a blender and blend on high for 2 to 3 minutes, or until very smooth. Strain through a nut milk bag, or pour into a clean dishtowel set over a clean container. "Milk" the bag or towel until all the liquid has passed through; discard the remaining pulp or reserve for another use. Store in an airtight container in the fridge for up to 5 days.
ALMOND MILK
MAKES 2 CUPS
•1 cup almonds
•1¼ cups water
Soak the almonds in filtered water to cover for 12 to 24 hours, then drain. In a blender, blend the soaked almonds and water on high for 2 to 3 minutes or until very smooth. Strain through a nut milk bag, or pour into a clean dishtowel set over a clean container. "Milk" the bag or towel until all the liquid has passed through; discard the remaining pulp or reserve for another use. Store in an airtight container in the fridge for up to 5 days.
CASHEW MILK
MAKES 2 CUPS
•1 cup cashews
•1¼ cups water
Soak the cashews in filtered water for 4 to 6 hours, then drain. In a blender, blend the soaked cashews and water on high for 2 to 3 minutes, or until very smooth. No need to strain. Store in an airtight container in the fridge for up to 5 days.
FRUIT
I specify in each recipe whether to use fresh or frozen fruit. In general, for the "fruity" pops, which rely on fruit as the superstar for texture and flavor, you'll want to buy fresh, while the rest of the pops will do fine with frozen. For the green smoothie pops in particular, I tend to use frozen berries, as they're cheaper and easier to find than fresh and you don't have to worry about their spoiling. Frozen berries are often healthier than fresh as well, as nutrients like vitamin C decline rapidly from the moment a berry is picked. Because frozen berries are flash-frozen just after they've been picked at peak ripeness, all that goodness is locked in, ready and waiting to be delivered to your body.
In general, for any fruit where you're using the skin, it's best to buy organic. For fruits like avocado and banana, conventional is fine; when in doubt, consult the Environmental Working Group's Dirty Dozen list for the most heavily sprayed produce. While conventional citrus is typically fine, many Glow Pop recipes call for the fruit to be zested, as citrus zest is packed with highly nutritious oils that are amazing for your skin. When using citrus skin, I strongly recommend buying organic. For any produce where you'll be eating the skin but can't find organic, soak the fruit in a 1:1 mix of apple cider vinegar and water for 10 minutes, then wash and dry well to remove any (not glow-worthy!) pesticide residue.
SUPERFOOD ADD-INS
The following are my go-to superfoods to ramp up the health factor of the pops even more. I want to stress that you don't _need_ these in any of the pops—even without them, all the Glow Pops are bursting with health benefits. That said, if you have specific health issues you're trying to address or simply want to take your pop to the next level, these add-ins are great. To use, simply add the superfood when you add whatever liquid you're using in the pop and blend until smooth. The quantities listed are appropriate for five or six 3-ounce pops.
2 tablespoons hulled hemp seeds—Hemp seeds offer one of the only forms of complete protein in the natural world. One tablespoon contains 5.3 grams protein, plus 13 percent of your daily iron and 8 percent of your daily vitamin A. Hemp adds a subtle nutty flavor and wonderful creaminess. Blend the hemp seeds with whatever liquid you're using, then proceed with the rest of the recipe as instructed. This would work well in any pops.
1 tablespoon lucuma powder—Lucuma powder is made from the South American lucuma fruit, which is dried at a low temperature and then ground into a powder. It's rich in vitamin A, which is fundamental for glowing skin, and vitamin B3, or niacin, which is mostly found in meat sources and in which many vegetarians are deficient. It has a lightly sweet, caramel flavor that's delicious in any pops that use maple syrup, coconut sugar, or dates as a sweetener.
1 tablespoon maca powder—This superfood is made from a South American root plant that is dried at low temperatures and then milled into a fine powder. It is noted for its hormone-balancing capabilities, and can be a great supplement for people dealing with low sex drive, menstrual cycle irregularities, hormonal skin conditions, and similar problems. It's also adaptogenic, which means that it adapts to the individualized needs of your body—you will respond differently to it than your friends and family might. It tends to make people feel more energized, but not in the buzzy way that's associated with caffeine. Outside of its adaptogenic properties, maca is rich in protein, vitamins, and minerals. It adds a malty quality to whatever you blend it into. I like it best in the Cookie Dough, Chocolate Fudge, and Chocolate Caramel Swirl pops and the Double Chocolate Brownie green smoothie pop. Because maca is a powerful herb, it can be best to start with a smaller amount (1 teaspoon per recipe) before working your way up to the full tablespoon.
1 tablespoon açai powder—Açai (pronounced _ah-SIGH-ee)_ is an Amazonian fruit that is freeze-dried and ground into powder. It has risen in popularity in recent years due to its incredibly high antioxidant content and its uniquely creamy texture from the high level of essential fatty acids. It has a sweet flavor not unlike that of blackberries and blueberries, and works well in any of the berry-based pops.
1 tablespoon camu camu powder—Camu camu is a fruit grown throughout the Amazon rain forest. Its claim to fame is its vitamin C content: it has more than any other food source on the planet! This is a great addition if you want to naturally boost your immune system. The powder has a tart berry flavor and works well in any of the berry-based pops or in pops with a slightly tart flavor, like Pink Lemonade or Lemon Ginger.
1 tablespoon whole flaxseeds—Fiber-rich flaxseeds are the richest source of plant-based omega-3s in the world, making them anti-inflammatory superstars. They're also rich in lignans _—_ phytonutrients that help regulate hormone levels, which positively affects stress levels and hormonal conditions like PMS, and may even help prevent certain types of cancer. The problem with flaxseeds is that they easily go rancid, losing their health benefits and developing an unappetizing flavor when exposed to light and air. This is even more true after they're ground—but they need to be ground well in order to be absorbed by our bodies, rather than simply passing through. This is why I love incorporating them in green smoothie pops—they're ground by the blender and then immediately frozen, which prevents oxidation and allows their awesome nutritional properties to stay intact. Flaxseeds have a very subtle nutty flavor, and work well in all the green smoothie pops. Store your flaxseeds in an opaque container in the freezer to preserve their freshness.
SPECIAL INGREDIENTS
There are a handful of ingredients included in these recipes that you may not have in your pantry already. They're not super hard to find, but they come up a bunch, so let's take a look before you dive in and get your glow on.
Medjool dates—Creamier and more caramel-y than Deglet Noors, Medjool dates are worth hunting down. I've found them at many natural food stores, at Whole Foods, and at Middle Eastern markets, where they are often sold in bulk at amazing prices. You can also find them online, usually for about $10 a pound. Just remember to remove the pit when using Medjool dates—they always come fully intact!
Culinary lavender—When a recipe calls for lavender, you want to be sure to buy the culinary variety, as the type you find in sachets or some gardens is sprayed with chemicals that aren't meant to be ingested. You can find culinary lavender at higher-end food and baking supply shops, like Williams-Sonoma and Dean & DeLuca; in the spice section at some higher-end supermarkets (Whole Foods usually has it); and, my personal favorite, online (just be sure to search for "food-grade" or "culinary" lavender).
Rose water—Rose water can be found at many Middle Eastern or Indian markets, usually for less than $5 a bottle. Because it's a common ingredient in many cocktail recipes, you'll also have good luck at higher-end liquor stores. It's also widely available online.
Matcha—A powdered green tea, matcha has become hugely popular in recent years and is now available in most grocery stores and online. Matcha comes in two types: culinary grade and the more expensive ceremonial grade. For pops, culinary grade will work just fine, although if you already have the more finely milled ceremonial grade on hand, that will work beautifully as well!
Salt—Salt isn't rare, but it's worth mentioning here. While any type of salt will work fine in these pops, I highly recommend using sea salt or pink Himalayan salt, which hasn't undergone the same bleaching and processing as typical white table salt, and has a much lower sodium and higher mineral content as a result. I also find unprocessed salt to be faintly sweet in a way that truly elevates a pop's flavor. You can find mineral-rich salt online or at most grocery stores for a very reasonable price.
Cacao—All the chocolate pops derive their flavor from some type of raw cacao, whether it be in the form of powder, nibs, or butter. Cacao nibs are made from breaking the cacao bean into small pieces and are the least processed form of cacao. Cacao powder is made from grinding nibs into a fine powder, usually using a low-heat or heatless process. Both cacao nibs and cacao powder retain the maximum health benefits of the cacao bean, which include minerals like potassium, magnesium, and chromium, and an incredibly high level of antioxidants. Raw cacao also contains a unique compound called theobromine, which raises energy and alertness levels without the buzzy effect of caffeine.
Cacao butter is the pressed oil of the cacao bean. To extract the cacao butter, the nibs are pressed to separate the oil from the fiber. Like cacao, cacao butter is rich in antioxidants and theobromine, as well as oleic acid, which has been shown to reduce the risk of heart disease.
Cocoa powder and cocoa butter are essentially the same as raw cacao and raw cacao butter, except that they've been processed at high temperatures that eliminate many of cacao's health benefits. In all recipes that call for cacao powder and cacao butter, you can sub in cocoa powder and cocoa butter in a 1:1 ratio; just try to get the highest-quality you can find, to retain the most nutrients.
Cacao nibs are used for texture in addition to their flavor, and can be replaced at a 1:1 ratio with mini dark chocolate chips or coarsely chopped dark chocolate.
A NOTE ABOUT TEXTURE
There are three things that account for a pop avoiding a rock-hard texture: fat, fiber, and sugar. Traditional pops typically rely on sugar—especially simple syrup. Without getting too scientific, simple syrup wedges itself between the water molecules of the pop, so they don't bond to one another hard and tight, whereas water molecules without sugar in between them are simply ice cubes—which you definitely wouldn't want to bite into! Glow Pops never use simple sugars, so we have to get creative to achieve that perfect pop texture, relying instead on fiber and fat.
FIBER
You'll find that many other pop recipes (typically the fruity ones) call for you to strain your mixture through a fine-mesh strainer to avoid any errant texture in the pops: persistent seeds, or bits of pulp that don't quite blend up. Generally speaking, though, for both health and texture reasons I prefer not to strain Glow Pops. Straining removes quite a bit of the fruit or vegetable's fiber, which is one of the things that will prevent your pops from having an icy consistency. Leaving in the bulk from the fruit or vegetable membrane gives the batter a weight that prevents iciness. Fiber is also a wonderfully healthy ingredient, glow-worthy all on its own. It helps slow down the rate at which you digest your food, which ensures you'll stay fuller longer and avoid the nasty blood-sugar roller coaster that comes from eating sugary, fiberless foods. It also acts like a gentle scrub brush on your intestinal tract, helping move along waste and ensuring you have productive bathroom visits (a vital part of getting that glow, as accumulated waste is bloating and inflammatory in the body).
FAT
Fat is a great way to get that satisfying, creamy texture. While you might shy away from fat for fear of your health or an expanding waistline, I implore you to reconsider. Researchers are finding more and more that a number of fats are not only not bad for you, but actually have amazing health benefits.
Glow Pops rely only on fats that are 100 percent nourishing for your body. Full-fat coconut milk, as discussed, is my go-to base. While many people avoid full-fat products, the type of fat found in coconut is mostly in the form of medium-chain fatty acids (MCFAs) and, in particular, one called lauric acid _._ Unlike saturated fats found in animal products, MCFAs are almost immediately converted into energy by the body, and are unlikely to be stored as fat. Lauric acid is also considered one of nature's most potent superfoods (outside of coconut, breast milk is one of the few places it's found in nature, which proves its importance in human health and development), and is renowned for its antiviral, antibacterial, and antifungal properties.
The majority of fat in both almonds and cashews is monounsaturated fat, similar to the type found in olive oil. Monounsaturated fat is known for its heart-healthy properties, and is often recommended as a dietary inclusion for people with heart disease and diabetes. A recent study found that people who eat nuts at least twice a week are much _less_ likely to gain weight than those who almost never eat nuts, so bring on the nut milks and nut butters!
The final fat I rely on is avocado, one of my favorite fruits (that's right—it's a fruit!). The majority of fat in avocados comes from oleic acid, a monounsaturated fat that reduces inflammation (the root of many chronic diseases) in the body, and has been linked with activating genes thought to be responsible for fighting many cancers, making them essentially anticarcinogenic. Avocados have a creamy, decadent texture and a mild flavor that works well as a subtle background note in many pops, although they do get their 15 minutes of fame in the Avocado Chile Lime pops.
The bottom line: Don't be afraid of fat. When it comes to both optimal health and optimal texture for your Glow Pops, fat is not only ideal, but necessary. Just pay attention to the _type_ of fat you select, making sure the ones you choose are nourishing your body.
THE FIBER-FAT DOUBLE WHAMMY: CHIA SEEDS
I use chia seeds in a number of pop recipes for their consistency: once blended, they can make a pop super creamy and wonderful to bite into. They're rich in omega-3s, and one serving contains more than 30 percent of your daily requirement for manganese, magnesium, and phosphorous. They're packed with protein and are made of almost 40 percent fiber, which makes them one of the most fiber-rich foods in the world. This fact, combined with the gelatinous coating that forms on each seed when submerged in liquid, makes them incredibly soothing to the digestive system. I love to use these if I am recovering from any sort of stomach bug or have been traveling, when my digestion is often quite backed up. Chia seeds are one of my favorite ways to achieve a creamy texture in my pops without relying on higher-fat milks. Because they're so satiating, I'll also add them to any pop I'm looking to turn into a meal replacement. They have almost no flavor and work well in any of the pops in this book.
To use, stir in 3 tablespoons chia seeds for every 1 cup of the pop mixture. You can then blend again or process in a food processor (for thicker mixtures) for optimal smoothness, or simply freeze with the chia seeds intact for a pop with bits of crunch in each bite. You may need to add a bit more liquid to get the pop to blend—do so 1 tablespoon at a time, if necessary.
APPLE PIE
CUCUMBER MINT MOJITO
WATERMELON LIME
CHAMOMILE CANTALOUPE MINT
MANGO CHILE
ROSEMARY STRAWBERRY
HONEYED PEACH THYME
PINK LEMONADE
LAVENDER BLUEBERRY
BLACKBERRY ROSE
CARAMELIZED PINEAPPLE
These pops are all about taking in the bounty of beautiful fruit in season during the spring, summer, and fall and letting it shine. Refreshing is the primary word that comes to mind, from thirst-quenching Watermelon Lime to spa-ready Cucumber Mint Mojito pops. Because fruit is the star flavor here, it pays to find the best, freshest fruit you can. In the summer, I love to go to farms to pick my own, or hit up my local farmers' market, where the vendors can help me select the highest-quality produce. Ripe fruit will have a powerful, faintly sweet smell that's noticeable even through a rind or peel. To keep more delicate fruit like berries fresh, first pick through to remove any moldy or rotting ones, then soak your remaining berries in a 1:4 mixture of distilled white or apple cider vinegar and water for 10 minutes before rinsing them and laying them out on a clean dishtowel to dry. Once dry, you can store them in the fridge for up to four days—the vinegar bath will have killed any bacteria on their flesh, staving off mold growth.
APPLE PIE
MAKES 5 OR 6 (3-OUNCE) POPS
Recent studies have found that people whose diets regularly include apples actually _do_ visit the doctor less. It's likely due to a flavonoid called quercetin, a potent antioxidant that may help protect against heart disease and cancer, in addition to having antihistamine and anti-inflammatory effects. Here they come together with cinnamon and just enough maple or coconut syrup to bring out their natural sweetness. These are a favorite among kids and any other apple pie lovers out there, as they taste just like a big bite of pie filling. You can use any common type of apple here, but I find the sweetness of Fujis and the tartness of Granny Smiths especially pleasing.
•2 tablespoons coconut oil
•5 or 6 apples, unpeeled, cut into 1-inch cubes (about 6 cups)
•½ cup water
•½ cup maple syrup or coconut syrup
•2 teaspoons ground cinnamon
•½ teaspoon ground ginger
•Generous pinch of salt
1.In a medium pot, melt the coconut oil over medium-high heat. Add the apples and cook, stirring occasionally, for 5 minutes. Reduce the heat to low and add the water, maple syrup, cinnamon, ginger, and salt. Cover and cook for 10 minutes, until the apples are very soft, almost falling apart.
2.Remove from the heat; let the mixture cool until warm to the touch before blending until smooth.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
Almost all the quercetin (and vitamins) in apples is found in the skin, so if you're consuming apples for their health benefits, be sure not to peel them.
CUCUMBER MINT MOJITO
MAKES 5 OR 6 (3-OUNCE) POPS
The perfect way to cool down and quench your thirst in the midst of summer heat, this pop is all about hydration. Cucumber has the highest water content of any food, and contains an antioxidant called quercetin, which has been found to be anti-inflammatory and bloat-reducing. The lime adds sunscreen-like qualities (citrus zest has been shown to be a skin protectant!) as well as an awesome zip in the flavor, while the mint complements it with an icy burst of refreshment.
•1 medium cucumber, peeled and coarsely chopped
•Zest and juice of 1 lime
•¼ cup lightly packed fresh mint leaves
•1 cup water
•2 tablespoons honey
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
WATERMELON LIME
MAKES 5 OR 6 (3-OUNCE) POPS
Is there anything more refreshing than icy-cold watermelon on a hot summer day? There isn't much to this pop, but there doesn't need to be: the zesty acidity of the lime is just enough to brighten the sweet and juicy watermelon. There's a reason we crave watermelons when we're sweating: they're 91 percent water and, therefore, super hydrating. That other 9 percent packs a heavy nutritional punch as well—watermelons contain more lycopene than tomatoes, and an amino acid called L-citrulline that's been found to relieve muscle pain. Fun fact to wow your friends: Did you know watermelon is both a fruit _and_ a vegetable, and is a cousin to cucumber, pumpkin, and squash?
•3 cups watermelon, in 1-inch cubes (about 20 ounces)
•Zest and juice of 1 lime
1.Blend together all the ingredients until smooth (you can leave some chunks of watermelon if you'd like).
2.Pour the mixture into pop molds and freeze for 1 hour. Insert sticks and freeze for at least 4 hours more, or until solid.
CHAMOMILE CANTALOUPE MINT
MAKES 5 OR 6 (3-OUNCE) POPS
This pop is relaxation on a stick, soothing and nourishing at once. I love to use teas in pop recipes, both for their ability to imbue flavor and for their medicinal qualities—and chamomile is no exception. This caffeine-free remedy calms each and every part of your body, from muscle spasms and menstrual cramps to insomnia and anxiety. The mint contributes further tummy-taming properties, while cantaloupe offers a hefty dose of vitamins A and C.
•1 cup water
•¼ cup coconut sugar
•½ cup lightly packed fresh mint leaves
•2 chamomile tea bags, or 2 heaping teaspoons loose-leaf chamomile tea
•½ large cantaloupe, peeled, seeded, and cut into chunks
•⅛ teaspoon salt
1.In a small pot, bring the water to a boil over high heat. Stir in the coconut sugar, half the mint, and the chamomile, then remove the pot from the heat. Let the mixture steep for 15 minutes, covered, then strain out and discard the mint and chamomile.
2.In a blender, combine the cantaloupe, salt, and chamomile-mint water and blend until very smooth. Finely chop the remaining mint leaves, then pulse them into the cantaloupe mixture until well distributed.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
MANGO CHILE
MAKES 5 OR 6 (3-OUNCE) POPS
I spent much of my childhood in Tucson, Arizona, about an hour from the Mexican border. On the weekends, my friends and I would pile into our parents' cars and head down to Nogales, where we'd bargain for silver and eat way too many Chiclets. My favorite thing about these trips, though, was always the mango sold on the street corners, with a squeeze of lime and a pinch of chili powder. The sweet and juicy mango was offset by the acidic lime and brought to life by the faintly spicy chili powder, and I've captured the same perfect combination in this Glow Pop. Not only do the ingredients balance each other, but they also pack quite a nutritional punch: the mango contains zeaxanthin, an antioxidant that promotes eye health and aids vision; the lime zest offers protection against skin cancer; and the chili powder ramps up your metabolism.
•2¼ cups cubed fresh mango (from about 2 mangoes)
•¾ teaspoon chili powder
•Zest and juice of 1 lime
•⅛ teaspoon salt
•½ cup water
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
I've tried this pop with frozen mango and it doesn't work nearly as well—the frozen mango has lost much of the moist juiciness it needs to bring this pop to life.
ROSEMARY STRAWBERRY
MAKES 5 OR 6 (3-OUNCE) POPS
While rosemary is typically found in savory dishes, I find its richly woody notes a grounding counterpoint to strawberry's airy sweetness; it infuses this pop with unforgettable flavor. Rosemary is one of the main herbs consumed by residents of so-called blue zones in the Greek islands, named because the people there are among the longest living in the world. Rosemary is powerfully anti-inflammatory and has been found to reduce both the severity and frequency of asthma attacks, while strawberries are rich in vitamin C.
•3 cups hulled fresh strawberries
•¼ cup water
•3 sprigs fresh rosemary
•½ teaspoon pure vanilla extract
•⅛ teaspoon salt
•3 tablespoons honey
1.In a small pot, combine the strawberries, water, and rosemary and cook over low heat, mashing the strawberries with a wooden spoon, until the berries have collapsed and released their juices, about 10 minutes.
2.Turn off the heat; cover, and let cool for 10 minutes. Remove and discard the rosemary sprigs. Transfer the mixture to a blender and add the vanilla, salt, and honey. Blend until well combined.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
HONEYED PEACH THYME
MAKES 5 OR 6 (3-OUNCE) POPS
Thyme's delicately sweet, slightly earthy taste complements rather than overpowers other flavors, and adds a subtle depth and uniqueness. It's also antiviral and antibacterial, and has been used for centuries to ward off everything from simple coughs to the plague. In fact, whenever I feel like I'm getting sick, I'll steep some fresh thyme in a cup of hot water and inhale the steam before drinking the resulting herbal tisane—and often, I'll find I'm better by the next morning. Here I've paired the herb with fresh summer peaches, roasted in the oven to concentrate their sweetness.
•4 or 5 medium peaches, halved and pitted
•3 tablespoons honey
•1 teaspoon pure vanilla extract
•1 teaspoon fresh lemon juice
•⅛ teaspoon salt
•1 teaspoon fresh thyme leaves
1.Preheat the oven to 350ºF. Place the peaches cut-side down on a rimmed baking sheet and bake for 20 minutes. Remove from the oven and let rest for 10 minutes, until cool to the touch.
2.Transfer the peaches, skin still on, to a blender or food processor and add the honey, vanilla, lemon juice, and salt. Blend until mostly smooth. Add the thyme and blend for 1 minute more to incorporate.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
PINK LEMONADE
MAKES 5 OR 6 (3-OUNCE) POPS
Pink lemonade is always so much better than the regular kind, isn't it? There's something superior about the sweetness of raspberries contrasted with the tartness of the lemon—that, and the aesthetic appeal of pretty bright pink, of course. Raspberries are one of the most fiber-filled fruits, meaning they'll help keep you regular and eliminate bloating, while lemon helps support your liver, the body's natural detoxifier. But for me, these pops are all about the flavor—they're just the refreshing, juicy hit I crave on a hot day.
•¾ cup fresh lemon juice (from 4 or 5 lemons)
•⅓ cup honey
•1½ cups fresh or frozen raspberries
1.Blend together all the ingredients until smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
LAVENDER BLUEBERRY
MAKES 5 OR 6 (3-OUNCE) POPS
My friend Nicola has a theory that foods of similar colors will often taste delicious together. I don't know how well this applies to, say, white chocolate and cauliflower, but here it works brilliantly. Lavender adds a delicate floral note to the sweet and tart blueberries. Lavender has been used for thousands of years to soothe both the body and the mind, for everything from ailments of the digestive tract to anxiety. Blueberries are well known for their superfood status, and have been proven to lower blood pressure, support heart health, and help cell regeneration and brain protection, with recent studies even pointing to their ability to stave off Alzheimer's!
•½ cup water
•1 cup coconut sugar
•1 tablespoon dried culinary lavender (see this page)
•4 cups fresh blueberries
1.In a small saucepan, whisk together the water and coconut sugar. Bring to a boil over medium heat, then reduce the heat to medium-low and simmer for 10 minutes, until the mixture begins to thicken. Add the lavender and simmer for 10 minutes more, until the mixture has reduced by about half and looks thick and syrupy. Remove the pot from the heat and strain through a fine-mesh strainer into a bowl. Discard the lavender. Let the syrup cool for 15 minutes. Transfer the syrup to a food processor, add the blueberries, and process until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
BLACKBERRY ROSE
MAKES 5 OR 6 (3-OUNCE) POPS
Ah, rose—how I do love thee. While rose water is found commonly in Middle Eastern and Indian dishes, it's more of a rarity in the United States—but it shouldn't be. Besides having a delicate floral flavor that complements the juicy blackberries in this recipe, rose water is rich in flavonoids, antioxidants, tannins, and essential vitamins like A, C, D, E, and B3. It's also a mild relaxant and mood balancer—even just the aroma is known to calm a restless or stressed mind. It's made by steam-distilling fresh rose petals and can be found in the international section of most grocery stores, Indian or Middle Eastern markets, and many liquor stores (as it's a common ingredient in cocktails). This pop oozes sophistication, and is a lovely go-to when you want something outside the norm.
•6 cups fresh blackberries
•1 tablespoon rose water (see this page)
•1 teaspoon pure vanilla extract
•1 teaspoon fresh lemon juice
•¼ cup honey
•⅛ teaspoon salt
1.Place all the ingredients in a food processor and pulse until smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CARAMELIZED PINEAPPLE
MAKES 5 OR 6 (3-OUNCE) POPS
Pineapple is truly the best thing to barbecue—the heat caramelizes the fruit, deepening its flavor in a way that's truly mind-blowing. This recipe allows you to capture this summer sensation even if you don't have access to a grill, as cooking the pineapple on the stovetop creates the same Maillard reaction. Pineapples are one of my favorite healthy foods, as they're the only natural source of bromelain, a digestive enzyme so potent that many people supplement their diets with it. This makes these pops the perfect accompaniment to a summer feast, as it'll help your body digest whatever else you eat and eliminate bloat before you hit the pool later.
•2 tablespoons coconut oil or ghee
•¼ cup coconut sugar
•5 cups fresh pineapple, in 1-inch chunks (about 1 medium pineapple)
•⅛ teaspoon salt
1.In large sauté pan, melt the coconut oil over high heat. Sprinkle the coconut sugar generously all over the pineapple, then add the pineapple to the pan and sear, stirring frequently, until rich brown on all sides, about 10 minutes.
2.Let cool completely before transferring the pineapple to a food processor. Add the salt and process until smooth.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
If you want to grill the pineapple instead of sautéing it, cut your pineapple lengthwise into 5 or 6 long strips. Rub with the coconut oil and sprinkle with coconut sugar before placing directly on a grill that has been heated to 400 degrees. Sear until grill marks appear, 2 to 3 minutes per side, before letting it cool completely and transferring it to a food processor. Follow the rest of the recipe as instructed.
COCONUT CHAI
CINNAMON ORANGE & CREAM
STRAWBERRY CARDAMOM ROSE LASSI
MEXICAN HORCHATA
COOKIE DOUGH
BLUEBERRY & CREAM
MATCHA LATTE
TURMERIC GOLDEN MILK
PEANUT BUTTER & JELLY
LAVENDER LONDON FOG
WHITE CHOCOLATE CHIA STRAWBERRY
These pops will take you around the world, from grassy Japanese matcha tea to a healthier take on a cinnamon-y Mexican Horchata to Indian Turmeric Golden Milk, an Ayurvedic remedy that's been used to bolster the immune system for thousands of years. The secret to a creamy pop is all in the base. For most of the recipes in this chapter, I like to use full-fat canned coconut milk, which will give you that perfect, smooth result. Light canned coconut milk also works well, albeit with slightly less creamy results, and the homemade milks included in Getting Started (see this page) are good choices, too. Whatever you do, don't use store-bought nut milks, as they're generally much too thin and will result in a watery, icy pop (see this page for more information).
COCONUT CHAI
MAKES 5 OR 6 (3-OUNCE) POPS
Generally, chai is a specific blend of black tea, spices, milk, and sweetener. The spices vary in type and amount (some families guard their chai recipe with their lives), but tend to include cinnamon, cardamom, cloves, black peppercorns, and ginger. I've swapped the traditional dairy for coconut milk, which is commonly used in Indian cooking and plays beautifully with the mélange of spices. All the spices here are health superstars as well: cinnamon balances blood sugar, fennel and ginger soothe the stomach and decrease bloat, black pepper helps increase absorption of other nutrients and helps burn fat, and cloves are highly antibacterial. This pop does contain caffeine from the black tea, but you can replace it with caffeine-free rooibos tea if you prefer.
•2 cups full-fat coconut milk
•10 cardamom pods, gently crushed
•1 (1½-inch) piece fresh ginger, peeled and sliced
•2 cinnamon sticks
•¼ teaspoon whole cloves
•½ teaspoon fennel seeds
•¼ teaspoon whole black peppercorns
•3 tablespoons coconut sugar
•1 teaspoon pure vanilla extract
•2 black tea bags
1.In a medium saucepan, combine all the ingredients except the tea bags. Bring to a boil over medium-high heat, then reduce the heat to low, cover, and simmer for 30 minutes. Turn off the heat, add the tea bags, and steep, covered, for 5 minutes more. Remove and discard the tea bags. Let the mixture cool for 20 minutes, then pour it through a fine-mesh strainer into a bowl. Discard any solids left in the strainer.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CINNAMON ORANGE & CREAM
MAKES 5 OR 6 (3-OUNCE) POPS
I first discovered the combination of cinnamon and orange in a tiny village in Morocco, where orange slices were served with a sprinkle of the spice on top. Seemingly simple, it's become one of my favorite flavor profiles—the cinnamon lends a spicy, aromatic quality while bringing out the zesty freshness of the orange, and the sweetness of each teases and complements the other. Cinnamon is a wonder spice: numerous studies point to its ability to stabilize blood sugar, lower LDL (or "bad") cholesterol, fight fungal and bacterial infections, and even protect against cancer. Oranges are vitamin C powerhouses, making your skin glow and helping keep your immune system in top-notch shape. If you'd prefer the taste of the classic Creamsicle of your childhood, simply omit the cinnamon.
•1 cup full-fat coconut milk
•Zest and juice of 2 oranges
•3 tablespoons honey
•2 teaspoons ground cinnamon
•1 teaspoon pure vanilla extract
•⅛ teaspoon salt
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
STRAWBERRY CARDAMOM ROSE LASSI
MAKES 5 OR 6 (3-OUNCE) POPS
Lassis are popular drinks in Indian cuisine, typically made with some type of fruit (mango is common) blended with sugar and yogurt to make a creamy, refreshing concoction. Here I elevated the traditional flavors to something slightly more exotic, with a blend of strawberry and rose that's downright transporting. Rose water is made by steam-distilling fresh rose petals and has a delicately sweet flavor. It's rich in vitamins A and C, which boost collagen and increase cell turnover in your skin, making it glow. It's also known for its relaxing properties—just a whiff is often enough to calm me down! No matter which kind of yogurt you use, it will provide a hefty dose of probiotics, which have been linked to clearer skin, reduced anxiety, and weight loss.
•1 cup hulled and halved fresh strawberries
•¾ cup unsweetened yogurt (coconut, Greek, and full-fat regular all work well)
•3 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•1 teaspoon rose water (see this page)
•1 teaspoon ground cardamom
•1 teaspoon pure vanilla extract
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
MEXICAN HORCHATA
MAKES 5 OR 6 (3-OUNCE) POPS
I'm obsessed with horchata, a sweet Mexican drink traditionally made with rice milk and cinnamon. To avoid the icy texture rice milk takes on when frozen, I swapped it out for coconut milk to make a creamy, dreamy pop with a perfect texture. Cinnamon stokes your metabolism and helps balance blood sugar, and I've long been a fan of dates' ability to mimic caramel, with a rich and buttery sweetness that works perfectly with the warmth of the cinnamon. Plus, they're rich in fiber and potassium. The ingredients here are simple but utterly unexpected. This recipe often winds up being people's favorite Glow Pop!
•2 cups full-fat coconut milk
•6 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•1 tablespoon ground cinnamon
•1 teaspoon pure vanilla extract
•Pinch of salt
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
COOKIE DOUGH
MAKES 5 OR 6 (3-OUNCE) POPS
I'm definitely one of those people who prefers cookie dough to actual cookies, so these pops always hit the spot—but in a much healthier way! The secret is in the mix of rich cashews and almond butter that come together to create that creamy, buttery texture that chocolate chip cookie dough is famous for. Almonds are rich in vitamin E, protein, and heart-healthy fat, while cashews contain largely heart-healthy monounsaturated fats, with a hefty dose of skin-beautifying copper to boot. Honey adds a lovely sweetness, while pure vanilla extract adds that just-out-of-the-oven aroma. I love to make these with raw cacao nibs, which offer potent antioxidants and a bitter, rich chocolate flavor that offsets the creamy sweetness, but they'd also be great with mini chocolate chips.
•1½ cups raw cashews, soaked for 1 to 2 hours, then drained
•1 tablespoon pure vanilla extract
•1½ tablespoons smooth almond butter
•¼ teaspoon salt
•⅓ cup honey
•1 cup water
•½ cup raw cacao nibs or mini chocolate chips
1.Blend together all the ingredients except the cacao nibs until very smooth. Add the nibs and pulse until just combined.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
You can use raw or roasted almond butter in this recipe—the roasted version will give a slightly richer flavor.
BLUEBERRY & CREAM
MAKES 5 OR 6 (3-OUNCE) POPS
These pops are just so pretty, with bright, colorful blueberries dotting a creamy white background like they're nature's confetti. I love not blending the cream with the blueberries, so they remain evenly suspended throughout the cream mixture and you get the separation of the sweet milkiness and the tart, juicy blueberries in every bite. Blueberries are one of those health foods that have garnered a reputation as a superfood, and for good reason. They have the highest antioxidant capacity of any commonly consumed fruit or vegetable, resulting in protection against heart disease and cell damage. They've also been shown to help increase memory function and prevent cognitive decline, and their compounds are currently being studied as a potential remedy for Alzheimer's. Similar to cranberries, they're also effective in preventing and treating UTIs.
•1 cup full-fat coconut milk
•¼ cup honey
•1 teaspoon pure vanilla extract
•⅛ teaspoon salt
•¾ cup frozen or fresh blueberries
1.Blend together the milk, honey, vanilla, and salt until smooth.
2.Spoon half the blueberries into pop molds, dividing them evenly. Top with the cream mixture, filling the molds about three-quarters of the way, then add the remaining blueberries. Fill any remaining space in the molds with a very small amount of the cream.
3.Freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
I like to use frozen blueberries in this recipe because they're much smaller, allowing you to more easily suspend them within the coconut milk "cream." Plus, they're generally cheaper than fresh, and available all year round. Fresh wild blueberries also work wonderfully.
MATCHA LATTE
MAKES 5 OR 6 (3-OUNCE) POPS
Matcha has taken the health food world by storm in recent years, and I must confess I haven't been immune to its charms. Fancy ceremonial rituals aside, matcha is simply finely ground green tea. Instead of steeping it and discarding the leaves, you dissolve the whole-leaf powder in whatever liquid you're using and drink it. Because you're consuming the whole tea leaf, rather than simply infusing liquid with it, the health benefits skyrocket: matcha has 137 more antioxidants than regularly brewed green tea, including EGCG, which has been found in numerous studies to have potent cancer-fighting properties. This pop has a slightly grassy, sweet flavor that will quickly become addictive.
•2 cups full-fat coconut milk
•2 teaspoons matcha powder
•3 tablespoons sweetener of choice (see this page)
1.Blend together all the ingredients until smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
TURMERIC GOLDEN MILK
MAKES 5 OR 6 (3-OUNCE) POPS
Long prized as a healing elixir in Ayurvedic tradition, turmeric golden milk is a mixture of turmeric, milk, and sweetener, and is typically drunk warm. It's one of my favorite winter beverages: the turmeric is earthy, the ginger and black pepper add a hint of spice, and the honey and coconut milk ground it all in creamy sweetness. All the healthy properties shine in a Glow Pop form. The superstar is turmeric, which is prized for its anti-inflammatory benefits and has been shown in numerous studies to match or even outperform over-the-counter NSAIDs. Some studies have also shown turmeric's effectiveness in treating conditions like IBS, rheumatoid arthritis, cystic fibrosis, and even cancer. Always be sure to consume turmeric with both black pepper and fat, as they make the spice 150 percent more bioavailable, dramatically increasing its healthful effects in your body.
•2 cups full-fat coconut milk
•3 tablespoons honey
•1 teaspoon ground turmeric
•1 teaspoon ground ginger
•⅛ teaspoon freshly ground black pepper
1.Blend together all the ingredients until smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
If you're an allergy sufferer like me, use local honey when you make this pop. Between the anti-inflammatory action of the turmeric and the local honey inoculating you against nearby pollens, your runny nose and itchy eyes will be gone in no time.
PEANUT BUTTER & JELLY
MAKES 5 OR 6 (3-OUNCE) POPS
This pop evokes childhood in such a playful way while looking super grown-up—the swirls of smooth tan peanut butter and chia seed–studded raspberry jelly are absolutely stunning. Chia seeds are one of my favorite ingredients to make a quick and easy jelly, thanks to their seemingly magic ability to make any liquid thick and gelatinous (see this page). They're incredibly filling and full of fiber, protein, and good fat. Here they're swirled with protein- and vitamin E–filled peanut butter for a sweet-and-salty combo that's equally good for breakfast, a light lunch, or dessert.
JELLY LAYER
•4 cups frozen raspberries
•2 teaspoons pure vanilla extract
•2 teaspoons honey
•Pinch of salt
•2 tablespoons chia seeds
PEANUT BUTTER LAYER
•¾ cup canned coconut milk or homemade milk of choice
•½ cup peanut butter (creamy and crunchy both work well)
•3 tablespoons honey
1.Make the jelly layer: In a small saucepan, cook the frozen raspberries over low heat, squishing and stirring them with a wooden spoon, for 10 to 15 minutes, until they defrost and release their juices. Remove from the heat and stir in the vanilla, honey, and salt. Sprinkle the chia seeds over the mixture, then stir to combine. Let sit for 15 to 20 minutes, or until the mixture becomes jelly-like.
2.Meanwhile, make the peanut butter layer: blend together all the ingredients until smooth (or with a bit of texture, if you're using crunchy peanut butter).
3.Place a teaspoon of the peanut butter mixture in the tip of each mold, then add a few spoonfuls of the jelly mixture. Continue to alternate peanut butter and jelly until your molds are full. If you'd like a swirled look, stir a chopstick or knife through the mixture once, until peanut butter and jelly layers are swirled but not fully mixed.
4.Freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
Any leftover raspberry jelly can be spread on toast, spooned onto oatmeal or ice cream, or eaten plain for a quick energy boost!
LAVENDER LONDON FOG
MAKES 5 OR 6 (3-OUNCE) POPS
Essentially an Earl Grey latte, a London Fog's milky sweetness perfectly complements bitter orange bergamot and tannic black tea. Here, I upped the ante even further by adding lavender, which makes the entire pop taste surprising, luxurious, and perfectly balanced. Black tea has been found to ward off certain types of cancer, reduce the risk of stroke and type 2 diabetes, and defend against heart disease. Combined with calming, anti-inflammatory lavender, it's a perfect wellness elixir.
•2 cups full-fat coconut milk
•3 tablespoons coconut sugar
•3 Earl Grey tea bags or 1 tablespoon loose-leaf Earl Grey tea
•1 teaspoon dried culinary lavender (see this page)
•½ teaspoon pure vanilla extract
1.In a small pot, bring the milk just to a boil over medium-high heat, then immediately remove from the heat. Stir in the coconut sugar until dissolved, then add the tea and the lavender. Cover and steep for 10 minutes.
2.Pour the mixture through a fine-mesh strainer into a bowl. Discard the tea and lavender. Stir in the vanilla. Let the mixture cool to room temperature, about 15 minutes, then stir.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
WHITE CHOCOLATE CHIA STRAWBERRY
MAKES 5 OR 6 (3-OUNCE) POPS
This pop is one of my favorite summer breakfasts—the good fats and fiber in the chia seeds and strawberries offer more than enough substance to power me through to lunch. Cacao butter, the pressed oil of the cacao bean, creates the white chocolate flavor. Like cacao powder and nibs, both long recognized as superfoods, the butter is rich in antioxidants; oleic acid, which has been shown to reduce the risk of heart disease; and theobromine, which raises energy and alertness levels.
•2 tablespoons cacao butter
•4 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•1 cup canned coconut milk or homemade milk of choice
•1 teaspoon pure vanilla extract
•⅛ teaspoon salt
•3 tablespoons chia seeds
•2 cups hulled fresh strawberries
•¼ cup water
1.In a small saucepan, melt the cacao butter over low heat, stirring frequently. Transfer to a blender; add the dates, milk, vanilla, and salt; and blend until very smooth.
2.Place the chia seeds in a large glass jar or container, then pour the milk mixture over them. Stir thoroughly, then let sit for 30 minutes until set into a gel. Meanwhile, puree the strawberries in a blender until smooth, adding water 1 tablespoon at a time as necessary to reach the desired texture.
3.In alternating layers, fill the molds with the chia mixture and strawberry puree. Freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
You can use cocoa butter in place of the cacao butter if you prefer. They're essentially the same thing, but cocoa butter has been processed at higher temperatures, causing it to lose some of its antioxidants.
MEXICAN HOT CHOCOLATE
CHOCOLATE FUDGE
CHOCOLATE-COVERED BANANA
CHOCOLATE HAZELNUT
COLD-BREW MOCHA
NEAPOLITAN
CHOCOLATE ORANGE
PEANUT BUTTER CUP
CHOCOLATE CHIA LAVENDER
CHOCOLATE CARAMEL SWIRL
OLIVE OIL CHOCOLATE ROSEMARY
Is there anything better than a frozen chocolate treat on a hot summer's day...or really, anytime? These Glow Pops allow you to enjoy your sweets and nourish yourself, too, forgoing the heavily processed chocolate of typical desserts for raw cacao, a pure form of the cacao bean that's regarded as a superfood packed with vitamins, minerals, and antioxidants (see this page). From the Chocolate Fudge pop, which uses avocado to mimic the creamy texture of the childhood favorite Fudgsicle, to the decidedly more grown-up Olive Oil Chocolate Rosemary to the decadent Chocolate Caramel Swirl that will surprise everyone with its secret, healthy ingredient, these chocolate pops are easy wins with any crowd, whether you're interested in health or not.
MEXICAN HOT CHOCOLATE
MAKES 5 OR 6 (3-OUNCE) POPS
Mexican hot chocolate is one of my favorite flavor combinations: sweet chocolate with spicy chiles and warm cinnamon. I'm guilty of trying it in everything from green smoothies to energy balls to pancakes. I often find myself staring wistfully at plain chocolate bars, wondering why they're so naked and spiceless. Spices are the built-in superfoods of any kitchen cabinet, so punching up the flavor punches up the nutrition factor as well. Cinnamon helps keep your blood sugar stable, making it a great option for anyone with blood sugar issues. Cayenne boosts your metabolism, and nutmeg contains specific antimicrobial compounds that work to freshen breath (yes, that is permission to eat one of these before that important meeting!). A word of warning: Once you try this kicked-up version of chocolate, you'll never want to go back to the basic stuff again.
•1½ cups full-fat coconut milk
•¼ cup maple syrup
•2 teaspoons ground cinnamon
•6 tablespoons raw cacao powder
•¼ teaspoon freshly grated nutmeg
•⅛ teaspoon cayenne
•1 teaspoon pure vanilla extract
•⅛ teaspoon salt
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CHOCOLATE FUDGE
MAKES 5 OR 6 (3-OUNCE) POPS
This pop tastes exactly like a Fudgsicle. The trick is in the texture: you want that super-creamy, easy-to-bite, ice-free consistency. I accomplish that here with a secret ingredient: avocado. While the strong chocolate flavor hides any hint of avocado taste, its inclusion creates that perfect smoothness you remember from childhood—but without the chemical nasties, and with plenty of good fats to nourish your skin and hair. Full-fat coconut milk adds to the texture, and a dose of antioxidant-rich cacao powder ensures a rich, chocolatey flavor.
•¾ cup full-fat coconut milk
•1 ripe avocado, pitted and peeled
•6 tablespoons raw cacao powder
•1 teaspoon pure vanilla extract
•4 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•2 tablespoons honey
•¼ teaspoon salt
1.Blend together all the ingredients until very smooth; the mixture should have the consistency of pudding. If it becomes too thick, add water 1 tablespoon at a time as needed to thin it.
2.Pour the mixture into pop molds and tap them on the counter to remove any air bubbles. Freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CHOCOLATE-COVERED BANANA
MAKES 5 OR 6 (3-OUNCE) POPS
In healthy cooking, bananas are a superstar fruit, pulling weight to hold together pancake batter and to sweeten and add creaminess to smoothies and pops (you'll see them in many of the green pops). They've even been turned into soft-serve ice cream! After so many supporting roles, they get their turn as the star here, in a pop that highlights their creamy sweetness by draping it in a decadent chocolate shell. Besides being one of the cheapest fruits on the market (I once bought six bananas for less than $1), bananas are rich in potassium, fiber, and vitamin C. Be sure to let them ripen completely before using them—brown spots mean their starch has converted into sugar, making the fruit easier to digest, more blendable, and better tasting!
BANANA POPS
•3 ripe bananas
•1 cup canned coconut milk or homemade milk of choice
•1 teaspoon pure vanilla extract
•⅛ teaspoon salt
CHOCOLATE SHELL
•6 ounces coarsely chopped dark chocolate
•¼ cup coconut oil
1.Make the pops: Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
3.Once completely frozen, unmold the pops and lay them on a parchment- or waxed paper–lined plate or baking sheet. Store them in the freezer while you make the chocolate shell.
4.Make the chocolate shell: Melt the chocolate (see Tips and on this page) until completely liquid. Stir in the coconut oil until the mixture is very smooth, then let it sit for 5 to 10 minutes, or until it's still liquid but cool enough to touch.
5.Remove the pops from the freezer. Working quickly, dip each frozen pop into the chocolate, twisting the pop around to coat evenly. Alternatively, drizzle the chocolate shell over the pops in a fun pattern for a festive touch.
6.Return the chocolate-covered pops to the plate or baking sheet and freeze until ready to eat.
To melt chocolate using a microwave, place the chopped chocolate in a wide, shallow bowl. Microwave in 15- to 20-second intervals, stirring between each, until the chocolate is completely melted and smooth.
To melt chocolate on the stovetop, heat 2 to 3 inches of water in a medium saucepan over medium heat until it's boiling. Place a tightly fitting bowl on top—make sure the bottom of the bowl doesn't touch the water and no steam can escape from the sides where the bowl meets the saucepan. Add the chopped chocolate to the bowl and stir frequently, until the chocolate is completely melted and smooth.
CHOCOLATE HAZELNUT
MAKES 5 OR 6 (3-OUNCE) POPS
I became addicted to Nutella the first time I went to Europe. With little effort, the creamy spread proved that chocolate and hazelnut are the perfect flavor combination. This Chocolate Hazelnut pop takes the best of Nutella, keeping its healthy, whole-food ingredients and leaving out the less desirable processed ones. Hazelnuts are no slouch in the health department, boasting the most folate of any tree nut (for all you expectant mamas out there, that's your excuse to eat as many of these pops as you'd like), plus high quantities of skin- and hair-beautifying manganese and copper. Soaking the nuts overnight makes them easier to blend, as well as decreases the quantity of phytic acid, which impedes mineral absorption in the body.
•2 cups raw hazelnuts
•⅛ teaspoon salt, plus a pinch
•2 cups water
•3 tablespoons raw cacao powder
•3 tablespoons maple syrup
1.Soak the hazelnuts in water to cover with a pinch of salt for 12 hours, then drain and transfer to a blender. Add the remaining ingredients and blend until very smooth. If desired, let sit for 30 minutes to allow the foam and air bubbles to dissipate.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
The base of this pop is essentially a very creamy Nutella-flavored milk, which I also love to drink straight or, if I'm feeling really decadent, pour over a bowl of granola or cereal.
COLD-BREW MOCHA
MAKES 5 OR 6 (3-OUNCE) POPS
For those days when you wish your coffee could just make itself, now you can just grab a mocha pop from the freezer and go! Chocolate and coffee are a classic combination—plus, this recipe utilizes cold-brew coffee for an extra health boost. Cold-brew coffee is, as the name suggests, brewed for twelve to twenty-four hours in _cold_ water, rather than by the typical hot water–steeping method. It is less bitter and acidic than traditional coffee (by as much as 67 percent), and highly concentrated. Coffee—cold brew or otherwise—is also rich in antioxidants, magnesium, and chromium, and has been found in studies to reduce the risk of type 2 diabetes as well as Alzheimer's. You can find cold-brew coffee in the refrigerated section at most grocery stores, or you can make it yourself (see recipe below).
•¾ cup full-fat coconut milk
•¾ cup undiluted cold-brew coffee concentrate
•¼ cup coconut syrup
•1 tablespoon raw cacao powder
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
MAKE YOUR OWN COLD-BREW COFFEE
MAKES 2 CUPS
•1 cup coarsely ground coffee beans
•2 cups filtered water
In a large bowl or glass jar, stir together the ground coffee beans and water. Cover and refrigerate for 12 to 24 hours. Strain the mixture into another bowl or jar by pouring it through a fine-mesh strainer lined with a traditional coffee filter or cheesecloth. Discard the coffee grounds. You can store any leftover cold-brew coffee in a covered container or jar in the fridge. To drink, rather than use in pops, dilute with 2 cups water for every 1 cup of cold brew.
NEAPOLITAN
MAKES 5 OR 6 (3-OUNCE) POPS
Neapolitan ice cream technically refers to the combination of several flavors. The original version from Naples was usually more like spumoni, with pistachio, chocolate, and cherry or strawberry ice cream, but this recipe riffs off the American classic flavor combination: smooth and buttery vanilla, sweet and fruity strawberry, and rich chocolate. Chia seeds are the secret that allows this pop's layers to remain distinct, while also adding tons of fiber, protein, and good fat. I love to use chia seeds in pop recipes because they give the pops a creamy, almost gelato-like texture, and these are no exception—the final result is as decadent as a childhood ice cream sandwich. Strawberries, beyond being the pretty, fruity counter to the vanilla and chocolate, are rich in vitamin C, which helps create skin-boosting collagen. You can put aside any leftover strawberry compote to use as an oatmeal or toast topping, or mix chia seeds into it to make a jam.
•1½ cups frozen strawberries
•⅓ cup plus 1¼ cups full-fat coconut milk
•5½ tablespoons chia seeds
•⅛ teaspoon salt
•2 teaspoons vanilla bean powder (or pure vanilla extract)
•2½ tablespoons plus 1¼ teaspoons honey
•4 teaspoons raw cacao powder
1.In a small saucepan, cook the strawberries over low heat, breaking them up with the back of a wooden spoon, for about 20 minutes, or until completely thawed. Bring to a simmer and cook, stirring occasionally, for 10 minutes more.
2.Transfer ⅓ cup of the strawberry compote to a small bowl and add the ⅓ cup milk and 2 tablespoons of the chia seeds. In a separate small bowl, stir together the remaining 1¼ cups milk and 3½ tablespoons chia seeds. Let both sit for 15 minutes, or until a thick gel forms.
3.Transfer the milk-chia mixture to a blender. Add half the salt, 1½ teaspoons of the vanilla, and the 2½ tablespoons honey. Blend until smooth, stopping to scrape down the sides of the blender as needed. Set aside half the mixture. Add the cacao powder to the mixture in the blender and blend until combined. Transfer to a bowl and rinse out the blender.
4.Blend together the strawberry-chia mixture and the remaining 1¼ teaspoons honey, salt, and ½ teaspoon vanilla extract until smooth.
5.Spoon the chocolate mixture into each pop mold until it is one-third full, then add the strawberry mixture until the molds are two-thirds full, and fill the remaining space in the molds with the vanilla mixture. Tap the molds on the counter between each layer to remove any air bubbles.
6.Freeze for 1 hour, then insert sticks and freeze for 6 hours more or overnight, until solid.
Want to keep your layers distinct and your pops looking perfect? The easiest way to make sure the mixture doesn't scrape the sides is to pour each one into a zip-top bag or a piping bag, then cut off the tip and squeeze the mixture into the center of the mold. This is only for appearances, to keep the colors totally separate—feel free to skip it.
CHOCOLATE ORANGE
MAKES 5 OR 6 (3-OUNCE) POPS
When I was a child, I'd rush downstairs every Christmas to find a chocolate orange stuffed into the tip of my stocking. It was the type you had to smash against a hard surface to reveal a dozen slim wedges—and it was the beginning of my love affair with this flavor combination. Can you blame me? Rich, creamy dark chocolate beautifully complements the zesty brightness of orange. While chocolate you find in the store is highly processed and so far removed from the cacao bean that many of the precious antioxidants and minerals are destroyed, raw cacao is one of the purest forms. I personally love it for the steady energy it gives (much gentler than caffeine) and the simultaneous calming effect of the magnesium it contains. This pop is also packed with vitamin C, of course, but by using the orange _zest_ , we amp up the flavor, add additional antioxidants, and protect against skin cancer.
•1 cup full-fat coconut milk
•Zest and juice of 2 oranges
•3 tablespoons honey
•6 tablespoons raw cacao powder
•1 teaspoon pure vanilla extract
•⅛ teaspoon salt
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
PEANUT BUTTER CUP
MAKES 5 OR 6 (3-OUNCE) POPS
Peanut butter cups were my absolute favorite childhood candy—I remember rifling through my plastic pumpkin every Halloween so I could eat them all first. There's something about the combination of salty peanut butter and smooth, rich chocolate that sends my taste buds to high heaven—and with these pops, I get all the pleasure with none of the artificial flavors, high-fructose corn syrup, or unhealthy fat. Peanut butter, when consumed in its whole state, is actually quite healthy—full of good fats, protein, and skin-boosting vitamin E. Here, a creamy peanut butter pop hides inside a dark chocolate shell for a show-stopping treat that's as decadent as a candy bar. Just be sure to look for peanut butter with only one ingredient—peanuts—and no highly processed oils or additives.
PEANUT BUTTER POPS
•1¼ cups canned coconut milk or homemade milk of choice
•⅔ cup peanut butter (creamy and crunchy both work well)
•⅛ teaspoon salt
•3 tablespoons honey
CHOCOLATE SHELL
•6 ounces coarsely chopped dark chocolate
•¼ cup coconut oil
1.Make the peanut butter pops: blend together all the peanut butter pop ingredients until very smooth (or with some texture, if using crunchy peanut butter).
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid. Once completely frozen, unmold the pops and lay them on a parchment- or waxed paper–lined plate or baking sheet and store in the freezer while you make the chocolate shell.
3.Make the chocolate shell: Melt the chocolate (see Tips) until completely liquid. Stir in the coconut oil until the mixture is very smooth, then let it sit for 5 to 10 minutes, or until it's still liquid but cool enough to touch.
4.Remove the pops from the freezer. Working quickly, dip each frozen pop into the chocolate, twisting the pop around to coat evenly. Alternatively, drizzle the chocolate shell over the pops in a fun pattern for a festive touch.
5.Return the chocolate-covered pops to the plate or baking sheet and freeze until ready to eat.
CHOCOLATE CHIA LAVENDER
MAKES 5 OR 6 (3-OUNCE) POPS
In this pop, the richness of the chocolate is heightened by the lavender notes—an unexpected twist. Lavender has long been renowned for its ability to soothe both body and mind, and is used as a remedy for anxiety, depression, insomnia, restlessness, in stomachaches, and indigestion. Loaded with omega-3s, fiber, and protein-rich chia seeds as well as antioxidant-packed raw cacao powder, this pop makes a great healthy dessert, and is filling enough to be a hearty breakfast.
•1¾ cups canned coconut milk or homemade milk of choice
•1 tablespoon dried culinary lavender (see this page)
•6 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•1 teaspoon pure vanilla extract
•¼ cup raw cacao powder
•⅛ teaspoon salt
•¼ cup chia seeds
1.In a medium pot, bring the milk to a boil over medium-high heat, then remove from the heat and add the lavender. Cover and steep for 20 minutes. Pour the mixture through a fine-mesh strainer into the blender. Discard the lavender. Add the dates, vanilla, cacao, and salt and blend until very smooth.
2.Transfer the mixture to a large glass jar or bowl and stir in the chia seeds. Let sit for 20 to 30 minutes, or until a gel forms. If you want a smooth and creamy pop, blend the mixture again; if you want to leave some texture, skip this step.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CHOCOLATE CARAMEL SWIRL
MAKES 5 OR 6 (3-OUNCE) POPS
This might be the most decadent Glow Pop in the book. It's my go-to recipe when people say they don't like healthy desserts (although I often don't even tell people they're healthy until after they eat one—and wait for their wide eyes and dropped jaws). The secret is in the sauce—the caramel sauce, that is. Made from the simple, dairy-free combination of coconut milk, coconut sugar, coconut oil, and vanilla, this caramel is absolutely heavenly—though do keep in mind that it needs a good chill before using. It's every bit as good as the heavy cream–based caramel sauces you're used to, and I'll often use the leftovers from this recipe to drizzle over ice cream or dunk apples into when I want a sweet treat. The chocolate base is made with protein- and fiber-rich chia seeds, which lend heft and creaminess that allow the caramel to swirl through, rather than sinking straight to the bottom. These take a bit longer to freeze solid, but they're worth it!
CARAMEL
•¾ cup full-fat coconut milk
•¼ cup coconut sugar
•½ teaspoon pure vanilla extract
•1½ teaspoons coconut oil
CHOCOLATE MIXTURE
•1¾ cups homemade cashew milk or coconut milk of choice
•5 tablespoons raw cacao powder
•⅛ teaspoon salt
•1 teaspoon pure vanilla extract
•6 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•5 tablespoons chia seeds
1.Make the caramel: in a small saucepan, combine the coconut milk and coconut sugar and heat over medium heat, stirring, until the sugar has dissolved. Bring just to a boil, then immediately reduce the heat to low and simmer, stirring occasionally, for 30 minutes. Remove the pot from the heat and vigorously stir in the vanilla and coconut oil. Refrigerate for at least 1 hour before using.
2.Meanwhile, make the chocolate mixture: Blend together all the ingredients for the chocolate mixture except the chia seeds until very smooth. Transfer the mixture to large jar or bowl and stir in the chia seeds. Let sit for 30 minutes, or until the mixture has thickened. Return the mixture to the blender and blend until very smooth. If necessary, add more milk 1 teaspoon at a time if you're having trouble blending.
3.Beginning with the chocolate mixture, in alternating layers, fill the molds with the chocolate mixture and caramel, one spoonful at a time. If desired, use a pop stick or knife to gently swirl the chocolate and caramel to create a marbled effect (but don't stir too much or they'll simply combine). Freeze for 1 hour, then insert sticks and freeze for at least 12 hours more, or until completely solid.
OLIVE OIL CHOCOLATE ROSEMARY
MAKES 5 OR 6 (3-OUNCE) POPS
The chocolate and rosemary here hit your tongue first—sweet, rich, and earthy—before being grounded by the savory olive oil. The result is rich, creamy, and luxurious. This is a pop you can serve as dessert after a swanky dinner party. Afterward, you can tell your friends they consumed tons of heart-healthy fat, plenty of antioxidants, and even a bit of antiviral and antibacterial essential oils. They might thank you, or they might be too busy asking for more.
•6 tablespoons coconut sugar
•1⅔ cups full-fat coconut milk
•3 sprigs fresh rosemary
•6 tablespoons raw cacao powder
•2 tablespoons olive oil
•Flaky sea salt, for garnish
1.In a small saucepan, combine the coconut sugar and coconut milk and heat over medium heat, stirring, until the sugar has dissolved. Add the rosemary and bring to a boil, then remove from the heat. Cover and let steep for 20 minutes.
2.Remove and discard the rosemary, then transfer the mixture to a blender. Add the cacao and olive oil and blend until completely smooth.
3.Pour the mixture into pop molds. Freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid. Just before serving, unmold the pops and sprinkle generously with flaky sea salt.
TOMATO BEET BLOODY MARY
AVOCADO CHILE LIME
ZUCCHINI BASIL
CURRIED BUTTERNUT SQUASH
SWEET-&-SPICY CORN
BALSAMIC BEET & ROSEMARY
SWEET PEA & MINT
CARDAMOM CINNAMON SWEET POTATO
PUMPKIN PIE
THAI PEANUT CILANTRO GINGER
ROASTED CARROT THYME
The pops in this chapter are for creative-minded people who like to play around with their food, finding surprise and delight in unexpected flavor combinations. While all these pops do have an element of sweetness, many of them contain vegetable bases, like zucchini or sweet pea, or unexpected seasonings. Some are more accessible (kids go crazy for the Cardamom Cinnamon Sweet Potato) and some are a bit more daring (I'm looking at you, Balsamic Beet & Rosemary), but all are delicious and packed with nutrients. Pops are one of my favorite places to play with flavor, since, like ice cream and doughnuts, they're the perfect vehicles for all sorts of crazy combinations.
TOMATO BEET BLOODY MARY
MAKES 5 OR 6 (3-OUNCE) POPS
How about a Bloody Mary that _actually_ cures hangovers? Unlike the typical brunch drink, this version is loaded with vitamins and antioxidants, using real, whole-food ingredients (like—gasp!—real tomatoes) for a spicy, hearty, Sunday morning mocktail pop (or cocktail pop, if you add the vodka). This pop, based on a Bloody Mary invented by my friend Hogan, boasts a salad's worth of nutrients, but the real superstar here is beet, which, besides adding a lovely, can't-put-your-finger-on-it sweetness, is renowned for its blood- and liver-cleansing properties—exactly the parts of your body that take a beating when you drink. Garlic gets your immune system back into fighting shape, while the hot sauce kicks your metabolism into high gear to quell any day-after nausea.
•1 cup chopped peeled beet (about 1 medium beet)
•1½ cups pure tomato juice (I like Sacramento)
•2 garlic cloves
•Zest and juice of 1 lime
•2 teaspoons hot sauce (or 3, if you'd like more heat)
•¼ teaspoon celery salt
•½ teaspoon tamari
•⅓ cup vodka (optional)
1.Put the beets in a microwave-safe dish with 1 inch of water, cover, and microwave until fork-tender, 3 to 5 minutes. Alternatively, cook the beets in a stovetop steamer for 15 minutes, or until fork-tender. Let cool, then transfer to a blender, add the remaining ingredients, and blend until smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
If the smell of garlic sticks to your molds, pour a bit of distilled white vinegar into each mold after it's clean then fill them to the brim with water. Let sit for 30 minutes, then wash and rinse.
AVOCADO CHILE LIME
MAKES 5 OR 6 (3-OUNCE) POPS
Avocado is often treated as a background ingredient: a spread to add butteriness to toast, an add-in for creaminess in a smoothie. In this pop, though, the avocado's subtly grassy, rich flavor truly shines, brought to life by a faintly acidic sweetness from the lime and a kick of umami spice from the chili powder. Avocado is a superstar for both health and beauty, prized for its good fats and antioxidants that will keep you full, quell inflammation, and make your skin absolutely glow.
•1 medium avocado, pitted and peeled
•⅛ teaspoon salt
•½ teaspoon chili powder, plus more (optional) for garnish
•Zest and juice of 1 lime
•1 cup canned coconut milk or homemade milk of choice
•1 tablespoon honey
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid. Just before serving, unmold the pops and sprinkle each with chili powder, if desired.
ZUCCHINI BASIL
MAKES 5 OR 6 (3-OUNCE) POPS
This pop is inspired by one of my favorite soups, created by famed Chicago chef Grant Achatz of Michelin-starred Alinea. I love how delicate it is; basil is a beautifully sweet, tender herb, and zucchini makes a wonderfully creamy base for its flavor to shine. The acid from the lemon and just a touch of honey bring the whole thing to life. Zucchini is one of my favorite vegetables. It is packed with vitamin C and bone-building manganese, and is rich in fiber and incredibly filling. This is one of my favorite pops to eat when I'm feeling peckish, as the flavor is interesting enough to stave off boredom (which, to be honest, is one of the main reasons I often find myself eating) and satiating enough to eliminate my hunger for hours.
•1¾ cups coarsely chopped zucchini
•¾ cup packed fresh basil leaves
•3 tablespoons honey
•¾ cup full-fat coconut or almond milk
•2 tablespoons fresh lemon juice
1.Put the zucchini in a microwave-safe dish with 1 inch of water, cover, and microwave until fork-tender, 2 to 4 minutes. Alternatively, cook the zucchini in a stovetop steamer for 10 to 15 minutes, or until fork-tender. Let cool until warm to the touch, then transfer to a blender, add the remaining ingredients, and blend until smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CURRIED BUTTERNUT SQUASH
MAKES 5 OR 6 (3-OUNCE) POPS
This pop is like an exotic vacation for your tongue. Curry, a staple in Indian cuisine, brings to life sweet, lightly steamed butternut squash. Although you buy it as a single spice, curry powder refers to a blend that varies by region and producer, but typically includes turmeric, coriander, cumin, fenugreek, and chile peppers. Lightly spicy and richly earthy, it packs a flavor and nutritional punch, with myriad antiaging and anti-inflammatory benefits from its many components. Feel free to buy ready-made curry or make your own, playing around with the various spices until you've found the perfect flavor to suit your taste buds. Either way, this pop will surprise and delight!
•3 cups coarsely chopped peeled butternut squash (about 1 medium butternut squash)
•2 teaspoons curry powder
•⅛ teaspoon salt
•2 tablespoons maple syrup
•¼ cup coconut milk of choice
1.Put the squash in a microwave-safe dish with 1 inch of water, cover, and microwave until very soft, 2 to 4 minutes. Alternatively, cook the squash in a stovetop steamer for 10 to15 minutes, or until very soft. Let cool until warm to the touch, then transfer to a blender, add the remaining ingredients, and blend until smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
SWEET-&-SPICY CORN
MAKES 5 OR 6 (3-OUNCE) POPS
In this pop, the sweetness of the corn plays off the spiciness of the serrano chile, and the lime adds a zesty brightness. Chili powder, isn't actually that spicy, but it adds a beautiful savory flavor here. Different peppers have unique flavors and health benefits, which range from boosting metabolism to boosting immunity. Several politicians are rumored to eat a pepper a day to avoid getting sick on the mega-germ-filled campaign trail!
•1½ cups fresh or frozen corn kernels
•1 cup full-fat coconut milk
•3 tablespoons coconut sugar
•1 serrano chile, halved lengthwise, stemmed, and seeded
•Zest and juice of 1 lime
•Chili powder, for garnish (optional)
1.In a medium pot, bring 3 cups water to a boil over high heat. Add the corn and boil for 2 minutes (if fresh) or 3 minutes (if frozen). Drain, then submerge in ice water to stop the cooking.
2.In the now empty pot, combine the coconut milk, coconut sugar, and chile and bring just to a simmer over medium-low heat. Reduce the heat to low and cook, stirring occasionally, for 15 minutes. Remove from the heat and let cool for 10 minutes, then remove and discard the chile.
3.Transfer the milk mixture to a blender. Drain the corn and add it to the milk mixture along with the lime zest and juice. Blend until smooth.
4.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid. Just before serving, unmold the pops and sprinkle each with chili powder, if desired.
BALSAMIC BEET & ROSEMARY
MAKES 5 OR 6 (3-OUNCE) POPS
This pop always feels so sophisticated to me, the equivalent of a swanky cocktail party on a stick (although the sweetness, and, to be frank, its messiness, are quite appealing to children as well). Beets are often relegated to salads, but they have a wonderfully complex, mildly sweet flavor that works well in everything from root vegetable mash to soups to this very pop. The trick is to roast them: this dulls the "dirty" flavor that beet-haters often react to, and causes the outsides to become beautifully caramelized. Paired with a rosemary balsamic reduction that swirls throughout the pop, the beet is elevated with every bite from humble salad accoutrement to the main star—and oh, what a star it is. Beets get their gorgeous pigment from betalains, a potent and somewhat rare class of antioxidants that have been linked to the support of nervous system health and the prevention and treatment of a number of cancers.
•3 cups coarsely chopped peeled beets (2 or 3 beets)
•2 tablespoons coconut oil, melted
•⅛ teaspoon salt
•1 cup balsamic vinegar
•2 sprigs fresh rosemary
•2 cups water
•2 tablespoons honey
1.Preheat the oven to 375ºf. Toss the beets with the coconut oil and salt, then arrange them in an even layer on a parchment paper–lined baking sheet. Roast for 40 to 50 minutes, or until fork-tender.
2.Meanwhile, in a small saucepan, combine the balsamic vinegar and rosemary and bring just to a simmer over medium-high heat; then reduce the heat to low and simmer, stirring occasionally, until the mixture reduces by half and has a thick, syrupy consistency. Remove from the heat and set aside.
3.Let the beets cool until warm to the touch, then transfer to a blender. Add the water and honey and blend until very smooth.
4.In alternating layers, fill the molds with the beet mixture and balsamic syrup (it's okay if they swirl together—you're looking for a marbled effect). Freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
SWEET PEA & MINT
MAKES 5 OR 6 (3-OUNCE) POPS
Mint and peas go together like, well, two peas in a pod; the mint is incredibly refreshing, balancing and elevating the starchy sweetness of the peas. Peas also offer a ton of health benefits (good to hear, I'm sure, as your mother was likely always making you finish yours). Besides being rich in vitamin K, manganese, fiber, and B vitamins (including folate, which is great for pregnant women), they're also one of the few dietary sources of pisumsaponins I and II and pisumosides A and B, phytonutrients that have been found to help combat type 2 diabetes. Studies also indicate that eating a serving or two of peas daily helps reduce the risk of stomach cancer. While fresh peas do taste minutely brighter, the difference is slight enough that I'll often reach for frozen.
•1½ cups fresh or frozen peas
•¾ cup packed fresh mint leaves
•3 tablespoons honey
•¾ cup full-fat coconut milk
•2 tablespoons fresh lemon juice
1.In a medium pot, bring 3 cups water to a boil over high heat. Add the peas and cook for 2 minutes (if fresh) or 3 minutes (if frozen). Drain and submerge in ice water, then drain again.
2.Transfer the peas to a blender. Add the remaining ingredients and blend until smooth.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CARDAMOM CINNAMON SWEET POTATO
MAKES 5 OR 6 (3-OUNCE) POPS
Kids go nuts for these sweet and creamy pops. Cardamom has a citrusy, spicy-sweet flavor, and is commonly found in Middle Eastern and Scandinavian cooking in both savory dishes and desserts. While it's packed with vitamins and minerals, particularly iron, potassium, and manganese (one teaspoon contains 150 percent of your daily requirement), many of its health benefits are thanks to its essential volatile oils, which work in the body in a more potent, medicinal way. Recent studies have shown that consumption of cardamom reduces the incidence of colorectal cancer, and helps to treat it as well. It also controls hypertension, lowers cholesterol, and aids in digestion. Sweet potatoes are no health slouch either, with each of these pops boasting 100 percent of your recommended daily vitamin A, a potent antioxidant that boosts your immune system and helps with eye health. The only sugar in this recipe comes from the fiber-packed sweet potatoes and dates, making it a great light breakfast or hearty snack.
•1 large sweet potato
•¾ cup canned coconut milk or homemade milk of choice
•1 teaspoon ground cinnamon
•½ teaspoon ground cardamom
•4 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•1 teaspoon pure vanilla extract
1.Preheat the oven to 400ºF. Place the sweet potato on a parchment paper–lined baking sheet and roast for 45 to 60 minutes, or until very soft. Remove and let cool completely.
2.Halve the sweet potato and scoop the flesh into a blender, discarding the skin. Add the remaining ingredients and blend until very smooth.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
Roasting a sweet potato helps caramelize its sugars and bring out its innate sweetness. My following described approach is super simple—I don't even wash the sweet potato first, as I'll be discarding the skin. I love to eat leftover sweet potato mash with a pat of butter or ghee, a dash of vanilla, and a generous shake of cinnamon—it tastes like pumpkin pie and is a great, filling snack.
PUMPKIN PIE
MAKES 5 OR 6 (3-OUNCE) POPS
This pop makes pumpkin pie flavor accessible anytime. Pumpkins are rich in vitamin A—one serving contains more than 200 percent of your recommended daily intake. Vitamin A is important for both skin and eye health, and should be consumed with fat (such as coconut milk) to be absorbed properly. Pumpkins are also rich in potassium, and their fiber will keep you full while helping to cleanse your intestinal tract. Their beta-carotene has been found to reduce bodily inflammation, and protect against heart disease, lung cancer, and skin cancer. Mixed together with the health benefits of the spices, this pop packs quite a health punch.
•¾ cup canned coconut milk or homemade milk of choice
•1 cup fresh or canned pumpkin puree (see Tip)
•5 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•1 teaspoon pure vanilla extract
•¾ teaspoon ground cinnamon
•½ teaspoon ground ginger
•½ teaspoon freshly grated nutmeg
•¼ teaspoon ground allspice
•⅛ teaspoon ground cloves
•⅛ teaspoon salt
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
To make your own pumpkin puree, scoop out a 4- to 6-pound pumpkin, brush with neutral oil, and sprinkle with sea salt. Bake cut-side down on a parchment–lined baking sheet at 400°F. Let cool, then scrape out the flesh and puree until smooth. You should have 2 to 3 cups pumpkin puree. Store in an airtight container in the fridge for up to 1 week or in the freezer for up to 5 months.
THAI PEANUT CILANTRO GINGER
MAKES 5 OR 6 (3-OUNCE) POPS
Based on the sauce used in a traditional Thai satay, this peanut-buttery pop is hearty enough for a light meal and packed with good-for-you ingredients. The cilantro, in addition to adding a citrusy, herby zip, is one of nature's best chelators, renowned for clearing the body of chemicals and heavy metals (accumulated from the environment). Peanuts, which are not actually nuts at all, but rather legumes, are rich in monounsaturated fatty acids like oleic acid, which helps to lower LDL (or "bad") cholesterol and increase HDL (or "good") cholesterol in the blood. The ginger, honey, and coconut milk help these pops straddle the sweet and savory line, while also adding further antibacterial benefits.
•1 cup coconut milk of choice
•3 tablespoons honey
•¼ cup unsalted peanut butter (creamy and crunchy both work well)
•¼ cup packed fresh cilantro leaves
•Zest and juice of 1 lime
•1 (1-inch) piece fresh ginger, peeled and grated
•¼ teaspoon salt (omit if using salted peanut butter)
•¼ cup roasted salted peanuts
1.Blend together all the ingredients except the roasted peanuts until very smooth. Add the peanuts and pulse until well distributed but not completely blended.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
ROASTED CARROT THYME
MAKES 5 OR 6 (3-OUNCE) POPS
This pop is rich enough to be a light meal, but sweet enough to be a real treat. The health benefits of carrots multiply substantially when the vegetable is cooked; plus, roasting them caramelizes their sugars. One of these pops contains more than 100 percent of the recommended daily amount of vitamin A, as well as a host of other vitamins and minerals. Thyme adds a subtle, herbaceous sweetness, but it's also powerfully antibacterial, even in very small doses, making this a wonderful choice to fight off—or prevent—colds.
•1 cup coarsely chopped carrots
•1 teaspoon melted coconut oil
•⅛ teaspoon salt
•1 cup canned coconut milk or homemade milk of choice
•2 tablespoons maple syrup
•1 teaspoon fresh thyme leaves
1.Preheat the oven to 400ºF. Toss the carrots with the coconut oil and salt and arrange them in a single layer on a parchment paper–lined baking sheet. Transfer to the oven and roast for 45 minutes, or until the edges are browned. Remove and let cool for 15 to 20 minutes, until cool to the touch.
2.Transfer the carrots to a blender and add the milk and maple syrup. Blend until very smooth. Add the thyme and pulse until the pieces are small and well distributed but still visible.
3.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
STRAWBERRY SPINACH BASIL
SPICY ARUGULA JALAPEÑO PINEAPPLE
TURMERIC MANGO SUNRISE
MANGO ARUGULA CILANTRO
CINNAMON BERRY
MINT CHOCOLATE CHIP
LEMON GINGER
CHOCOLATE-COVERED STRAWBERRY
THE ULTIMATE GREEN SMOOTHIE
DOUBLE CHOCOLATE BROWNIE
PIÑA COLADA
The pops in this chapter put health at the absolute forefront. While the others forgo simple sugars for mineral-rich honey, maple syrup, or coconut syrup, the pops in this section are 100 percent sweetener-free, deriving their sweetness only from fruits like bananas, dates (see this page), mangoes, and strawberries. Each pop also contains a complete serving of vegetables in the form of leafy greens, which are some of nature's healthiest foods. They are packed with fiber, chlorophyll (which serves as a cell builder), and tons of vitamins and nutrients, detailed in each recipe. These health benefits don't come at the expense of flavor, however. The pops are designed to either highlight the flavor of the greens (in the case of basil and cilantro, they'll knock your socks off), or completely mask them (you'd never know there was more than a cup of spinach packed into the Chocolate-Covered Strawberry pops). They're great for children who might turn their nose up at a salad, but will always say yes to a Mint Chocolate Chip pop.
STRAWBERRY SPINACH BASIL
MAKES 5 OR 6 (3-OUNCE) POPS
Basil is one of my favorite herbs to use in desserts, as it has a lovely sweetness that's brought to life when paired with the naturally occurring sugars found in fruits. Strawberry, in my opinion, is its soulmate: together, they're sweet and just a bit savory, which adds a wonderful depth to the resulting pop. The health benefits of basil come from its flavonoids—which have been found to be cell protective—and volatile oils, which contain estragole, linalool, cineole, eugenol, sabinene, myrcene, and limonene _._ These oils are strongly antibacterial (this pop is, in fact, great to eat if you're recovering from any type of food poisoning), as well as anti-inflammatory, working the same way in the body as over-the-counter NSAIDs. So, the next time you feel a headache coming on, reach for this pop instead of the medicine cabinet—your body and your taste buds will thank you!
•½ cup packed fresh basil leaves
•½ cup packed spinach
•⅔ cup homemade almond or cashew milk (see this page)
•1½ cups hulled fresh or frozen strawberries
•2 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
1.Blend together the basil, spinach, and milk until smooth. Add the strawberries and dates and blend again until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
SPICY ARUGULA JALAPEÑO PINEAPPLE
MAKES 5 OR 6 (3-OUNCE) POPS
Attention, spicy food fans: This pop is for you. A play on a salsa from one of my favorite Mexican restaurants, this pop is a sweet and spicy dream. The pineapple cuts through the fiery jalapeño, and the arugula, which has a peppery, earthy flavor, acts as a spicy complement. These are amazing if you feel yourself coming down with a cold, as one jalapeño contains 18 percent of the recommended daily vitamin C, and its spiciness can help clear mucus from your nose and throat. Like all hot peppers, jalapeños contain capsaicin, which has been shown to help with weight loss, especially in the stubborn belly fat area, in addition to offering protection from cancer and heart disease. Be careful to remove the seeds and all the white membrane from your pepper before blending, though, or your tongue might get a bit more heat than it's ready for!
•Zest and juice of 1 lime
•½ banana
•1 cup arugula
•1 cup fresh or frozen cubed pineapple
•1 jalapeño, stemmed, seeded, and deveined
•¾ cup coconut water
•⅛ teaspoon salt
•½ teaspoon pure vanilla extract
1.Blend together all the ingredients until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
TURMERIC MANGO SUNRISE
MAKES 5 OR 6 (3-OUNCE) POPS
This green smoothie pop is all about getting that glowing skin. Turmeric, one of my absolute favorite superfoods, quells any inflammation in your body, while mango is rich in vitamins A and C. These are _the_ vitamins to look for in topical skin care (retinol, which is derived from vitamin A, is considered the holy grail of antiaging products), and they work similarly inside your body, helping to build collagen, which keeps your skin plump and youthful. I love using citrus zest in summery pops: besides adding a wonderful, lively flavor (these pops taste like a sunny day), they help protect against the type of cell damage that causes skin cancer. So go ahead—grab your sunscreen and your Turmeric Mango Sunrise Glow Pop, and head out to catch some of those luscious rays!
•1½ cups lightly packed spinach
•¾ cup canned coconut milk or homemade milk of choice
•1½ cups fresh or frozen mango
•1 teaspoon ground turmeric
•⅛ teaspoon freshly ground black pepper
•⅛ teaspoon salt
•1 teaspoon pure vanilla extract
•Zest and juice of 1 orange
1.Blend together the spinach and milk until smooth. Add the remaining ingredients and blend until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
MANGO ARUGULA CILANTRO
MAKES 5 OR 6 (3-OUNCE) POPS
While it's often relegated to the role of garnish, I like to use cilantro as a star ingredient. I love its fresh, green flavor and citrusy bite. Combined with arugula's spicy, peppery notes and mellowed by mango's creamy sweetness, this pop is like a beach vacation in your mouth. It's great to include cilantro as a regular part of your diet, because it's one of the best natural chelators, which means that it cleans your blood, ridding it of heavy metal buildup and other toxins we accumulate naturally as part of our modern lifestyles. This is one of my favorite pops to eat after a weekend spent engaging in a bit of debauchery, whether it's eating poorly, drinking too much, or simply not taking care of myself, as it cleans out my system and readies me to start fresh.
•½ cup packed arugula
•½ cup packed fresh cilantro, woody ends removed (see Tip)
•1 cup canned coconut milk or homemade milk of choice
•1½ cups coarsely chopped fresh or frozen mango
•½ banana
•3 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
1.Blend together the arugula, cilantro, and milk until smooth. Add the remaining ingredients and blend until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
When blending cilantro for pops, feel free to leave most of the stem after discarding the woody ends—the stems contain much of the flavor!
CINNAMON BERRY
MAKES 5 OR 6 (3-OUNCE) POPS
This is a pop version of my own personal go-to green smoothie. There are no fancy superfoods or exotic fruits in it, which means I usually have everything on hand. The creamy milk and warm, earthy cinnamon highlight the sweetness in all berries, so you can really play around based on your taste preferences: I like to use a mix of blackberries, raspberries, and blueberries, which are all antiaging, fiber-packed powerhouses, and incredibly delicious. This recipe also includes kale, a leafy green that's gotten quite a lot of press in the last few years. Despite being too trendy for its own good, kale is an amazing choice to include in green pops: a cruciferous vegetable, it supports your body's own detoxification processes.
•1½ cups lightly packed kale leaves
•1⅓ cups canned coconut milk or homemade milk of choice
•2 cups fresh or frozen mixed berries
•½ banana
•2 teaspoons ground cinnamon
•⅛ teaspoon sea salt
1.Blend together the kale and milk until smooth. Add the remaining ingredients and blend until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
MINT CHOCOLATE CHIP
MAKES 5 OR 6 (3-OUNCE) POPS
In every bite of this pop, the silky texture gives way to crunchy bits of cacao that explode into chocolate goodness. This pop gets its flavor from two types of mint: fresh leaves and peppermint extract. Both are incredibly soothing to your stomach (and to your mind as well: on a hot day, a few bites of mint are like an icy gust of fresh air for your insides). Cacao nibs are, essentially, chocolate in its purest form, created by simply roasting and then crushing the beans. They're incredibly dense in the antioxidants and minerals that chocolate is known for. Because you can't taste the spinach in these at all, they're fun pops to give to people who think they don't like "health food," and kids absolutely go wild for them. Be sure to use only the mint leaves, as the stems are very bitter.
•½ cup packed fresh mint leaves
•½ cup packed spinach
•¾ cup homemade cashew milk
•2 bananas
•½ teaspoon peppermint extract (optional; see Tip)
•½ teaspoon pure vanilla extract
•⅛ teaspoon sea salt
•¼ cup raw cacao nibs or dark chocolate chips
1.Blend together the mint, spinach, and milk until smooth. Add the bananas, peppermint extract (if using), vanilla, and salt and blend until very smooth. Add the cacao nibs and pulse just until well distributed but still intact.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
While the peppermint extract will add a stronger, more multilayered mint flavor, these pops are absolutely delicious without it, so feel free to omit it if you don't have it on hand.
LEMON GINGER
MAKES 5 OR 6 (3-OUNCE) POPS
Ginger is one of my favorite flavors to pair with lemon; I'll often make a ginger-lemon tea when I'm feeling under the weather, and ginger lemonade is a summer staple for me. There's a reason the combination feels so good when you're sick: ginger is one of the most powerful anti-inflammatory foods around. It quells nausea and indigestion and helps with arthritis, post-workout pain, and other inflammation-related conditions, while lemons offer a powerful dose of immunity-boosting vitamin C. This pop utilizes the nutrient- and flavor-rich zest of the lemon, but be sure to avoid the white membrane underneath the bright-yellow, citrus oil–packed exterior, as it's incredibly bitter. I like to use mild-tasting spinach in this pop to really let the ginger and lemon shine, while still getting all the benefits of the leafy green's digestion-boosting fiber and skin-loving vitamins C and K.
•2 cups lightly packed spinach
•1 cup coconut milk of choice
•Zest and juice of 2 lemons
•1 ripe banana
•1 (1-inch) piece fresh ginger, peeled and grated
1.Blend together the spinach and milk until smooth. Add the remaining ingredients and blend until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
CHOCOLATE-COVERED STRAWBERRY
MAKES 5 OR 6 (3-OUNCE) POPS
Chocolate-covered strawberries might be my favorite dessert. The rich, velvety chocolate so perfectly sets off the delicate, fruity sweetness of the berries. Raw cacao, prized by the Aztecs as the bean of the gods, is one of the highest known natural sources of magnesium, copper, zinc, iron, chromium, and manganese—all minerals in which many people are deficient. In addition, raw cacao contains a unique alkaloid chemical called theobromine, a known mood booster, and provides a mild stimulatory effect without the buzzy feeling of caffeine. Spinach is the perfect fiber-rich base for this pop, as its mild flavor lets the decadent strawberry-chocolate goodness shine through.
•1½ cups lightly packed spinach
•1 cup canned coconut milk or homemade milk of choice
•1½ cups hulled fresh or frozen strawberries
•5 tablespoons raw cacao powder
•½ banana
•⅛ teaspoon salt
•1 teaspoon pure vanilla extract
•3 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
1.Blend together the spinach and milk until smooth. Add the remaining ingredients and blend until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
THE ULTIMATE GREEN SMOOTHIE
MAKES 5 OR 6 (3-OUNCE) POPS
If you've ever bought a bottled green juice at your local juice bar or grocery store, you'll have a good idea of what this pop tastes like. That classic smoothie gets quite a bit of its flavor from apple, with the greens brightening and freshening the overall taste. I love this one for breakfast or as an afternoon pick-me-up. The greens bulk it out with fiber to keep you full and satiated, while the apple and banana give sweetness to add extra bounce to your day. This is a great green smoothie pop to get started with, as you likely have most—if not all—of the ingredients in your kitchen right now!
•1 cup lightly packed mixed greens, like romaine, kale, and spinach
•1 cup canned coconut milk or homemade milk of choice
•1 ripe banana
•1 medium apple, cored and cut into cubes
•⅛ teaspoon sea salt
1.Blend together the greens and milk until smooth. Add the remaining ingredients and blend until very smooth.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
I prefer to use Granny Smith apples in this smoothie, due to their low sugar content and tart, delicious flavor, but any type of apple will work well.
DOUBLE CHOCOLATE BROWNIE
MAKES 5 OR 6 (3-OUNCE) POPS
If you're one of those people who likes your chocolate with a side of chocolate, this pop is for you. The greens-filled base is like a healthy frozen brownie batter. You know all those studies about chocolate's health benefits? Raw cacao is the kind they're talking about, as many of the vitamins and minerals are lost during roasting and processing. This pop contains cacao in two forms: the powder is blended with creamy, sweet bananas and dates to make a brownie batter base, which is then studded with crunchy cacao nibs. The avocado and spinach (the flavors of which are overpowered completely by all that chocolate deliciousness) add enough good fat and fiber to make this an awesome breakfast—filling enough to keep you going through lunch, and tasty enough to make you feel like you're sneaking in a decadent dessert.
•1 cup lightly packed spinach
•1⅓ cups canned coconut milk or homemade milk of choice
•5 tablespoons raw cacao powder
•2 bananas
•½ avocado
•⅛ teaspoon salt
•1 teaspoon pure vanilla extract
•3 Medjool dates, pitted, soaked in boiling water for 10 minutes, and drained
•3 tablespoons cacao nibs
1.Blend together the spinach and milk until smooth. Add the remaining ingredients except the cacao nibs and blend until very smooth. Add the cacao nibs and pulse until well distributed but still intact.
2.Pour the mixture into pop molds and freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
PIÑA COLADA
MAKES 5 OR 6 (3-OUNCE) POPS
Is there any drink that screams fun more than a piña colada? A virgin version of the tropical cocktail works perfectly as a green smoothie pop: the sweet pineapple and creamy coconut beautifully mask the flavor of the spinach (a neutral mixed green blend would also work well here). Pineapple is the only food source of an enzyme called bromelain, which helps you digest and maximize nutrient absorption from your food—and it's so potent that it's often sold in supplement form. Am I saying you should start every meal with a piña colada Glow Pop? Well...let's just say I'm not saying you _shouldn't._
•1½ cups lightly packed spinach
•1 cup coconut milk of choice
•1½ cups cubed fresh or frozen pineapple
•½ banana
•⅛ teaspoon salt
•1 teaspoon pure vanilla extract
•3 tablespoons unsweetened flaked coconut (optional)
1.Blend together the spinach and milk until smooth. Add the remaining ingredients except the coconut flakes and blend until very smooth.
2.Pour the mixture into pop molds and sprinkle with the coconut flakes, if desired. Freeze for 1 hour, then insert sticks and freeze for at least 4 hours more, or until solid.
APPENDIX
CANCER FIGHTING
Apple Pie
Mango Chile
Rosemary Strawberry
Honeyed Peach Thyme
Lavender Blueberry
Blackberry Rose
Coconut Chai
Cinnamon Orange & Cream
Cookie Dough
Blueberry & Cream
Matcha Latte
Turmeric Golden Milk
Lavender London Fog
Mexican Hot Chocolate
Chocolate Fudge
Chocolate-Covered Banana
Chocolate Hazelnut
Cold-Brew Mocha
Chocolate Orange
Chocolate Chia Lavender
Olive Oil Chocolate Rosemary
Tomato Beet Bloody Mary
Avocado Chile Lime
Curried Butternut Squash
Balsamic Beet & Rosemary
Cardamom Cinnamon Sweet Potato
Pumpkin Pie
Roasted Carrot Thyme
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
Chocolate-Covered Strawberry
The Ultimate Green Smoothie
Double Chocolate Brownie
Piña Colada
HEART HEALTHY
Apple Pie
Watermelon Lime
Mango Chile
Rosemary Strawberry
Pink Lemonade
Lavender Blueberry
Mexican Horchata
Matcha Latte
Lavender London Fog
White Chocolate Chia Strawberry
Chocolate-Covered Banana
Chocolate Hazelnut
Cold-Brew Mocha
Neapolitan
Chocolate Orange
Olive Oil Chocolate Rosemary
Tomato Beet Bloody Mary
Avocado Chile Lime
Curried Butternut Squash
Balsamic Beet & Rosemary
Cardamom Cinnamon Sweet Potato
Pumpkin Pie
Thai Peanut Cilantro Ginger
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
The Ultimate Green Smoothie
Double Chocolate Brownie
BRAIN BOOSTING
Mango Chile
Lavender Blueberry
Blueberry & Cream
Matcha Latte
Turmeric Golden Milk
Cold-Brew Mocha
Chocolate Orange
Olive Oil Chocolate Rosemary
Avocado Chile Lime
Balsamic Beet & Rosemary
Pumpkin Pie
Strawberry Spinach Basil
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Chocolate-Covered Strawberry
The Ultimate Green Smoothie
Double Chocolate Brownie
BLOAT REDUCING
Cucumber Mint Mojito
Watermelon Lime
Chamomile Cantaloupe Mint
Pink Lemonade
Lavender Blueberry
Caramelized Pineapple
Strawberry Cardamom Rose Lassi
Chocolate-Covered Banana
Neapolitan
Avocado Chile Lime
Sweet Pea & Mint
Pumpkin Pie
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
The Ultimate Green Smoothie
Double Chocolate Brownie
Piña Colada
GOOD FOR THE GUT
Watermelon Lime
Chamomile Cantaloupe Mint
Pink Lemonade
Lavender Blueberry
Caramelized Pineapple
Cinnamon Orange & Cream
Strawberry Cardamom Rose Lassi
Peanut Butter & Jelly
White Chocolate Chia Strawberry
Chocolate Fudge
Neapolitan
Chocolate Chia Lavender
Chocolate Caramel Swirl
Tomato Beet Bloody Mary
Avocado Chile Lime
Curried Butternut Squash
Sweet Pea & Mint
Cardamom Cinnamon Sweet Potato
Pumpkin Pie
Thai Peanut Cilantro Ginger
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
Chocolate-Covered Strawberry
The Ultimate Green Smoothie
Double Chocolate Brownie
Piña Colada
DETOXIFYING
Chamomile Cantaloupe Mint
Mango Chile
Pink Lemonade
Lavender Blueberry
Caramelized Pineapple
Coconut Chai
Turmeric Golden Milk
Olive Oil Chocolate Rosemary
Tomato Beet Bloody Mary
Avocado Chile Lime
Curried Butternut Squash
Cardamom Cinnamon Sweet Potato
Thai Peanut Cilantro Ginger
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
The Ultimate Green Smoothie
Piña Colada
PROMOTES WEIGHT LOSS
Apple Pie
Chamomile Cantaloupe Mint
Mango Chile
Coconut Chai
Cinnamon Orange & Cream
Strawberry Cardamom Rose Lassi
Mexican Horchata
Neapolitan
Avocado Chile Lime
Curried Butternut Squash
Sweet-&-Spicy Corn
Cardamom Cinnamon Sweet Potato
Pumpkin Pie
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
Chocolate-Covered Strawberry
The Ultimate Green Smoothie
Double Chocolate Brownie
Piña Colada
GLOWING SKIN
Cucumber Mint Mojito
Watermelon Lime
Chamomile Cantaloupe Mint
Mango Chile
Rosemary Strawberry
Pink Lemonade
Lavender Blueberry
Blackberry Rose
Cinnamon Orange & Cream
Strawberry Cardamom Rose Lassi
Cookie Dough
Turmeric Golden Milk
Chocolate Fudge
Chocolate-Covered Banana
Chocolate Hazelnut
Neapolitan
Chocolate Orange
Peanut Butter Cup
Olive Oil Chocolate Rosemary
Avocado Chile Lime
Cardamom Cinnamon Sweet Potato
Pumpkin Pie
Roasted Carrot Thyme
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
Chocolate-Covered Strawberry
The Ultimate Green Smoothie
Double Chocolate Brownie
Piña Colada
CALMING
Chamomile Cantaloupe Mint
Lavender Blueberry
Blackberry Rose
Strawberry Cardamom Rose Lassi
Lavender London Fog
Chocolate Fudge
Chocolate Orange
Chocolate Chia Lavender
Mint Chocolate Chip
Lemon Ginger
Chocolate-Covered Strawberry
Double Chocolate Brownie
ENERGIZING
Mango Chile
Coconut Chai
Cinnamon Orange & Cream
Matcha Latte
Turmeric Golden Milk
Lavender London Fog
White Chocolate Chia Strawberry
Mexican Hot Chocolate
Chocolate Fudge
Chocolate-Covered Banana
Chocolate Hazelnut
Cold-Brew Mocha
Neapolitan
Chocolate Orange
Chocolate Chia Lavender
Chocolate Caramel Swirl
Olive Oil Chocolate Rosemary
Chocolate-Covered Strawberry
Double Chocolate Brownie
MEAL REPLACING
Peanut Butter & Jelly
White Chocolate Chia Strawberry
Chocolate Fudge
Neapolitan
Peanut Butter Cup
Chocolate Chia Lavender
Chocolate Caramel Swirl
Avocado Chile Lime
Zucchini Basil
Curried Butternut Squash
Sweet Pea & Mint
Cardamom Cinnamon Sweet Potato
Pumpkin Pie
Thai Peanut Cilantro Ginger
Roasted Carrot Thyme
Strawberry Spinach Basil
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
Chocolate-Covered Strawberry
The Ultimate Green Smoothie
Double Chocolate Brownie
Piña Colada
PAIN RELIEVING
Watermelon Lime
Mango Chile
Coconut Chai
Turmeric Golden Milk
Mexican Hot Chocolate
Tomato Beet Bloody Mary
Curried Butternut Squash
Sweet-&-Spicy Corn
Pumpkin Pie
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
Piña Colada
ALLERGY RELIEVING
Honeyed Peach Thyme
Coconut Chai
Turmeric Golden Milk
Olive Oil Chocolate Rosemary
Zucchini Basil
Curried Butternut Squash
Sweet Pea & Mint
Roasted Carrot Thyme
Turmeric Mango Sunrise
Lemon Ginger
IMMUNE BOOSTING
Apple Pie
Watermelon Lime
Mango Chile
Rosemary Strawberry
Honeyed Peach Thyme
Lavender Blueberry
Blackberry Rose
Coconut Chai
Cinnamon Orange & Cream
Blueberry & Cream
Matcha Latte
Turmeric Golden Milk
Mexican Hot Chocolate
Neapolitan
Olive Oil Chocolate Rosemary
Tomato Beet Bloody Mary
Curried Butternut Squash
Sweet-&-Spicy Corn
Cardamom Cinnamon Sweet Potato
Pumpkin Pie
Thai Peanut Cilantro Ginger
Roasted Carrot Thyme
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Turmeric Mango Sunrise
Mango Arugula Cilantro
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
Chocolate-Covered Strawberry
The Ultimate Green Smoothie
Double Chocolate Brownie
Piña Colada
HYDRATING
Cucumber Mint Mojito
Watermelon Lime
Chamomile Cantaloupe Mint
Pink Lemonade
Lavender Blueberry
Caramelized Pineapple
Peanut Butter & Jelly
Neapolitan
Tomato Beet Bloody Mary
Avocado Chile Lime
Strawberry Spinach Basil
Spicy Arugula Jalapeño Pineapple
Cinnamon Berry
Mint Chocolate Chip
Lemon Ginger
The Ultimate Green Smoothie
Piña Colada
GLOW GLOSSARY
Almonds: Almonds are rich in skin-smoothing vitamin E, bone-building and heart-supporting calcium, and biotin, which is great for your hair, skin, and nails. They're also packed with protein and heart-healthy fats.
Apples: Apple contain quercetin, a potent antioxidant that may help protect against heart disease and cancer, in addition to having antihistamine and anti-inflammatory effects. They also boast a ton of soluble fiber, which helps cleanse your intestines and leaves you feeling slim and detoxified.
Arugula: Arugula contains high levels of blood-pressure-lowering nitrates, along with sulfuric compounds that have been found to lower the risk of many types of cancer. Like all leafy greens, it's rich in cleansing, filling fiber and vitamin K.
Avocado: Avocado is a superstar for both health and beauty, prized for its good fats and antioxidants that will keep you full, quell inflammation, and make your skin absolutely glow.
Bananas: Bananas are rich in potassium, fiber, and skin-brightening vitamin C, which boosts both youth-enriching collagen and general immunity.
Basil: The benefits of basil come from its flavonoids—which have been found to be cell-protective—and volatile oils, including estragole, linalool, cineole, eugenol, sabinene, myrcene, and limonene. These oils are strongly antibacterial as well as anti-inflammatory, and work the same way in the body as over-the-counter NSAIDs.
Beet: Beets get their pigment from betalains, a potent and somewhat rare class of antioxidants that has been linked to supporting nervous system health and prevention and treatment of a number of cancers. They're also high in immune-boosting vitamin C and, if consumed before exercising, have been found to enhance both energy and stamina.
Black Pepper: Black pepper contains piperine, a compound that dramatically increases the body's absorption of other nutrients.
Black Tea: Black tea contains two types of antioxidants, catechins and polyphenols, which have been found to ward off certain types of cancer, reduce the risk of stroke and type 2 diabetes, and defend against heart disease.
Blackberry: Just one serving of blackberries has more than a third of your daily dose of vitamin C, and both soluble and insoluble fiber, which clean out your intestines. These are also rich in phytochemicals that protect against cancer, aging, and inflammation.
Blueberry: Blueberries have the highest antioxidant capacity of any commonly consumed fruit or vegetable, resulting in protection against heart disease and cell damage. They're also shown to help increase memory function and prevent cognitive decline, and their compounds are being studied as a potential Alzheimer's treatment. Similar to cranberries, they're also effective in preventing and fighting UTIs.
Butternut Squash: Butternut squash gets its color from carotenoids, the same powerful antioxidants found in carrots and sweet potatoes that help enhance eye health and protect against heart disease. One cup of butternut squash contains half your daily recommended intake of vitamin C, which is wonderful for both glowing skin and boosting the immune system. It's also rich in fiber, potassium, and vitamin B6.
Cacao (Nibs and Powder): Cacao is one of the greatest known natural sources of magnesium, copper, zinc, iron, chromium, and manganese—all minerals that many people are deficient in. In addition, raw cacao contains a unique alkaloid chemical called theobromine, which provides a mild stimulatory effect without the buzzy feeling of caffeine.
Cacao Butter, Raw: Cacao butter is the pressed oil of the cacao bean. Like cacao powder and nibs, cacao butter is rich in antioxidants; oleic acid, which has been shown to reduce the risk of heart disease; and theobromine, which raises energy and alertness levels.
Cantaloupe: One cup of cantaloupe has 75 percent of your recommended daily intake of vitamin C, which helps build skin collagen and boost immunity. It's also rich in vitamin A and potassium, and, like all melons, is incredibly hydrating.
Cardamom: While it's packed with vitamins and minerals—particularly iron, potassium, and manganese (one teaspoon contains 150 percent of your daily requirement)—many of cardamom's health benefits come from its essential volatile oils, which work in the body in a more potent, medicinal way. Recent studies have shown that consumption of cardamom reduces incidents of colorectal cancer and helps to treat it as well. It also controls hypertension, lowers cholesterol, and aids in digestion.
Cashews: Cashews contain high levels of lutein and zeaxanthin, antioxidants that particularly benefit eye health. They also contain arginine, an amino acid that helps with heart health, and have been shown in studies to reduce the risk of type 2 diabetes.
Cayenne: Cayenne breaks up mucus, making it ideal to consume when you have a cold or the flu, and it speeds up your metabolism. It also turns on your body's natural pain management system—so the next time you hurt yourself, eat some cayenne instead of popping a pain killer!
Chamomile: Chamomile tea is made from dried chamomile flower buds. It's caffeine-free and known for calming each and every part of your body, from muscle spasms and menstrual cramps to insomnia and anxiety.
Chile Peppers: Chile peppers contain capsaicin, a potent anti-inflammatory and metabolism-boosting compound. They've been shown to reduce LDL ("bad") cholesterol and lower the risk of strokes and heart attacks.
Cilantro: While cilantro is rich in vitamins A, C, and K, it's more notably one of nature's best chelators, renowned for clearing chemicals and heavy metals (accumulated from the environment) out of the body.
Cinnamon: Cinnamon is a wonder spice. Numerous studies point to its ability to stabilize blood sugar, lower LDL ("bad") cholesterol, fight fungal and bacterial infections, and even protect against cancer.
Cloves: Cloves contain a compound called eugenol, which is an anesthetic and antiseptic. In the Middle Ages, cloves were regularly used to numb toothache pain! They are great for increasing gut secretions, which helps relieve indigestion and constipation.
Coconut (Milk and Oil): The type of fat found in coconut is mostly in the form of medium-chain saturated fatty acids (MCFAs) and, in particular, lauric acid. Unlike saturated fats from animal sources, MCFAs are almost immediately converted into energy by the body, and are unlikely to be stored as fat. Lauric acid is also considered one of nature's most potent superfoods (outside of coconut, breast milk is one of the few places it's found in nature, which proves its importance in human health and development), and is renowned for its antiviral, antibacterial, and antifungal properties.
Coconut Sugar: Coconut sugar is made from evaporating the water out of the sap of the coconut palm. It retains many of the palm's nutrients, and is rich in minerals like iron, zinc, calcium, and potassium. It also contains inulin fiber, which accounts for its low ranking on the glycemic index.
Coffee: Coffee is rich in antioxidants, magnesium, and chromium, and has been found in studies to reduce the risk of type 2 diabetes and Alzheimer's.
Cucumber: Cucumber has the highest water content of any food, and contains an antioxidant called quercetin, found to be anti-inflammatory and bloat reducing.
Curry Powder: While purchased as a single spice, curry powder refers to a blend that varies by region and producer, but typically includes turmeric, coriander, cumin, fenugreek, and chile peppers. Together or alone, these spices are powerful anti-inflammatories and help boost your overall immunity.
Dates: Dates are rich in soluble fiber, which helps keep your blood sugar steady and cleanses your intestines. They're also rich in many minerals, including selenium, manganese, copper, and magnesium, all of which help build strong bones, teeth, and nails.
Fennel: Fennel contains a phytonutrient called anethole, which has been found to reduce inflammation and protect against cancer. It has long been known as a stomach soother, and its high vitamin C levels increase immunity while making skin glow.
Garlic: Garlic is one of the most powerful antiviral and antibacterial foods in existence—studies have found it to be more effective against many bacteria than commonly prescribed antibiotics!
Ginger: Ginger is one of the most powerful anti-inflammatory foods around. It quells nausea and indigestion and helps with arthritis, post-workout pain, and other inflammation-related conditions.
Hazelnuts: Hazelnuts boast the most folate of any tree nut. Folate is famous for helping pregnant moms nurture a healthy fetus, but it has a host of other benefits, including reducing the risk of heart attack and stroke by helping metabolize excess homocysteine in your system. Hazelnuts also contain high quantities of skin- and hair-beautifying manganese and copper.
Honey: Honey has been found to be as effective for cough suppression as over-the-counter remedies. Consuming local honey can even help fight allergies by providing microdoses of local pollen consumed by the bees; when you ingest these tiny amounts of the pollen, your immune system can develop defenses against the allergens without being thrown into overdrive. Honey's antibacterial properties ensure that it's a powerful natural antibiotic, both internally and topically.
Jalapeño: One jalapeño contains 18 percent of your recommended daily dose of vitamin C, and its spiciness can help clear mucus from your nose and throat. Like all hot peppers, jalepeños also contain capsaicin, which has been shown to help with weight loss, especially in the stubborn belly fat area, and offer protection from cancer and heart disease.
Kale: Kale is a cruciferous vegetable in the same family as arugula, Brussels sprouts, and cabbage. All these vegetables contain sulfuric compounds that support your body's detoxification system, helping it eliminate any unwanted toxins. Its dark green color is evidence of high levels of cell-nourishing chlorophyll; plus, it has ample amounts of blood- and bone-aiding vitamin K.
Lavender: Lavender has long been renowned for its ability to soothe both body and mind, and is used as a remedy for anxiety, depression, insomnia, and restlessness, in addition to stomachaches and pains and indigestion.
Lemon: Lemon juice is loaded with vitamin C—one cup contains almost double your daily needs! It helps shift your digestive system into high gear (great if you're ever constipated!), and aids your liver in its own natural detoxification processes. The zest of lemons and other citrus fruit has also been found to be protective against skin cancer.
Lime: The nutrient profile of limes is fairly similar to that of lemons. They contain high amounts of vitamin C, and their acidity matches that of your stomach and gets your digestion going. Lime zest also helps protect against skin cancer.
Mango: Mangoes contain zeaxanthin, an antioxidant that promotes eye health and aids vision. They're also high in vitamins A and C. These are the vitamins to look for in topical skin care (retinol, which is derived from vitamin A, is considered the holy grail of antiaging products). They work similarly inside your body, helping to build the collagen that keeps your skin plump and youthful.
Maple Syrup: Because maple syrup is essentially the sap from the maple tree, all its vitamins and minerals are still intact (unlike processed cane sugar). Just one tablespoon contains one-third of your daily recommended dose of manganese, which helps build healthy skin and bones, and twenty-four free radical-reducing antioxidants. It also has a lower glycemic index than cane sugar, and will help maintain steady blood sugar.
Matcha: Matcha has 137 more antioxidants than regularly brewed green tea (which is already a health superstar in itself!). One of these antioxidants is EGCG, which has been found in numerous studies to have potent cancer-fighting properties.
Mint: The menthol oil found in mint is a much-lauded stomach soother—it's wonderful to cleanse the palate, aid digestion, and fight off nausea. It also helps clear respiratory channels and is a mild natural stimulant, a great way to help wake up without the aid of caffeine. Because of its digestive enzymes and stimulant properties, it can also be an effective weight-loss aid.
Orange: Oranges are vitamin C powerhouses, making your skin glow and helping keep your immune system in top-notch shape. Like lemons and limes, orange zest has been found to be protective against skin cancer.
Peanut Butter: Peanuts, which are not actually nuts but rather legumes, are rich in monounsaturated fatty acids like oleic acid, which help to lower LDL ("bad") cholesterol and increase HDL ("good") cholesterol in the blood. They also pack a protein punch.
Peas: Besides being rich in vitamin K (which reduces your risk of stroke and heart disease), bone-building manganese, filling fiber, and B vitamins (including folate, which is great for pregnant women), peas are one of the few dietary sources of pisumsaponins I and II and pisumosides A and B, phytonutrients that have been found to help combat type 2 diabetes. Studies also indicate that eating a serving or two of peas every day helps reduce the risk of stomach cancer.
Pineapple: Pineapple is the only food source of bromelain, an enzyme that helps you digest and maximize nutrient absorption from your food—it's so potent that it's often sold in supplement form.
Pumpkin: As you might have guessed from their bright orange color, pumpkins are rich in vitamin A—one serving actually contains more than 200 percent of the recommended daily intake. Vitamin A is important for both skin and eye health, and is best consumed with fat (such as coconut milk) to be absorbed properly. Pumpkins are even richer in the heart-healthy electrolyte potassium than bananas, and their fiber will keep you full for hours while also helping to cleanse your intestinal tract. Beta-carotene, an antioxidant abundant in pumpkins, has been found to reduce bodily inflammation and protect against heart disease, lung cancer, and skin cancer.
Raspberries: Raspberries are one of the most fiber-filled fruits, meaning they'll help keep you regular and eliminate bloating. They also contain phytonutrients called raspberry ketones, which increase fat burning and aid liver function. They are highly anti-inflammatory, and have been found to have cancer-preventing properties.
Rose Water: Rose water, made by steam-distilling fresh rose petals, is rich in free radical–annihilating antioxidants and skin-brightening vitamins A and C. It is also a mild relaxant and mood balancer; rose essential oil has been used for centuries for its calming properties—even just sniffing it is effective.
Rosemary: Rosemary is one of the main herbs consumed by residents of certain areas in the Greek islands, where people are among the longest living in the world. Rosemary is powerfully anti-inflammatory, and has been found to reduce both the severity and the frequency of asthma attacks.
Sea Salt: While typical table salt contains only sodium chloride, sea salt and Himalayan salt retain all their nutrients, including potassium, iron, and zinc. Consuming sea salt is a wonderful way to rehydrate yourself and replenish lost electrolytes post-workout.
Spinach: Spinach is rich in vitamin K (one serving contains 1,000 percent of your recommended daily intake) and vitamin A, both of which nourish your cells and skin. As made famous by Popeye, it also contains high amounts of iron, which helps give you energy. Plus, it's packed with fiber, which helps clean out your intestines.
Strawberries: Strawberries boast high quantities of vitamin C, which builds immunity in your body and collagen in your skin. Mashed and mixed with baking soda, they'll even whiten your teeth.
Sweet Potato: The bright orange color of sweet potatoes is a good tip-off that they contain large amounts of vitamin A, a potent antioxidant that boosts your immune system and helps with eye health.
Thyme: Thyme is antiviral and antibacterial, and has been used for centuries to ward off everything from simple coughs to the plague.
Turmeric: Turmeric is prized for its anti-inflammatory benefits—it has been shown in numerous studies to match or even outperform over-the-counter NSAIDs. Turmeric can also be effective in treating diseases like IBS, rheumatoid arthritis, cystic fibrosis, and even cancer.
Watermelon: Watermelons are 91 percent water and, therefore, are super hydrating. That other 9 percent packs a heavy nutrition punch as well—watermelons contain more lycopene, an antioxidant known for ramping up skin protection against the sun, than tomatoes, and an amino acid called L-citrulline that's been found to relieve muscle pain.
Yogurt: Yogurt provides a hefty dose of probiotics, which have been linked to clearer skin, reduced anxiety, better digestion, and weight loss.
Zucchini: Zucchini are packed with collagen- and immunity-boosting vitamin C and bone-building manganese. They're also rich in fiber, which fills you up and helps get your digestive system moving.
ACKNOWLEDGMENTS
There are so many people who have supported me throughout my life, and throughout this book. I feel so lucky to have known each of you.
To Zack, my partner in everything. It feels silly to thank you for this book, so much of it was your doing as much as mine. I would list everything you've done for it and for me but, quite simply, it is everything. You're a husband in a way that surpasses my expectations for the word. I can't imagine this book without you—I can't imagine my life without you—and I'm so thankful to have you by my side. I hope all of the taste testing hasn't ruined pops for you forever.
To my dad, my indefatigable recipe tester, who has believed in me doing great things with my life even when there was absolutely no evidence. To have someone so sure that your life path is the right one, even when you're not, is an amazing gift. Plus, every time you actually made one of my recipes, I got a little zing of excitement.
To my mom, who has always taken such fervent interest in what I'm doing; who always likes all of my photos; who will probably win for "most copies of this cookbook on her shelf." It's so wonderful to have you as my cheerleader.
Susan and Leslie: this book is a direct result of your generosity. _Sprouted Routes_ is a direct result of your generosity. You were the best roommates a girl could ask for, the best parents-in-law a wife could dream of, and the best pop testers a chef could hope for. Thank you for always believing in me, and for taste testing Glow Pops before there even _were_ Glow Pops.
To Uncle Kev and Aunt Joan, Steve and Aunt Val: you're the closest thing to parents without being parents that exist in my life. I love you bunches, and so appreciate your support over the years.
Thank you, Kim and Jane, for testing so many of these recipes, time and time again. Reader, if you think these pops are delicious, these women are the ones you want to applaud (if you think they're not great, well—we can talk about that some other time). I don't know why you were so gracious with your time, but it's so appreciated.
Alia, my wonderful agent: there would be no Glow Pops without you. From coming up with the best name in history to guiding me through the crazy world of publishing, I couldn't have done this without you—and I'm so glad I didn't have to. You're truly the crème de la crème of agents.
Amanda: you stood out from every other editor I met from the beginning. I hesitate to use the word _spitfire_ , but it's truly what you are. You're a fighter, and you're brilliant, and you get me and Glow Pops in a way I'm not sure I even deserve. You're the reason I came to Potter, and I'm so glad I did. Danielle, you've been so wonderful to work with, and I feel so lucky to have you on my team.
To the rest of the team at Potter, Natasha Martin, Carly Gorga, Cathy Hennessy, and Kim Tyner—you guys are all so wonderful. I'm in awe of how beautiful _Glow Pops_ is, and it truly has taken a village to take this book from my brain into the hands of people all over the United States. The Potter village is clearly the best one, and I appreciate you all letting me hang out in it for a little while.
To my photo shoot crew: Maeve Sheridan, with her sweet smile and spectacular surfaces; Mariana Velasquez, for making everything look so stunningly beautiful (and artfully messy!), even when working with such tricky subject matter. To the incomparable Lauren Volo, a photographer so much better than I could've even hoped for. These photos are stunning, and it's because of you. I'm honored to have my recipes share these pages with your beautiful images. And to Stephanie Huntwork, the eagle-eyed overseer of it all: your vision so elevated _Glow Pops,_ taking it from what could've been a simple little book about ice pops, to a beautiful work of art. Thank you for seeing what it could be, and guiding it to become that.
To my mindbodygreen family, for making our Dumbo office the best place to work (and making my health-nut tendencies seem normal!). To Bella, who helps me write by laying her head on my keyboard. To Bri, for her crazy late night ideas and willingness to talk about pops long after others would be bored. To all of my wonderful family and friends, who never judge my crazy pursuits, and always find the patience to support me and support me, and support me. I'm so lucky to know all of you.
And last but definitely not least—infinite gratitude to all of my wonderful _Sprouted Routes_ readers, who send me photos of my recipes, who share their stories, and who keep me going, even when I want to toss my mixing bowl across the room. You let me do what I love, and share what I'm passionate about. Thank you, thank you, thank you to each and every one of you.
ABCDEFGHIJKLMNOPQRSTUVWXYZ
INDEX
Note: References in _italics_ indicate recipe photographs.
A
Açai powder
Almond(s)
health benefits
Milk, _fm.1_ , fm.2
Apple(s)
health benefits
Pie
The Ultimate Green Smoothie, 5.1, _5.2_
Arugula
health benefits
Jalapeño Pineapple, Spicy, _5.1_ , 5.2
Mango Cilantro
Avocado(s)
Chile Lime, _4.1_ , 4.2
Chocolate Fudge
Double Chocolate Brownie
health benefits
healthy fats in
B
Balsamic Beet & Rosemary
Banana(s)
Chocolate-Covered, 3.1, _3.2_
Double Chocolate Brownie
health benefits
Mint Chocolate Chip, _5.1_ , 5.2
Basil
health benefits
Strawberry Spinach
Zucchini
Beet(s)
health benefits
& Rosemary, Balsamic
Tomato Bloody Mary
Berry, Cinnamon
Blackberry(ies)
health benefits
Rose
Black pepper
Blenders
Blueberry
& Cream, _2.1_ , 2.2
health benefits
Lavender, 1.1, _1.2_
C
Cacao butter
about
health benefits
White Chocolate Chia Strawberry, _2.1_ , 2.2
Cacao nibs
about
Cookie Dough
Double Chocolate Brownie
health benefits
Mint Chocolate Chip, _5.1_ , 5.2
Cacao powder
about
Chocolate Caramel Swirl, 3.1, _3.2_
Chocolate Chia Lavender
Chocolate-Covered Strawberry
Chocolate Fudge
Chocolate Hazelnut
Chocolate Orange
Cold-Brew Mocha
Double Chocolate Brownie
health benefits
Mexican Hot Chocolate
Neapolitan, 3.1, _3.2_
Olive Oil Chocolate Rosemary, 3.1, _3.2_
Camu camu powder
Cantaloupe
Chamomile Mint, _1.1_ , 1.2
health benefits
Cardamom
Cinnamon Sweet Potato, #104.1, _4.1_
Coconut Chai
health benefits
Strawberry Rose Lassi
Carrot, Roasted, Thyme, _4.1_ , 4.2
Cashew(s)
Cookie Dough
health benefits
Milk, _fm.1_ , fm.2
Cayenne
Chamomile
Cantaloupe Mint, _1.1_ , 1.2
health benefits
Chia seeds
about
Chocolate Caramel Swirl, 3.1, _3.2_
Chocolate Chia Lavender
Neapolitan, 3.1, _3.2_
Peanut Butter & Jelly, 2.1, _2.2_
White Chocolate Chia Strawberry, _2.1_ , 2.2
Chile/chili powder
Avocado Chile Lime, _4.1_ , 4.2
Mango Chile
Sweet-&-Spicy Corn, 4.1, _4.2_
Chile peppers
Chocolate. _See also_ Cacao butter; Cacao nibs; Cacao powder
Chip, Mint, _5.1_ , 5.2
-Covered Banana, 3.1, _3.2_
Peanut Butter Cup
replacing cacao nibs with
Cilantro
health benefits
Mango Arugula
Thai Peanut Ginger
Cinnamon
Berry
Cardamom Sweet Potato, #104.1, _4.1_
Coconut Chai
health benefits
Mexican Horchata
_Mexican Hot Chocolate_
Orange & Cream
Cloves
Cocoa butter
Cocoa powder
Coconut
Chai
healthy fats in
oil, health benefits
sugar, glycemic index
sugar, health benefits
syrup, about, fm.1, fm.2
Syrup, Homemade
Coconut Milk
canned, notes about
health benefits
homemade, _fm.1_ , fm.2
Coffee
cold-brew, preparing
Cold-Brew Mocha
health benefits
Cookie Dough
Corn, Sweet-&-Spicy, 4.1, _4.2_
Cucumber(s)
health benefits
Mint Mojito, _1.1_ , 1.2
Curried Butternut Squash
Curry powder
D
Dates, fm.1, fm.2, fm.3, bm.1
E
Equipment
F
Fats, dietary
Fennel
Fiber
Flaxseeds
Food processor
Fruit. _See also specific fruits_
for recipes
G
Garlic
Ginger
Coconut Chai
health benefits
Lemon
Thai Peanut Cilantro
Greens. _See also_ Arugula; Kale; Spinach
The Ultimate Green Smoothie, 5.1, _5.2_
H
Hazelnut(s)
Chocolate
health benefits
Hemp seeds
Honey
health benefits
for recipes, fm.1, fm.2
Horchata, Mexican
I
Ingredients
J
Jalapeño(s)
Arugula Pineapple, Spicy, _5.1_ , 5.2
health benefits
K
Kale
Cinnamon Berry
health benefits
The Ultimate Green Smoothie, 5.1, _5.2_
L
Lavender
about
Blueberry, 1.1, _1.2_
Chocolate Chia
health benefits
London Fog
Lemon
Ginger
health benefits
Pink Lemonade, _1.1_ , 1.2
Lime
Avocado Chile, _4.1_ , 4.2
health benefits
Watermelon, 1.1, _1.2_
Lucuma powder
M
Maca powder
Mango
Arugula Cilantro
Chile
health benefits
Turmeric Sunrise, 5.1, _5.2_
Maple syrup, fm.1, bm.1
Matcha
about
health benefits
Latte, 2.1, _2.2_
Mexican Horchata
Mexican Hot Chocolate
Milks, nondairy
Mint
Chamomile Cantaloupe, _1.1_ , 1.2
Chocolate Chip, _5.1_ , 5.2
Cucumber Mojito, _1.1_ , 1.2
health benefits
& Sweet Pea
Molds, fm.1, fm.2
N
Neapolitan, 3.1, _3.2_
O
Olive Oil Chocolate Rosemary, 3.1, _3.2_
Orange
Chocolate
& Cream, Cinnamon, 2.1
health benefits
P
Peach Thyme, Honeyed
Peanut Butter
Cup
health benefits
& Jelly, 2.1, _2.2_
Thai Peanut Cilantro Ginger
Pea(s)
health benefits
Sweet, & Mint
Piña Colada
Pineapple
Arugula Jalapeño, Spicy, _5.1_ , 5.2
Caramelized
health benefits
Piña Colada
Pumpkin
health benefits
Pie
R
Raspberries
health benefits
Peanut Butter & Jelly, 2.1, _2.2_
Pink Lemonade, _1.1_ , 1.2
Rosemary
Balsamic Beet &
health benefits
Olive Oil Chocolate, 3.1, 3.2
Strawberry
Rose water
about
Blackberry Rose
health benefits
Strawberry Cardamom Rose Lassi
S
Salt, fm.1, bm.1
Spinach
Chocolate-Covered Strawberry
Double Chocolate Brownie
health benefits
Lemon Ginger
Mint Chocolate Chip, _5.1_ , 5.2
Piña Colada
Strawberry Basil
Turmeric Mango Sunrise, 5.1, _5.2_
The Ultimate Green Smoothie, 5.1, 5.2
Squash. _See also_ Pumpkin; Zucchini
butternut, health benefits
Curried Butternut
Strawberry(ies)
Cardamom Rose Lassi
Chocolate-Covered
health benefits
Neapolitan, 3.1, _3.2_
Rosemary
Spinach Basil
White Chocolate Chia, _2.1_ , 2.2
Superfood add-ins
Sweeteners, types of
Sweet Potato(es)
Cardamom Cinnamon, #104.1, _4.1_
health benefits
T
Tea. _See also_ Chamomile; Matcha
black, health benefits
Coconut Chai
Lavender London Fog
Texture, note about
Thai Peanut Cilantro Ginger
Thyme
health benefits
Honeyed Peach
Roasted Carrot, _4.1_ , 4.2
Tomato Beet Bloody Mary
Turmeric
Golden Milk, _2.1_ , 2.2
health benefits
Mango Sunrise, 5.1, _5.2_
W
Watermelon
health benefits
Lime, 1.1, _1.2_
White Chocolate Chia Strawberry, _2.1_ , 2.2
Y
Yogurt
health benefits
Strawberry Cardamom Rose Lassi
Z
Zucchini
Basil
health benefits
ABCDEFGHIJKLMNOPQRSTUVWXYZ
ABOUT THE AUTHOR
Liz Moody is a writer, recipe developer, and the woman behind _Sprouted Routes,_ the healthy recipe and wellness blog. A California native, she spent years traveling the globe as a syndicated newspaper columnist, and has contributed recipes to _Clean Eating_ , _Cooking Light,_ _Goop_ , Buzzfeed, mindbodygreen, _Women's Health,_ _Glamour,_ and many more. After making homes in San Francisco and London, she's currently based in Brooklyn, where she lives with her cat, Isabella, and husband Zack.
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## Contents
1. Cover
2. Title Page
3. Copyright
4. Contents
5. Introduction
6. Getting Started
1. Coconut Syrup
2. Coconut Milk
3. Almond Milk
4. Cashew Milk
7. Fruity
1. Apple Pie
2. Cucumber Mint Mojito
3. Watermelon Lime
4. Chamomile Cantaloupe Mint
5. Mango Chile
6. Rosemary Strawberry
7. Honeyed Peach Thyme
8. Pink Lemonade
9. Lavender Blueberry
10. Blackberry Rose
11. Caramelized Pineapple
8. Creamy
1. Coconut Chai
2. Cinnamon Orange & Cream
3. Strawberry Cardamom Rose Lassi
4. Mexican Horchata
5. Cookie Dough
6. Blueberry & Cream
7. Matcha Latte
8. Turmeric Golden Milk
9. Peanut Butter & Jelly
10. Lavender London Fog
11. White Chocolate Chia Strawberry
9. Chocolatey
1. Mexican Hot Chocolate
2. Chocolate Fudge
3. Chocolate-Covered Banana
4. Chocolate Hazelnut
5. Cold-Brew Mocha
6. Neapolitan
7. Chocolate Orange
8. Peanut Butter Cup
9. Chocolate Chia Lavender
10. Chocolate Caramel Swirl
11. Olive Oil Chocolate Rosemary
10. Savory
1. Tomato Beet Bloody Mary
2. Avocado Chile Lime
3. Zucchini Basil
4. Curried Butternut Squash
5. Sweet-&-Spicy Corn
6. Balsamic Beet & Rosemary
7. Sweet Pea & Mint
8. Cardamom Cinnamon Sweet Potato
9. Pumpkin Pie
10. Thai Peanut Cilantro Ginger
11. Roasted Carrot Thyme
11. Green
1. Strawberry Spinach Basil
2. Spicy Arugula Jalapeño Pineapple
3. Turmeric Mango Sunrise
4. Mango Arugula Cilantro
5. Cinnamon Berry
6. Mint Chocolate Chip
7. Lemon Ginger
8. Chocolate-Covered Strawberry
9. The Ultimate Green Smoothie
10. Double Chocolate Brownie
11. Piña Colada
12. Appendix
13. Glow Glossary
14. Acknowledgments
15. Index
16. About the Author
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
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1. Cover
2. Cover
3. Title Page
4. Table of Contents
5. Start
6. Index
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We pardon your usual Sunday morning Instagram ritual of scrolling through everyone's brunch photos to tell you that Behati Prinsloo and Adam Levine just shared an adorable photo of their new baby girl, and you need to see it!
The model and her husband/super-hunky lead singer of Maroon 5 welcomed their first child, Dusty Rose Levine, into the world on Wednesday. While some celebrities may wait weeks or months (or for a magazine cover) to reveal their little bundle of joy to the world, Prinsloo wasted no time, posting a photo of Dusty Rose and her husband just two days later.
The black and white photo, which you can see below, is of baby Dusty laying on Levine's shirtless chest, putting his many tattoos on full display. The heart-warming and adorable photo is accompanied by the simple caption "Words can't describe, Dusty Rose Levine 9/21/16" with a heart emoji.
This is the first time that fans are getting to see the couple's first child, but Us Weekly reveals that the couple has been so excited about their new family member that they are showing her off to friends and family constantly. Gwen Stefani, a former judge on The Voice with Levine, revealed to Ellen DeGeneres while stopping by The Ellen Show last week that she has already seen Dusty Rose. DeGeneres replied that she had seen the baby, too.
"Adam actually has been FaceTiming me all day because he just had his baby. She's so cute!" said Stefani.
"I know! I saw the baby," replied DeGeneres. "She's adorable. She's cute."
An insider also revealed to Us Weekly that everything went well for Prinsloo during the delivery and the couple could not be more excited.
"They're a very happy couple and excited to start their family," said the source. "I can tell you they make very beautiful babies together."
Prinsloo and Levine got engaged in July 2013 with a 1930's vintage engagement ring. They got married in a private ceremony in Mexico in 2014 and spent their honeymoon in South America. Dusty Rose is the first child for the couple, but we bet they won't wait long before giving her a little brother or sister! | {
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You are here: Home / Offices / Press Office / Press Releases / 2017 / July / Student balloon flight tests innovative descent system and astrobiology experiment
Student balloon flight tests innovative descent system and astrobiology experiment
Posted by ap507 at Jul 05, 2017 09:29 AM | Permalink
University of Leicester student project takes to the skies to help in developing future stratospheric flights
Issued by University of Leicester Press Office on 5 July 2017
Images of the balloon and E.coli strain used as part of the astrobiology experiment are available here: https://www.dropbox.com/sh/x9qqdm3rak7youj/AADReIcDtaj_eKkVqQq3E0S0a?dl=0
Physics students from the University of Leicester have flown a new high altitude balloon, achieving a successful test of their innovative descent system which could help in the development of stratospheric flights in the future.
The system kept the balloon airborne for over 7 hours 15 minutes before a gentle and controlled landing in the North Sea.
Developed by students from the University of Leicester's Department of Physics and Astronomy, the system has the significant advantage of providing greater opportunities for scientific research across many fields.
The system may also enable future balloons to be reused after recovery.
The flight was launched from near to Tewkesbury, Gloucestershire on Tuesday 27 June, and reached a maximum height of 23.5 km above Leicester where it hovered for several hours before drifting North East to land 20 km off the coast near the Humber estuary.
The flight duration was longer than expected due to unanticipated behaviour from the descent system. The jet of escaping helium acted as a thruster when the vent was initially opened, changing the ascent rate from 1.2 m/s to almost 5 m/s. This allowed the balloon to travel more than 250km across the country.
The students, in collaboration with the University of Leicester's Space Research Centre (SRC), also flew a biological experiment with a harmless growth limited (uracil auxotroph) E.coli strain called OP50 acting as a food source for a wild, N2, strain of C.elegans nematode worms.
The experiment planned to measure the C.elegans' growth rate when subjected to physiological stresses.
Student Ryan Bradley-Evans said: "Astrobiology experiments like these are very important to not only allow us to figure out if life could exist elsewhere in the universe but also, in order to push the boundaries of human space exploration, we need to understand how biological matter interacts with extreme environments such as space. The stratosphere provides a good testing ground due to its low pressures, low temperatures and an increased UV levels."
Robert Peck said: "We have gathered significant data on the performance of our descent system and we have high hopes for a much greater level of control in future flights."
Professor Paul Monks, Head of College of Science and Engineering at the University of Leicester, said: "Student innovation lies at the heart of the modern university experience. These experiments have real science and engineering at the heart of them and could lead to new ways of being able to look at growth under stress. As can be seen in science everything doesn't always go to plan and you discover new things."
The students involved were members of the University of Leicester's Astronomy and Rocketry Society (AstRoSoc) with Ryan Bradley-Evans as project leader and responsible for the biological experiment, Robert Peck responsible for flight control electronics, Oli Crask developing the radio communications systems, and several other team members including Vlad Verminskyi, William Schulz and Hannah Musson assisting with the project.
The students express their thanks to the amateur radio enthusiasts who helped to track the flight, as well as to John Holt of the SRC for his support throughout the project.
Two days after the balloon's loss at sea Sarah and Emily Pattison contacted the student team after having found the balloon on a beach just north of the Humber estuary.
All of the data logging and flight control systems aboard the balloon had survived.
The students are now analysing the stored data to improve their system for future flights.
For more information contact Ryan Bradley-Evans, project lead, on email rbe3@student.le.ac.uk
'Omnipresent' effects of human impact on England's landscape revealed by University of Leicester geologists | {
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When the citizens of Ivy Town suddenly begin shrinking at random, it's left up to the Atom and his mentor Ray Palmer to solve the riddle! But what happens when a frightened populace places the blame for the shrinking syndrome on its famous size-reducing hero?
This entry was posted in Earth-1, Stories and tagged Andrea Wye, Atom, by Libbylawrence, Gil Baxter, I.Q., Jean Loring, Juliet Thayer, Mechano, Melanie Larvan, Mister Poseidon, Ray Palmer on March 23, 2004 by 5earths. | {
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Q: Resize inner dimension of 2D vector I want to create a vector with inner dimension having a initialize size of 10 and outer dimension can have any size which grows dynamically:
std::vector<std::vector<int>>;
The inner must have an initial size of 10. Is this possible ?
A: It is possible to initialize the size of the inner vectors to 10:
std::vector<std::vector<int>> foo(n, // will contain n inner vectors
std::vector<int>(10)); // each inner vector has size 10 initially
A: When you use vector of vectors, the inner vector is also dynamic in nature. You just declare and initialize it when defining the inner vector, then push it to outer vector.
const int col_size = 10;
vector<vector<int> > table;
vector<int> row(col_size, 0);
table.push_back(row);
Code below defines an array of size n*m and filled with 0: (noting that this does not restrict the dimension of vector appended in future)
const int m = 5, n = 4;
vector<vector<int> > arr(n, vector<int>(m,0));
An alternative approach is to define your own structure for inner dimension using static type:
const int col_size = 3;
struct Row {
int val[col_size];
// to behave like an array.
int operator[](int x) { return val[x]; }
};
vector<Row> table;
A: As others here, mentioned that while using vectors you probably dont have to worry about allocation/de-allocation, std::vector can grow their sizes if required. if you want to use fixed (some what fixed, which which change if some new elements are pushed) size vector you can use can reserve slots by using std::vector::reserve
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} | 5,914 |
\section{Introduction}
The use of machine learning techniques for estimation and control has gained increasing attention in recent years \cite{Busoniu,Hewing_review,special_issue_CSS_letters}. Learning based control holds the promise of enabling the solution of problems which are difficult or even intractable using traditional control design techniques.
In this paper we focus on a remote estimation problem where the channel statistics are unknown. As a motivating example, consider the use of unmanned aerial vehicles (UAVs) for providing wireless communication capabilities \cite{ZengZhangLim,Mohamed_UAV}, with the UAV acting as a mobile aerial base station collecting information from wireless sensors. The use of such systems can allow for monitoring of processes which are difficult to access otherwise, e.g., in a hostile or contested environment. The UAV would transmit collected information to a remote estimator or controller via one of $M$ different channels, e.g., using $M$ different frequency bands. Characteristics of the individual channels can vary substantially at different locations/environments~\cite{ZengZhangLim} over complex terrain, which makes it difficult to have accurate knowledge of the channel statistics whenever the UAV moves to a different location.
Learning of transmission schedules for a single system over a channel with unknown packet reception probability has been studied in~\cite{WuRen_learning}, while learning of power allocations for multiple control systems connected over an unknown non-stationary channel is considered in~\cite{Eisen_TSP}.
With multiple channels, channel allocation for multiple processes using deep reinforcement learning have been considered in \cite{LeongRamaswamy} and \cite{RedderRamaswamy}. While the use of deep reinforcement learning techniques can find schedules which perform well, in general stability of the learned policies is not guaranteed. Furthermore, many training samples may be needed during learning.
In this paper, we consider a remote estimation problem where, at every time instant, a sensor can choose one of $M$ unknown channels for transmission. The aim is to learn the best channel while guaranteeing stability of the remote estimator and being \emph{sample efficient} (via maintaining \emph{low regret}).
We will view the channel selection task as a multi-armed bandit type problem \cite{Slivkins_book,Lattimore_book}. The multi-armed bandit problem is a simple example of a reinforcement learning problem which captures the exploration vs.\ exploitation tradeoff, between basing decisions only on knowledge currently obtained about a system (``exploit'') vs.\ trying out alternative decisions which may potentially be better (``explore''). In contrast to classical multi-armed bandit problems, here the underlying process is a dynamical system, which raises additional challenges on stability and performance. In this paper, we will study the use of the $\varepsilon$-greedy algorithm \cite{SuttonBarto}, as well as several algorithms based on the Thompson Sampling technique \cite{Russo_tutorial}, to carry out channel selection in the context of remote state estimation.
Thompson Sampling is a sampling based algorithm for the multi-armed bandit problem, which has been shown to be optimal with respect to the accumulated regret in certain problems.
Interestingly, Thompson sampling has been previously used to address problems in stochastic \cite{Kim_TS} and linear quadratic control \cite{OuyangGagraniJain}.
\emph{Summary of contributions}:
The main contributions of the current work are as follows:
\begin{itemize}
\item We propose the use of bandit algorithms for channel selection in the context of remote state estimation, including a new stability-aware Bayesian sampling algorithm.
\item We prove that the considered algorithms can guarantee stability of the remote estimation process.
\item We study performance of these algorithms by introducing the notion of estimation regret, and prove that the $\varepsilon$-greedy algorithm has linear estimation regret, while the sampling based algorithms can achieve logarithmic estimation regret.
\end{itemize}
The remainder of the paper is organized as follows: Section \ref{sec:system_model} provides the system model. A selection of bandit algorithms for channel selection are presented in Section \ref{sec:bandit_algorithms}. Stability and performance bounds of these algorithms are studied in Sections \ref{sec:stability} and \ref{sec:regret_bounds} respectively. Section \ref{sec:proof} presents the proof of the main logarithmic regret result. Numerical studies are given in Section \ref{sec:numerical}. Section \ref{sec:conclusion} draws conclusions.
\emph{Notation}: Given a matrix $X$, we say that $X \geq 0$ if $X$ is positive semi-definite. For matrices $X$ and $Y$, we say that $X \leq Y$ if $Y - X \geq 0$. The beta distribution with parameters $\alpha$ and $\beta$ is denoted by $\textnormal{Beta}(\alpha,\beta)$, with expected value $\alpha/(\alpha+\beta)$. $\mathds{1}(\cdot)$ is the indicator function that returns 1 if its argument is true and 0 otherwise.
\section{System Model}
\label{sec:system_model}
\begin{figure}[t!]
\centering
\includegraphics[scale=0.4]{system_model.pdf}
\caption{Remote state estimation over unknown channels}
\label{fig:system_model}
\end{figure}
We consider a discrete-time linear dynamical process
\begin{equation}
\label{eqn:sys}
x_{k+1} = A x_k + w_k
\end{equation}
with sensor measurements
$$y_k = C x_k + v_k$$
where $w_k \sim \mathcal{N}(0,Q)$, $v_k \sim \mathcal{N}(0,R)$, see Fig.\ \ref{fig:system_model}.
The sensor runs a local Kalman filter, with local state estimates $\hat{x}_k^s \triangleq \mathbb{E}[x_k|y_0,\dots,y_k]$ and estimation error covariances $P_k^s \triangleq \mathbb{E}[(x_k - \hat{x}_k^s)(x_k - \hat{x}_k^s)^T|y_0,\dots,y_k]$. We assume that the pair $(A,C)$ is observable and the pair $(A, Q^{1/2})$ is controllable, which implies that $P_k^s$ converges to some $\overline{P}$ as $k \rightarrow \infty$ \cite{andersonMoore}. We will assume steady state, so that $P_k^s = \overline{P}, \forall k$.
As depicted in Fig.\ \ref{fig:system_model}, the local state estimates are transmitted over lossy channels to a remote estimator. There are $M$ such channels, each i.i.d. Bernoulli with packet reception probabilities $\theta_m, m=1,\dots,M$. For notational simplicity, we will assume that all the packet reception probabilities are different, i.e. $\theta_i \neq \theta_j, \forall i\neq j$.
\par At every time step $k$, we can choose one of the $M$ channels to transmit over. A key issue is that none of the packet reception probabilities are known. Ideally, we would like to always use the best channel (i.e., the channel with the highest packet reception probability). But since the packet reception probabilities are unknown, this requires us to find/learn the best channel by trying out different channels and observing their outcomes. The situation resembles a multi-armed bandit type problem \cite{Slivkins_book,Lattimore_book}, with the channels acting as the different arms. The difference between the situation considered in this paper and classical multi-armed bandit problems is that the underlying estimation process has dynamics. This raises additional issues such as stability, as investigated in the current work.
Before proceeding, we recall (see, e.g., \cite{ShiEpsteinMurray}) that the remote estimator has estimation error covariance
\begin{equation}
\label{remote_estimator_eqns}
P_k = \left\{\begin{array}{cc} \overline{P}, & \gamma_k = 1 \\ h(P_{k-1}), & \gamma_k=0 \end{array} \right.
\end{equation}
where
\begin{equation}
\label{eqn:h_defn}
h(X) \triangleq A X A^\top + Q.
\end{equation}
In the above, $\gamma_k = 1$ means that the transmission at time $k$ was successful, while $\gamma_k=0$ means that the transmission was lost. We assume that $P_0 = \overline{P}.$
\par When $A$ is unstable, we know that if the same channel $m$ is chosen at every time step, then the remote estimator has bounded expected estimation error covariance for all $k$ if and only if its packet reception probability satisfies $\theta_m > \theta_c$ \cite{XuHespanha,Schenato}, where
\begin{equation}
\label{eqn:crit}
\theta_c \triangleq 1 - \frac{1}{\rho(A)^2}
\end{equation}
and $\rho(A)$ is the spectral radius of $A$. Accordingly, in this paper, we will make the assumption that at least one of the channels has packet reception probability larger than $\theta_c$, i.e.,
\begin{equation}
\label{best_channel_condition}
\theta^* \triangleq \max_{m} \theta_m > \theta_c.
\end{equation}
\subsection{Preliminaries}
For later reference, we mention some properties of the operator $h(.)$ defined in \eqref{eqn:h_defn}.
\begin{lemma}
\label{lemma:h_properties}
The operator $h(.)$ defined in \eqref{eqn:h_defn} satisfies: \\
(i) $h(X) \leq h(Y) $ if $X \leq Y$ \\
(ii) $h(\overline{P}) \geq \overline{P}$\\
(iii) $\textnormal{tr} (h(\overline{P})) > \textnormal{tr} (\overline{P})$.
\end{lemma}
\begin{proof}
The proof of (i) follows easily from the definition \eqref{eqn:h_defn}. Proofs of (ii) and (iii) can be found in \cite{ShiZhang}.
\end{proof}
Note that Lemma \ref{lemma:h_properties} implies that $P_k \geq \overline{P}, \forall k$, for $P_k$ evolving according to \eqref{remote_estimator_eqns}.
The following recursion for the expected error covariances will also be needed later. Let $\theta(k)$ be the reception probability at time $k$. Then, it follows from
$$ \mathbb{E}[P_{k+1}|P_k] = \theta(k) \overline{P} + (1-\theta(k)) (A P_k A^\top + Q),$$
that
\begin{equation}
\label{eqn:EP_recursion}
\mathbb{E}[P_{k+1}] = \theta(k) \overline{P} + (1-\theta(k)) h(\mathbb{E}[P_k]).
\end{equation}
\section{Bandit Algorithms for Channel Selection}
\label{sec:bandit_algorithms}
In this section we present a selection of multi-armed bandit algorithms for carrying out channel selection, mostly based on the Thompson sampling approach \cite{Russo_tutorial}. Thompson sampling is computationally efficient and can be easily implemented, requiring very little tuning apart from the choice of a prior distribution. There exist other bandit algorithms such as those based on the upper confidence bound (UCB) approach, which are not considered in the current work. Numerical studies have however shown that the performance of Thompson sampling based approaches is highly competitive against even well-tuned UCB approaches in many situations \cite{Granmo,MayKorda,Fatale_AOI}.
\begin{savenotes}
\begin{algorithm}[t]
\caption{$\varepsilon$-greedy}
\label{alg:epsilon_greedy}
\begin{algorithmic}[1]
\State \textbf{Initialize}: $\alpha_{m,1} = 1, \beta_{m,1} = 1, \quad m=1,\dots,M$
\For{$k = 1, 2, \dots$}
\State Set $\hat{\theta}_{m,k}^{\varepsilon} = \frac{\alpha_{m,k}}{\alpha_{m,k}+\beta_{m,k}}, \quad m=1,\dots,M$
\State With probability $\varepsilon > 0$, choose $m_{k}^{\varepsilon}$ uniformly from $\{1,\dots,M\},$
else choose\footnote{Note that if multiple channels have the same maximum estimated packet reception probability, then a channel is chosen uniformly at random from these (maximizing) channels.} $m_{k}^{\varepsilon} = \textnormal{arg}\max_{m} \hat{\theta}_{m,k}^{\varepsilon}$
\State Observe $\gamma_k$
\State Update sample means
\begin{align*}
\alpha_{m,k+1} & = \alpha_{m,k} + \gamma_k, & \textnormal{ if } m = m_{k}^{\varepsilon} \\
\beta_{m,k+1} &= \beta_{m,k} + 1 - \gamma_k, & \textnormal{ if } m = m_{k}^{\varepsilon} \\
\alpha_{m,k+1} & = \alpha_{m,k}, \,
\beta_{m,k+1} = \beta_{m,k}, & \textnormal{ if } m \neq m_{k}^{\varepsilon}
\end{align*}
\EndFor
\end{algorithmic}
\end{algorithm}
\end{savenotes}
\subsection{$\varepsilon$-greedy}
We first consider the well-known $\varepsilon$-greedy algorithm \cite{SuttonBarto}, which with probability $\varepsilon > 0$ chooses a channel uniformly at random (``explore''), and with probability $1-\varepsilon$ chooses the channel with the maximum estimated (via a sample mean) packet reception probability (``exploit''). The procedure is stated formally in Algorithm \ref{alg:epsilon_greedy}. In Algorithm \ref{alg:epsilon_greedy}, we have expressed the estimated packet reception probabilities in terms of the quantities $\alpha_{m,k}$ and $\beta_{m,k}$. This allows us to emphasize the similarities in the implementation with the other algorithms considered in this paper.
\subsection{Thompson Sampling}
Thompson sampling is a sampling based approach which draws samples $\hat{\theta}_{m,k}^{\textnormal{TS}}, m=1,\dots,M$ from the posterior distribution of the unknown packet reception probability of each arm/channel. The posterior distribution can be shown to have a beta distribution if the prior distribution is beta distributed \cite{Russo_tutorial}, as we will assume here. In particular, the prior beta distribution with $\alpha_{m,1} = 1, \beta_{m,1} = 1, \, m=1,\dots,M$ is uniformly distributed in $[0,1]$. The channel that is selected is then given by $$m^{\textnormal{TS}}_k = \textnormal{arg}\max_{m} \hat{\theta}_{m,k}^{\textnormal{TS}}.$$
After transmitting over this channel and observing $\gamma_k$, i.e. whether the transmission was successful, the posterior distribution is then updated for the currently chosen channel $m = m^{\textnormal{TS}}_k$, using the formulas $\alpha_{m,k+1} = \alpha_{m,k} + \gamma_k$ and $\beta_{m,k+1} = \beta_{m,k} + 1 - \gamma_k$. The procedure is summarized in Algorithm \ref{alg:TS}.
\begin{algorithm}[t]
\caption{Thompson Sampling}
\label{alg:TS}
\begin{algorithmic}[1]
\State \textbf{Initialize}: $\alpha_{m,1} = 1, \beta_{m,1} = 1, \quad m=1,\dots,M$
\For{$k = 1, 2, \dots$}
\State Sample $\hat{\theta}_{m,k}^{\textnormal{TS}} \sim \textnormal{Beta}(\alpha_{m,k}, \beta_{m,k}), \quad m=1,\dots,M$
\State Choose $m_{k}^{\textnormal{TS}} = \textnormal{arg}\max_{m} \hat{\theta}_{m,k}^{\textnormal{TS}}$ and observe $\gamma_k$
\State Update posterior distributions
\begin{align*}
\alpha_{m,k+1} & = \alpha_{m,k} + \gamma_k, & \textnormal{ if } m = m_{k}^{\textnormal{TS}} \\
\beta_{m,k+1} &= \beta_{m,k} + 1 - \gamma_k, & \textnormal{ if } m = m_{k}^{\textnormal{TS}} \\
\alpha_{m,k+1} & = \alpha_{m,k}, \,
\beta_{m,k+1} = \beta_{m,k}, & \textnormal{ if } m \neq m_{k}^{\textnormal{TS}}
\end{align*}
\EndFor
\end{algorithmic}
\end{algorithm}
\subsection{Optimistic Bayesian Sampling}
In \cite{MayKorda} a modification of Thompson sampling called ``Optimistic Bayesian sampling'' (OBS) was proposed and studied. Recall that in Thompson sampling, at time $k$ one would draw samples $\hat{\theta}_{m,k}^{\textnormal{TS}}, m=1,\dots,M$ and then select $m^{\textnormal{TS}}_k = \textnormal{arg}\max_{m} \hat{\theta}_{m,k}^{\textnormal{TS}}.$
In OBS we would also draw samples $\hat{\theta}_{m,k}^{\textnormal{TS}}, m=1,\dots,M$, but now select
\begin{align*}
m^{\textnormal{OBS}}_k &= \textnormal{arg}\max_{m} \hat{\theta}_{m,k}^{\textnormal{OBS}},
\end{align*}
where
\begin{align*}
\hat{\theta}_{m,k}^{\textnormal{OBS}} & \triangleq \max\left(\hat{\theta}_{m,k}^{\textnormal{TS}}, \mathbb{E}[\hat{\theta}_{m,k}^{\textnormal{TS}}] \right) = \max\left(\hat{\theta}_{m,k}^{\textnormal{TS}}, \frac{\alpha_{m,k}}{\alpha_{m,k} \!+\! \beta_{m,k}} \right).
\end{align*}
The procedure is summarized in Algorithm \ref{alg:OBS}.
\begin{algorithm}[t]
\caption{Optimistic Bayesian Sampling (OBS)}
\label{alg:OBS}
\begin{algorithmic}[1]
\State \textbf{Initialize}: $\alpha_{m,1} = 1, \beta_{m,1} = 1, \quad m=1,\dots,M$
\For{$k = 1, 2, \dots$}
\State Sample $\hat{\theta}_{m,k}^{\textnormal{TS}} \sim \textnormal{Beta}(\alpha_{m,k}, \beta_{m,k}), \quad m=1,\dots,M$
\State Set $\hat{\theta}_{m,k}^{\textnormal{OBS}} = \max\left(\hat{\theta}_{m,k}^{\textnormal{TS}}, \frac{\alpha_{m,k}}{\alpha_{m,k} \!+\! \beta_{m,k}} \right)$
\State Choose $m_{k}^{\textnormal{OBS}} = \textnormal{arg}\max_{m} \hat{\theta}_{m,k}^{\textnormal{OBS}}$ and observe $\gamma_k$
\State Update posterior distributions
\begin{align*}
\alpha_{m,k+1} & = \alpha_{m,k} + \gamma_k, & \textnormal{ if } m = m_{k}^{\textnormal{OBS}} \\
\beta_{m,k+1} &= \beta_{m,k} + 1 - \gamma_k, & \textnormal{ if } m = m_{k}^{\textnormal{OBS}} \\
\alpha_{m,k+1} & = \alpha_{m,k}, \,
\beta_{m,k+1} = \beta_{m,k}, & \textnormal{ if } m \neq m_{k}^{\textnormal{OBS}}
\end{align*}
\EndFor
\end{algorithmic}
\end{algorithm}
The motivation for OBS is to result in increased selection probabilities for arms which are uncertain \cite{MayKorda}. For classical multi-armed bandit problems, simulations have shown that OBS can outperform Thompson sampling in certain situations \cite{MayKorda,Mellor_thesis}.
\subsection{Stability-aware Bayesian Sampling}
In the context of selecting channels for the purpose of remote state estimation of dynamical systems as in \eqref{eqn:sys}, it is important to take into account the long term estimation performance. In particular, the fundamental bound \eqref{eqn:crit} suggests that, unlike when using OBS, one should not artificially stimulate the selection probabilities for channels which seem to be poor. More specifically, if the current mean-estimate $\alpha_{m,k}/(\alpha_{m,k} + \beta_{m,k})$ of the reception probability is less than the critical threshold $\theta_c$, then it is questionable whether this channel should be chosen with a probability higher than that prescribed by Thompson sampling.
Motivated by the above, in this paper, we propose a \emph{stability-aware modification} of OBS which uses the OBS sample $\hat{\theta}_{m,k}^{\textnormal{OBS}} $ only if $\alpha_{m,k}/(\alpha_{m,k} + \beta_{m,k}) > \theta_c$, while using the Thompson sample $\hat{\theta}_{m,k}^{\textnormal{TS}}$ otherwise.
The algorithm, here named \emph{Stability-aware Bayesian Sampling} (SBS), can thus be characterized by:
\begin{equation}
\label{eqn:OBS_mod}
\hat{\theta}_{m,k}^{\textnormal{SBS}} = \eta_{m,k} \hat{\theta}_{m,k}^{\textnormal{OBS}} + (1-\eta_{m,k}) \hat{\theta}_{m,k}^{\textnormal{TS}}, \quad m=1, \dots,M
\end{equation}
and
$$m_k^{\textnormal{SBS}} = \arg\max_m \hat{\theta}_{m,k}^{\textnormal{SBS}},$$
where \begin{equation}
\label{eqn:obs_mod}
\eta_{m,k} =
\begin{cases}
1&\text{if $\alpha_{m,k}/(\alpha_{m,k} + \beta_{m,k}) > \theta_c$}\\
0&\text{otherwise.}
\end{cases}
\end{equation}
The method is summarized in Algorithm \ref{alg:mod}.
\begin{algorithm}[t]
\caption{Stability-aware Bayesian Sampling (SBS)}
\label{alg:mod}
\begin{algorithmic}[1]
\State \textbf{Initialize}: $\alpha_{m,1} = 1, \beta_{m,1} = 1, \quad m=1,\dots,M$
\For{$k = 1, 2, \dots$}
\State Sample $\hat{\theta}_{m,k}^{\textnormal{TS}} \sim \textnormal{Beta}(\alpha_{m,k}, \beta_{m,k}), \quad m=1,\dots,M$
\State Set $\hat{\theta}_{m,k}^{\textnormal{OBS}} = \max\left(\hat{\theta}_{m,k}^{\textnormal{TS}}, \frac{\alpha_{m,k}}{\alpha_{m,k} \!+\! \beta_{m,k}} \right)$ and $\hat{\theta}_{m,k}^{\textnormal{SBS}} = \eta_{m,k} \hat{\theta}_{m,k}^{\textnormal{OBS}} + (1-\eta_{m,k}) \hat{\theta}_{m,k}^{\textnormal{TS}}$, where $\eta_{m,k} = \mathds{1}(\alpha_{m,k}/(\alpha_{m,k} + \beta_{m,k}) > \theta_c)$
\State Choose $m_{k}^{\textnormal{SBS}} = \textnormal{arg}\max_{m} \hat{\theta}_{m,k}^{\textnormal{SBS}}$ and observe $\gamma_k$
\State Update posterior distributions
\begin{align*}
\alpha_{m,k+1} & = \alpha_{m,k} + \gamma_k, & \textnormal{ if } m = m_{k}^{\textnormal{SBS}} \\
\beta_{m,k+1} &= \beta_{m,k} + 1 - \gamma_k, & \textnormal{ if } m = m_{k}^{\textnormal{SBS}} \\
\alpha_{m,k+1} & = \alpha_{m,k}, \,
\beta_{m,k+1} = \beta_{m,k}, & \textnormal{ if } m \neq m_{k}^{\textnormal{SBS}}
\end{align*}
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Stability of Remote State Estimation with Channel Selection}
\label{sec:stability}
In this section, we show that the $\varepsilon$-greedy algorithm will ensure stability of the remote estimator, provided the exploration rate $\varepsilon$ is not too high, while all of the sampling based channel selection schemes presented in the previous section (Thompson sampling, OBS, and SBS) will ensure stability of the remote estimator. It is important to emphasize that Theorems \ref{thm:stability_epsilon_greedy} and \ref{thm:stability_TS}, stated below, cover the challenging case where channel qualities are unknown, and where channel selection and learning are done in real-time.
Let $m_k$ be the channel chosen at time $k$, and define $$m^* \triangleq \textnormal{argmax}_m \theta_m.$$
\begin{theorem}
\label{thm:stability_epsilon_greedy}
Suppose condition \eqref{best_channel_condition} holds. Then under the $\varepsilon$-greedy algorithm, the expected estimation error covariance $\mathbb{E}[P_k]$ is bounded for all $k$ if and only if \begin{equation}
\label{epsilon_stability_condition}
0 < \varepsilon < \frac{\theta^* - \theta_c}{\theta^* - \frac{1}{M}\sum_{m=1}^M \theta_m}.
\end{equation}
\end{theorem}
\begin{proof}
Because of the exploration phase, where with probability $\varepsilon > 0$ a channel is chosen randomly, it is easy to see that each channel will be accessed infinitely often. Hence the estimates of the packet reception probabilities for each of the channels will converge to their true values as $k \rightarrow \infty$. Thus, during the exploitation phase we have
$$\mathbb{P}(m_k = m^* | \textnormal{exploit}) \rightarrow 1 \textnormal{ as } k \rightarrow \infty,$$
while during the exploration phase we have
$$\mathbb{P}(m_k = m^* | \textnormal{explore}) = \frac{1}{M}.$$
Then, given any $\delta > 0$, there exists a $K(\delta) < \infty$ such that
$$\mathbb{P}(\textnormal{reception at time } k) \geq (1-\varepsilon)\theta^*(1-\delta) + \frac{\varepsilon}{M} \sum_{m=1}^M \theta_m$$
for all $k > K(\delta)$.
Now let $$\theta' \triangleq (1-\varepsilon)\theta^*(1-\delta) + \frac{\varepsilon}{M} \sum_{m=1}^M \theta_m.$$ For $k > K(\delta)$, we have by Lemma \ref{lemma:h_properties} and a similar derivation to \eqref{eqn:EP_recursion} that
$$ \mathbb{E}[P_{k+1}] \leq \theta' \overline{P} + (1-\theta') h(\mathbb{E}[P_k]).$$
By induction, it follows that $\{\mathbb{E}[P_k]\}$ will be upper bounded by a sequence $\{V_k\}$ defined by
\begin{align*}
V_k & = \mathbb{E}[P_k], \quad k \leq K(\delta) \\
V_{k+1} & = \theta' \overline{P} + (1-\theta') h(V_k), \quad k \geq K(\delta).
\end{align*}
The sequence $\{V_k\}$ converges if and only if
$\theta' > \theta_c$.
If
\begin{equation}
\label{epsilon_stability_condition2}
(1-\varepsilon) \theta^* + \frac{\varepsilon}{M} \sum_{m=1}^M \theta_m > \theta_c,
\end{equation} then one can always find a sufficiently small $\delta > 0$ to satisfy $\theta' > \theta_c$. As \eqref{epsilon_stability_condition2} is equivalent to the condition \eqref{epsilon_stability_condition}, this proves the ``if'' direction of the theorem.
For the ``only if'' direction, first define
\begin{equation}
\label{eqn:opt_reception_prob_epsilon_greedy}
\tilde{\theta} \triangleq \theta^*(1-\varepsilon) + \frac{\varepsilon}{M} \sum_{m=1}^M \theta_m,
\end{equation}
and note that, for a given $\varepsilon$,
$$
\mathbb{P}(\textnormal{reception at time } k) \leq \tilde{\theta}$$
holds, because $\tilde{\theta}$ is the reception probability assuming the optimal $m^*$ is always chosen during the exploitation phase.
If $ \varepsilon \geq (\theta^* - \theta_c)/(\theta^* - \frac{1}{M}\sum_{m=1}^M \theta_m)$, then $
\mathbb{P}(\textnormal{reception at time } k) \leq \tilde{\theta} \leq \theta_c $, and
thus
$$ \mathbb{E}[P_{k+1}] \geq \theta_c \overline{P} + (1-\theta_c) h(\mathbb{E}[P_k]).$$
Define a sequence $\{\tilde{V}_k\}$ by
$$ \tilde{V}_{k+1} = \theta_c \overline{P} + (1-\theta_c) h(\tilde{V}_k), $$
which from the definition of $\theta_c$ is a divergent sequence.
An induction argument shows that $\mathbb{E}[P_k] \geq \tilde{V}_{k}, \forall k$. This implies that $\mathbb{E}[P_k]$ also diverges and completes the proof.
\end{proof}
\begin{remark}
If $(\theta^* - \theta_c)/(\theta^* - \frac{1}{M}\sum_{m=1}^M \theta_m) > 1$, then Theorem~\ref{thm:stability_epsilon_greedy} establishes that the $\varepsilon$-greedy algorithm provides estimation stability for all $\varepsilon > 0$.
\end{remark}
We next consider stability of the sampling based algorithms.
We first give a preliminary result.
\begin{lemma}
\label{lemma:infinite_exploration}
Under i) Thompson sampling, ii) OBS, and iii) SBS, all channels will be used infinitely often, and
\begin{equation}
\label{eqn:persist}
\mathbb{P}(m_k = m^*) \rightarrow 1 \textnormal{ as } k \rightarrow \infty.
\end{equation}
\end{lemma}
\begin{proof}
For Thompson sampling and OBS, this is proved in \cite{MayKorda} (see also \cite{Granmo} for the two armed bandit case under Thompson sampling).
For SBS we note that, in view of \eqref{eqn:OBS_mod}, it holds that $\hat{\theta}_{m,k}^{\textnormal{TS}} \leq \hat{\theta}_{m,k}^{\textnormal{SBS}} \leq \hat{\theta}_{m,k}^{\textnormal{OBS}}$. Thus, intuitively \eqref{eqn:persist} should also hold, as SBS in a sense lies in between Thompson sampling and OBS (and convergence holds for both of these schemes). A rigorous proof of Lemma~\ref{lemma:infinite_exploration} for SBS can be given using similar arguments as in \cite{MayKorda}. The details are omitted for brevity.
\end{proof}
\begin{theorem}
\label{thm:stability_TS}
Suppose condition \eqref{best_channel_condition} holds. Then under i) Thompson sampling, ii) OBS, and iii) SBS, the expected error covariance $\mathbb{E}[P_k]$ is bounded for all $k$.
\end{theorem}
\begin{proof}
From Lemma \ref{lemma:infinite_exploration}, we know that under all three sampling based schemes, given any $\delta > 0$, there exists a $K(\delta) < \infty$ such that
$$\mathbb{P}(\textnormal{reception at time } k) \geq \theta^*(1-\delta), \forall k > K(\delta).$$
Pick a sufficiently small $\delta$ such that
$\theta' \triangleq \theta^*(1-\delta) > \theta_c$ still holds.
Then for $k > K(\delta)$, we have by Lemma \ref{lemma:h_properties} and a similar derivation to \eqref{eqn:EP_recursion} that
$$ \mathbb{E}[P_{k+1}] \leq \theta' \overline{P} + (1-\theta') h(\mathbb{E}[P_k]).$$
Define a sequence $\{V_k\}$ by
\begin{align*}
V_{k} & = \mathbb{E}[P_k], \quad k \leq K(\delta)\\
V_{k+1} & = \theta' \overline{P} + (1-\theta') h(V_k), \quad k \geq K(\delta).
\end{align*}
Using an induction argument, we can show that $V_k$ upper bounds $\mathbb{E}[P_k]$ for all $k $. Since $\theta' > \theta_c$, the sequence $\{V_k\}$ converges, and in particular $V_k$ is bounded for all $k$. Hence $\mathbb{E}[P_k]$ is also bounded for all $k$.
\end{proof}
\section{Performance Bounds}
\label{sec:regret_bounds}
In the multi-armed bandit literature, it is common to analyze the performance of an algorithm via the notion of \emph{regret} \cite{Slivkins_book,Lattimore_book}, which is defined as the difference between the expected cumulative reward from always playing the optimal arm and the expected cumulative reward using a particular bandit algorithm.
For the remote estimation problem at hand, it is convenient to regard the trace of the estimation error covariance $\textnormal{tr} P_k$ as a \emph{cost}, or alternatively regard $-\textnormal{tr} P_k$ as a \emph{reward}, see also \cite{LeongRamaswamy}. Motivated by the notion of regret in multi-armed bandit problems, in this paper we define the \emph{estimation regret} over a horizon $T$ for the remote state estimation problem as
\begin{equation}
\label{eqn:regret_defn}
\begin{split}
\textnormal{regret}(T) & \triangleq \sum_{k=1}^T \big( -\textnormal{tr} \mathbb{E}[P_k^*] - (-\textnormal{tr}\mathbb{E}[P_k]) \big) \\
& = \sum_{k=1}^T \textnormal{tr} (\mathbb{E}[P_k]-\mathbb{E}[P_k^*]).
\end{split}
\end{equation}
In \eqref{eqn:regret_defn}, $\mathbb{E}[P_k^*]$ is the expected error covariance assuming that the optimal channel $m^*$ is always chosen, and $\mathbb{E}[P_k]$ is the expected error covariance when using a particular channel selection algorithm. Thus, \eqref{eqn:regret_defn} constitutes a measure of the degree of suboptimality incurred from using a particular algorithm. Stationary and transient performance of an algorithm can be quantified by inspecting how fast the regret grows in relation to the horizon length $T$.
To state our results, we first recall some order notation.
\begin{definition}
Given functions $f(.)$ and $g(.)$, we say that $f(T) = O(g(T))$ if there exist constants $c$ and $T_0$, such that $|f(T)| \leq c g(T), \forall T \geq T_0$.
We say that $f(T) = \Omega(g(T))$ if $g(T) = O(f(T))$, and that $f(T) = \Theta(g(T))$ if both $f(T) = O(g(T))$ and $f(T) = \Omega(g(T))$. We say that $f(T) = o(g(T))$ if $\lim_{T\rightarrow \infty} f(T)/g(T) = 0$.
\end{definition}
For $\varepsilon$-greedy, the regret scales linearly with $T$.
\begin{theorem}
\label{thm:regret_bound_epsilon_greedy}
Suppose conditions \eqref{best_channel_condition} and \eqref{epsilon_stability_condition} hold. Then under the $\varepsilon$-greedy algorithm, we have
$$\textnormal{regret}(T) = \Theta (T).$$
\end{theorem}
\begin{proof}
We first show that $\textnormal{regret}(T) = O (T)$.
By Theorem~\ref{thm:stability_epsilon_greedy}, there exists some bounded $\mathcal{P}^\varepsilon$ such that
$$\mathbb{E}[P_k] \leq \mathcal{P}^\varepsilon, \quad \forall k.$$
We have
\begin{align*}
\textnormal{regret}(T) & = \sum_{k=1}^T \textnormal{tr} (\mathbb{E}[P_k]-\mathbb{E}[P_k^*]) \\
& \leq \sum_{k=1}^T \textnormal{tr} \mathbb{E}[P_k] \\
& \leq \textnormal{tr} (\mathcal{P}^\varepsilon) T = O(T).
\end{align*}
We now show that $\textnormal{regret}(T) = \Omega (T)$. Recall the definition $\tilde{\theta}$ from \eqref{eqn:opt_reception_prob_epsilon_greedy}. Note that we have
\begin{align*}
\mathbb{E}[P_k] &\geq \tilde{\theta} \overline{P} + (1-\tilde{\theta}) h(\mathbb{E}[P_{k-1}]) \\
& \geq \tilde{\theta} \overline{P} + (1-\tilde{\theta}) h(\mathbb{E}[P_{k-1}^*]), \\
\mathbb{E}[P_k^*] &= \theta^* \overline{P} + (1-\theta^*) h(\mathbb{E}[P_{k-1}^*]).
\end{align*}
Then
\begin{align}
\textnormal{tr} (\mathbb{E}[P_k]-\mathbb{E}[P_k^*])
& \geq \textnormal{tr} \big( (\theta^* - \tilde{\theta}) ( h(\mathbb{E}[P_{k-1}^*]) - \overline{P} )\big) \nonumber \\
& \geq (\theta^* - \tilde{\theta}) \min_{k \geq 1} \textnormal{tr}\big( h(\mathbb{E}[P_{k-1}^*]) - \overline{P}\big) \nonumber \\
&\triangleq r. \label{eqn:r_epsilon_greedy}
\end{align}
By the definition \eqref{eqn:opt_reception_prob_epsilon_greedy} and Lemma \ref{lemma:h_properties}, we have $r > 0$. Hence
\begin{align*}
\textnormal{regret}(T) & = \sum_{k=1}^T \textnormal{tr} (\mathbb{E}[P_k]-\mathbb{E}[P_k^*]) \\
& \geq r T = \Omega (T).
\end{align*}
\end{proof}
For the Bernoulli bandit problem with per stage rewards in $\{0,1\}$, it is known that the regret scales logarithmically with $T$ under Thompson sampling \cite{AgrawalGoyal_ACM,KaufmannKordaMunos}. An ``age-of-information regret'' measure was recently considered in \cite{Fatale_AOI}, where the per stage cost is unbounded, and can increase by (at most) one at every time step. It is shown that this age-of-information regret also scales logarithmically with $T$. In the current work, it follows from \eqref{eqn:h_defn} that the per stage cost $\mathbb{E}[P_k]$ (or reward $-\mathbb{E}[P_k]$) may also become unbounded, and furthermore can increase at an exponential rate. Interestingly, for the estimation regret introduced in \eqref{eqn:regret_defn}, a bound that is logarithmic in $T$ still holds.
Let us refer to sub-optimal channels as those whose packet reception probabilities are not equal to the optimal $\theta^*$.
We begin with a preliminary result.
\begin{lemma}
\label{lemma:suboptimal_arms}
Let $N_{\textnormal{sub}}(T) $ denote the number of uses of sub-optimal channels over horizon $T$. Then, under i) Thompson sampling, ii) OBS, and iii) SBS, we have
$$\mathbb{E}[N_{\textnormal{sub}}(T)] = \Theta(\log T).$$
\end{lemma}
\begin{proof}
The property that $\mathbb{E}[N_{\textnormal{sub}}(T)] = O(\log T)$ is shown for Thompson sampling in \cite{AgrawalGoyal_ACM,KaufmannKordaMunos}. The corresponding result for OBS is proved in \cite{Mellor_thesis}, based on similar arguments as in \cite{AgrawalGoyal_ACM}. For SBS, $\mathbb{E}[N_{\textnormal{sub}}(T)] = O(\log T)$ can be shown by making use of the relation $\hat{\theta}_{m,k}^{\textnormal{TS}} \leq \hat{\theta}_{m,k}^{\textnormal{SBS}} \leq \hat{\theta}_{m,k}^{\textnormal{OBS}}$ and the arguments of \cite{Mellor_thesis}. The (rather lengthy) details are omitted for brevity.
We next recall a result from \cite{LaiRobbins}, namely that any policy with $\mathcal{R}(T) = o(T^a)$ for every $a>0$ satisfies $\mathbb{E}[N_{\textnormal{sub}}(T)] = \Omega(\log T)$, where $\mathcal{R}(T) \triangleq \mathbb{E} \big[\sum_{k=1}^T (\theta^* - \theta(k)) \big]$ is the classical notion of regret for the Bernoulli bandit problem, see e.g. \cite{AgrawalGoyal_ACM,KaufmannKordaMunos}. Since for Thompson sampling, OBS, and SBS:
$$ \mathcal{R}(T) = O\big(\mathbb{E}[N_{\textnormal{sub}}(T)]\big) = O(\log T) = o(T^a), \, \forall a > 0,$$
where the first equality follows from standard arguments \cite{AgrawalGoyal_ACM,KaufmannKordaMunos} and the second equality is what we have just shown, the property $\mathbb{E}[N_{\textnormal{sub}}(T)] = \Omega(\log T)$ therefore holds for all three schemes.
\end{proof}
\begin{theorem}
\label{thm:regret_bound_general}
Suppose that condition \eqref{best_channel_condition} holds. Then under i) Thompson sampling, ii) OBS, and iii) SBS, we have
$$\qquad\qquad\qquad\qquad \textnormal{regret}(T) = \Theta (\log T). \qquad\qquad\qquad\qquad\qquad\square$$
\end{theorem}
Before giving the formal proof of this result, we first provide a sketch of the proof strategy for the upper bound $ \textnormal{regret}(T) = O(\log T)$. Following the idea of \cite{Fatale_AOI}, which showed a logarithmic bound for the age-of-information regret, we will upper bound the estimation regret with the regret that is achieved using an alternative schedule, say $\mathcal{A}$, that replaces all uses of sub-optimal channels with the worst channel. In \cite{Fatale_AOI}, this schedule $\mathcal{A}$ is then further upper bounded by a ``worst-case'' schedule $\mathcal{B}$ that groups all the uses of the worst channel together. However, for the problem at hand, as the expected error covariance can increase exponentially fast if the worst channel does not stabilize the remote estimator, this argument will give the desired logarithmic regret bound only in problem instances where the worst channel (and hence all channels) is stabilizing.\footnote{See Remark \ref{remark:regret_special_case} included at the end of Section \ref{sec:proof}, for this argument.} To cover more interesting and challenging situations, to establish Theorem~\ref{thm:regret_bound_general}, we need to additionally make use of the result in Theorem~\ref{thm:stability_TS} that the expected estimation error covariance is always bounded. This will allow us to derive an upper bound on the number of consecutive uses of the worst channel. This will then divide up the time interval into ``epochs'', see Fig.\ \ref{fig:regret_proof}. We will show that the regret within each epoch is bounded, while the expected number of epochs is logarithmic in $T$. As a consequence the required logarithmic regret upper bound is established.
\begin{remark}
We note that some works on learning based control have made assumptions that the underlying system is stable \cite{Lale}, or can be stabilized for all possible controls \cite{OuyangGagraniJain}, when carrying out a regret analysis. This stands in contrast to Theorem~\ref{thm:regret_bound_general} which merely requires that at least one of the $M$ channels is stabilizing for the remote estimator, while some of the sub-optimal channels can cause the expected error covariance to diverge if they are used too often.
\end{remark}
\section{Proof of Theorem \ref{thm:regret_bound_general}}
\label{sec:proof}
We will first prove that $\textnormal{regret}(T) = O (\log T)$.
Define the reception probability of the worst channel as $$\theta_{\textnormal{min}} \triangleq \min_m \theta_m.$$
Given any schedule of channel selections, let schedule $\mathcal{A}$ denote an alternative schedule which replaces all uses of sub-optimal channels with the worst channel. Let $\mathbb{E}[P_{k}^\mathcal{A}] $ be the expected error covariance under this replacement procedure. Then from \eqref{eqn:EP_recursion} and Lemma \ref{lemma:h_properties}, we can easily show that
$$\mathbb{E}[P_k] \leq \mathbb{E}[P_k^\mathcal{A}],\quad \forall k.$$
In particular, this means that
\begin{equation}
\label{eqn:EPk_sum_bound}
\sum_{k=1}^T \textnormal{tr} \mathbb{E}[P_k] \leq \sum_{k=1}^T \textnormal{tr} \mathbb{E}[P_k^\mathcal{A}].
\end{equation}
Next, using a similar argument as in the proof of Theorem~\ref{thm:stability_TS}, we note that the process of replacing uses of sub-optimal channels with the worst channel will still result in the expected error covariance $\mathbb{E}[P_k^\mathcal{A}]$ being bounded for all $k$, i.e., there exists a bounded $\mathcal{P}$ such that
\begin{equation}
\label{eqn:EPk_bound}
\mathbb{E}[P_k^\mathcal{A}] \leq \mathcal{P}, \forall k.
\end{equation}
We now consider separate problem instances depending on whether $\theta_{\textnormal{min}} \leq \theta_c$ or $\theta_{\textnormal{min}} > \theta_c$.
\emph{Case $\theta_{\textnormal{min}} \leq \theta_c$}:
In this case, the number of consecutive uses of the worst channel, for any schedule $\mathcal{A}$, must be bounded by some finite value $\tau$, as otherwise it would contradict \eqref{eqn:EPk_bound}.
For a time interval of length $T$ with $N_w$ uses of the worst channel (and $T-N_w$ uses of the optimal channel), any feasible schedule $\mathcal{A}$ must then be of the form shown in Fig.~\ref{fig:regret_proof},
where one alternates between uses of the worst channel (denoted by $w$) and the best channel (denoted by $b$), and the leading and trailing blocks of $b$'s may have length 0. The total number of $w$'s is equal to $N_w$, and the total number of $b$'s is equal to $T-N_w$.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.17]{regret_proof_diagram.png}
\caption{Structure of feasible schedule $\mathcal{A}$ when $\theta_{\textnormal{min}} \leq \theta_c$}
\label{fig:regret_proof}
\end{figure}
Schedules of this form thus consist of a leading block of $b$'s followed by ``epochs''. Each epoch consists of between 1 and $\tau$ consecutive uses of the worst channel, followed by a variable number (non-zero except possibly in the last epoch) of uses of the optimal channel.\footnote{Note that the epochs are in general non-periodic.}
\par Let $N_e$ represent the number of epochs and note that $N_e \leq N_w$. Let $k_0^b$ denote the time instance when the leading block of $b$'s ends, $k_i^w, i=1,\dots,N_e$ denote the times when the block of $w$'s in the $i$-th epoch ends, and $k_i^b, i=1,\dots,N_e$ the time instances when the block of $b$'s in the $i$-th epoch ends (with $k_{N_e}^b = T$), see Fig.~\ref{fig:regret_proof}.
The regret \eqref{eqn:regret_defn} can then be upper-bounded by:
\begin{align}
\textnormal{regret}(T) & = \sum_{k=1}^{T} \textnormal{tr} (\mathbb{E}[P_k] - \mathbb{E}[P_k^*]) \nonumber \\
& \leq \sum_{k=1}^{T} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{A}] - \mathbb{E}[P_k^*]) \nonumber \\
& = \mathbb{E} \bigg[\sum_{k=1}^{k_0^b} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{A}] - \mathbb{E}[P_k^*]) \nonumber \\ & \quad \quad+ \sum_{i=1}^{N_e} \sum_{k=k_{i\!-\!1}^b+1}^{k_i^b} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{A}] - \mathbb{E}[P_k^*]) \bigg], \label{eqn:regret_split_up}
\end{align}
where the inequality uses \eqref{eqn:EPk_sum_bound}, and the outer expectation in the last equality is over $N_e$.
Since the optimal channel is used in the first $k_0^b$ time slots, we have:
\begin{equation}
\label{eqn:regret_initial}
\sum_{k=1}^{k_0^b} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{A}] - \mathbb{E}[P_k^*]) = 0.
\end{equation}
We will next bound the accumulated regret over each epoch. Over the at most $\tau$ uses of the worst channel, we have $\mathbb{E}[P_k^\mathcal{A}] \leq \mathcal{P}$, so the accumulated regret over the uses of the worst channel can be bounded as
\begin{equation}
\label{eqn:regret_epoch_w}
\sum_{k=k_{i\!-\!1}^b+1}^{k_i^w} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{A}] - \mathbb{E}[P_k^*]) \leq \tau \textnormal{tr}(\mathcal{P}).
\end{equation}
For the accumulated regret over the remaining uses of the optimal channel,
note first that we have
\begin{align*}
\mathbb{E}[P_{k_i^w+1}^\mathcal{A}] &\leq \theta^* \overline{P} + (1-\theta^*) h(\mathcal{P}) \\
\mathbb{E}[P_{k+1}^\mathcal{A}] &= \theta^* \overline{P} + (1-\theta^*) h(\mathbb{E}[P_{k}^\mathcal{A}]), \quad k > k_i^w \\
\mathbb{E}[P_{k+1}^*] &= \theta^* \overline{P} + (1-\theta^*) h(\mathbb{E}[P_{k}^*]), \quad k \geq k_i^w.
\end{align*}
Therefore,
\begin{align*}
&\mathbb{E}[P_{k_i^w+1}^\mathcal{A}] - \mathbb{E}[P_{k_i^w+1}^*] \\ & \quad \leq (1-\theta^*) (h(\mathcal{P}) - h(\mathbb{E}[P_{k_i^w}^*])) \\
& \quad = (1-\theta^*) A (\mathcal{P} - \mathbb{E}[P_{k_i^w}^*]) A^\top, \\
& \mathbb{E}[P_{k_i^w+2}^\mathcal{A}] - \mathbb{E}[P_{k_i^w+2}^*] \\ & \quad = (1-\theta^*) (h(\mathbb{E}[P_{k_i^w+1}^\mathcal{A}]) - h(\mathbb{E}[P_{k_i^w+1}^*])) \\
& \quad = (1-\theta^*) A (\mathbb{E}[P_{k_i^w+1}^\mathcal{A}] - \mathbb{E}[P_{k_i^w+1}^*]) A^\top \\
& \quad \leq (1-\theta^*)^2 A^2 (\mathcal{P} - \mathbb{E}[P_{k_i^w}^*]) (A^\top)^2, \\
& \quad \,\,\, \vdots
\end{align*}
and thus
\begin{align}
& \sum_{k=k_i^w+1}^{k_i^b} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{A}] - \mathbb{E}[P_k^*]) \nonumber \\ & \leq \sum_{k=k_i^w+1}^{k_i^b} (1-\theta^*)^{k-k_i^w} \textnormal{tr} \big( A^{k-k_i^w} (\mathcal{P} - \mathbb{E}[P_{k_i^w}^*]) (A^\top)^{k-k_i^w} \big) \nonumber
\\ & \leq \sum_{k=k_i^w+1}^{\infty} (1-\theta^*)^{k-k_i^w} \textnormal{tr} \big( A^{k-k_i^w} \mathcal{P} (A^\top)^{k-k_i^w} \big) \nonumber\\
& = \textnormal{tr} \Big( \sum_{j=1}^{\infty} (1-\theta^*)^{j} A^{j} \mathcal{P} (A^\top)^{j} \Big) \nonumber\\
& \triangleq r^b, \label{eqn:regret_epoch_b}
\end{align}
where $r^b < \infty$ since $\theta^* > \theta_c$.
From \eqref{eqn:regret_split_up}, \eqref{eqn:regret_initial}, \eqref{eqn:regret_epoch_w}, and \eqref{eqn:regret_epoch_b}, we have
\begin{align*}
\textnormal{regret}(T) & \leq \mathbb{E} \left[ 0 + N_e (\tau \textnormal{tr}(\mathcal{P}) + r^b) \right] \\
& = (\tau \textnormal{tr}(\mathcal{P}) + r^b) \mathbb{E} [N_e] \\
& \leq (\tau \textnormal{tr}(\mathcal{P}) + r^b) \mathbb{E} [N_w] \\
& = O (\log T),
\end{align*}
where the last line uses Lemma \ref{lemma:suboptimal_arms}. This proves that $\textnormal{regret}(T) = O(\log T)$ for the case $\theta_{\textnormal{min}} \leq \theta_c$.
\emph{Case $\theta_{\textnormal{min}} > \theta_c$}: In this case, as the worst channel will still result in a bounded expected error covariance, it is not clear whether an upper bound on the number of consecutive uses of the worst channel necessarily exists. Therefore, we will use a slightly different argument.
Again consider a time interval of length $T$ with $N_w$ uses of the worst channel. We can still express any schedule $\mathcal{A}$ as a leading block of $b$'s followed by a number of epochs, but now we don't upper bound the number of uses of the worst channel, i.e., feasible schedules are now of the form
\begin{equation*}
(\underbrace{b,\dots,b}_{\geq 0},\underbrace{w,\dots,w}_{\geq 1 },\underbrace{b,\dots,b}_{\geq 1},\dots\dots, \underbrace{w,\dots,w}_{\geq 1 },\underbrace{b,\dots,b}_{\geq 0})
\end{equation*}
The regret again satisfies \eqref{eqn:regret_split_up} and \eqref{eqn:regret_initial}.
Let $l_i$ denote the number of uses of the worst channel in epoch $i$, and note that $\sum_{i=1}^{N_e} l_i = N_w$.
Then by similar arguments as in the case $\theta_{\textnormal{min}} \leq \theta_c$, we can obtain
\begin{align}
\sum_{i=1}^{N_e} \sum_{k=k_{i\!-\!1}^b+1}^{k_i^b} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{A}] - \mathbb{E}[P_k^*]) & \leq \sum_{i=1}^{N_e} (l_i \textnormal{tr}(\mathcal{P}) + r^b) \nonumber \\
& = \textnormal{tr}(\mathcal{P}) N_w + r^b N_e \nonumber \\
& \leq ( \textnormal{tr}(\mathcal{P}) + r^b) N_w. \label{eqn:regret_epoch_stable}
\end{align}
Hence from \eqref{eqn:regret_split_up}, \eqref{eqn:regret_initial}, and \eqref{eqn:regret_epoch_stable}, we have
\begin{align*}
\textnormal{regret}(T) & \leq \mathbb{E} \left[ 0 + ( \textnormal{tr}(\mathcal{P}) + r^b) N_w \right] \\
& = (\textnormal{tr}(\mathcal{P}) + r^b) \mathbb{E} [N_w] \\
& = O (\log T).
\end{align*}
This completes the proof that $\textnormal{regret}(T) = O (\log T)$.
To conclude, we will now prove that $\textnormal{regret}(T) = \Omega (\log T)$. We have
\begin{align*}
\textnormal{regret}(T) & = \sum_{k=1}^{T} \textnormal{tr} (\mathbb{E}[P_k] - \mathbb{E}[P_k^*]) \\
& = \mathbb{E} \bigg[\sum_{k=1}^{T} \textnormal{tr} (\mathbb{E}[P_k] - \mathbb{E}[P_k^*]) \mathds{1}(\theta(k) = \theta^*) \\
& \quad + \sum_{k=1}^{T} \textnormal{tr} (\mathbb{E}[P_k] - \mathbb{E}[P_k^*]) \mathds{1}(\theta(k) \neq \theta^*) \bigg] \\
& \geq \mathbb{E} \bigg[ \sum_{k=1}^{T} \textnormal{tr} (\mathbb{E}[P_k] - \mathbb{E}[P_k^*]) \mathds{1}(\theta(k) \neq \theta^*) \bigg].
\end{align*}
Let $\theta^\circ$ denote the second largest packet reception probability. When $\theta(k) \neq \theta^*$, we can show similar to \eqref{eqn:r_epsilon_greedy} that
\begin{align*}
\textnormal{tr} (\mathbb{E}[P_k]-\mathbb{E}[P_k^*])
& \geq \textnormal{tr} \big( (\theta^* - \theta^\circ) ( h(\mathbb{E}[P_{k-1}^*]) - \overline{P} )\big) \nonumber \\
& \geq (\theta^* - \theta^\circ) \min_{k \geq 1} \textnormal{tr}\big( h(\mathbb{E}[P_{k-1}^*]) - \overline{P}\big) \nonumber \\
&\triangleq r^\circ > 0.
\end{align*}
Thus
\begin{align*}
\textnormal{regret}(T) & \geq r^\circ \mathbb{E} \bigg[ \sum_{k=1}^T \mathds{1}(\theta(k) \neq \theta^*) \bigg] \\
& = r^\circ \mathbb{E}[N_{\textnormal{sub}}(T)] \\
& = \Omega (\log T),
\end{align*}
where the last line follows from Lemma \ref{lemma:suboptimal_arms}.
This completes the proof that $\textnormal{regret}(T) = \Omega (\log T)$, and hence establishes Theorem \ref{thm:regret_bound_general}.
\begin{remark}
\label{remark:regret_special_case}
An alternative argument that $\textnormal{regret}(T) = O (\log T)$ for the special case where all channels are stabilizing ($\theta_{\textnormal{min}} > \theta_c$) can be obtained by making use of results from \cite{Qin_optimalDoS} as follows:
\par Consider a schedule, say $\mathcal{B}$, where the optimal channel is used for the first $\frac{T-N_w}{2}$ time slots, the worst channel is used for the next $N_w$ time slots, and the optimal channel is used for the last $\frac{T-N_w}{2}$ time slots. Then the results of \cite{Qin_optimalDoS} show that this schedule will maximize the sum of expected error covariances, among all schedules where there are $N_w$ uses of the worst channel in a time interval of length $T$. This corresponds to a leading block of $\frac{T-N_w}{2}$ uses of the best channel, followed by a single epoch consisting of $N_w$ uses of the worst channel and $\frac{T-N_w}{2}$ uses of the best channel. The bound in \eqref{eqn:regret_epoch_stable} then simplifies to
\begin{align*}
\sum_{i=1}^{N_e} \sum_{k=k_{i\!-\!1}^b+1}^{k_i^b} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{B}] - \mathbb{E}[P_k^*]) &= \sum_{k=k_{0}^b+1}^{k_1^b} \textnormal{tr} (\mathbb{E}[P_k^\mathcal{B}] - \mathbb{E}[P_k^*])\\ & \leq \textnormal{tr}(\mathcal{P}) N_w + r^b,
\end{align*}
which will again lead to a logarithmic regret upper bound. $\hfill\square$
\end{remark}
\section{Numerical Studies}
\label{sec:numerical}
We consider a situation with $M=4$ channels.
In Table \ref{table:algorithm_comparison} we show the simulated estimation regret for various different channel probabilities $$\bm{\theta}=(\theta_1, \theta_2, \theta_3, \theta_4)$$ and system matrices $A$, with $C = \begin{bmatrix} 1 & 1\end{bmatrix}$, $Q=I$, $R=1$, under $\varepsilon$-greedy, Thompson sampling (TS), OBS, and SBS. The regret is computed over horizon $T = 1000$, and averaged over 10000 runs. For $\varepsilon$-greedy, we considered different values of $\varepsilon$ in steps of 0.02, with the value giving the smallest simulated regret presented in Table~\ref{table:algorithm_comparison}.
\begin{table*}[t!]
\caption{Estimation regret of different algorithms}
\centering
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
$\bm{\theta}$ & $A$ & $\theta_c$ & $\varepsilon$ & $\varepsilon$-greedy & TS & OBS & SBS \\ \hline \hline
$(0.5, 0.4, 0.3, 0.2)$ & $\begin{bmatrix} 1.15 & 0.1 \\ 0 & 0.9 \end{bmatrix}$ & 0.244 & 0.08 & 946 & 635 & 584 & \textbf{579} \\
\hline
$(0.6, 0.5, 0.2, 0.1)$ & $\begin{bmatrix} 1.15 & 0.1 \\ 0 & 0.9 \end{bmatrix}$ & 0.244 & 0.06 & 437 & 251 & 237 & \textbf{232} \\ \hline
$(0.65, 0.55, 0.45, 0.35)$ & $\begin{bmatrix} 1.2 & 0.1 \\ 0.2 & 1.1 \end{bmatrix} $ & 0.408 & 0.12 & 886 & 517 & \textbf{473} & 483 \\ \hline
$(0.7, 0.6, 0.4, 0.3)$ & $\begin{bmatrix} 1.2 & 0.1 \\ 0.2 & 1.1 \end{bmatrix} $ & 0.408 & 0.10 & 582 & 295 & 274 & \textbf{273} \\ \hline
$(0.8, 0.7, 0.6, 0.5)$ & $\begin{bmatrix} 1.5 & 0.2 \\ 0.3 & 0.9 \end{bmatrix} $ & 0.603 & 0.18 & 803 & 403 & \textbf{354} & 363 \\ \hline
$(0.9, 0.8, 0.7, 0.5)$ & $\begin{bmatrix} 1.5 & 0.2 \\ 0.3 & 0.9 \end{bmatrix} $ & 0.603 & 0.14 & 290 & 104 & 98 & \textbf{97} \\ \hline
\end{tabular}
\label{table:algorithm_comparison}
\end{table*}
As expected, the $\varepsilon$-greedy algorithm is outperformed by the sampling based algorithms.
It also seems that OBS and SBS can achieve some improvement (on the order of 5-10 \%) in performance over Thompson sampling.\footnote{Note that when looking at individual simulation runs, we cannot say that any one scheme will outperform any other.} In our simulation study the performance of OBS and SBS are quite close to each other; in some scenarios SBS performs slightly better, while in other scenarios OBS performs better.
In Fig. \ref{fig:regret_comparison_plot} we plot the estimation regret over time for the $\varepsilon$-greedy and Thompson sampling algorithms. Plots for OBS and SBS are qualitatively similar to Thompson sampling and are omitted. The parameters used are the same as those in the fifth scenario of Table~\ref{table:algorithm_comparison}. We see a linear scaling (at larger times) for $\epsilon$-greedy and a logarithmic scaling in the case of Thompson sampling, in agreement with Theorems \ref{thm:regret_bound_epsilon_greedy} and \ref{thm:regret_bound_general}.
\begin{figure}[t!]
\centering
\includegraphics[scale=0.5]{regret_comparison_plot.pdf}
\caption{Estimation regret over time}
\label{fig:regret_comparison_plot}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We have studied a remote estimation problem where a sensor can choose between a number of channels with unknown channel statistics, with which to transmit over. We have made use of bandit algorithms to learn the statistics of the channels, while simultaneously carrying out the estimation procedure. Stability of the remote estimation process using these algorithms has been shown, and bounds on the estimation regret have been derived.
Future work will include the use of bandit algorithms in the allocation of channels to multiple processes.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,619 |
Description When I do a find via either *f or ctrl-f, the results flash onto the screen then immediately vanish. I see the results long enough to see that they are populated with correct-looking information.
Version: 0.14-a0-1337-g1537c4c on CAO, I have also reproduced this on CSO git version as well. This does -not- happen when I play 0.13.
Could potentially be an issue with old Firefox version.
Added a workaround in 0.14-a0-1415-ge301bcb.
I can confirm it does work now. Thanks. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,752 |
{"url":"https:\/\/questions.examside.com\/past-years\/jee\/question\/let-theta-in-left-0pi-over-4-right-and-t1-left-tan-theta-rig-jee-advanced-2006-marks-3-mczrhssowkvnsgsr.htm","text":"NEW\nNew Website Launch\nExperience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc...\n1\n\nIIT-JEE 2006\n\nLet $$\\theta \\in \\left( {0,{\\pi \\over 4}} \\right)$$ and $${t_1} = {\\left( {\\tan \\theta } \\right)^{\\tan \\theta }},\\,\\,\\,\\,{t_2} = \\,\\,{\\left( {\\tan \\theta } \\right)^{\\cot \\theta }}$$, $${t_3}\\, = \\,\\,{\\left( {\\cot \\theta } \\right)^{\\tan \\theta }}$$ and $${t_4}\\, = \\,\\,{\\left( {\\cot \\theta } \\right)^{\\cot \\theta }},$$then\nA\n$${t_1} > {t_2} > {t_3} > {t_4}$$\nB\n$${t_4} > {t_3} > {t_1} > {t_2}$$\nC\n$${t_3} > {t_1} > {t_2} > {t_4}$$\nD\n$${t_2} > {t_3} > {t_1} > {t_4}$$\n2\n\nIIT-JEE 2006 Screening\n\nThe values of $$\\theta \\in \\left( {0,2\\pi } \\right)$$ for which $$2\\,{\\sin ^2}\\theta - 5\\,\\sin \\theta + 2 > 0,$$ are\nA\n$$\\left( {0,{\\pi \\over 6}} \\right)\\, \\cup \\,\\left( {{{5\\pi } \\over 6},2\\pi } \\right)$$\nB\n$$\\,\\left( {{\\pi \\over 8},{{5\\pi } \\over 6}} \\right)$$\nC\n$$\\left( {0,{\\pi \\over 8}} \\right)\\, \\cup \\,\\left( {{\\pi \\over 6},{{5\\pi } \\over 6}} \\right)$$ v\nD\n$$\\,\\left( {{{41\\pi } \\over {48}},\\,\\pi } \\right)$$\n3\n\nIIT-JEE 2005 Screening\n\n$$\\cos \\left( {\\alpha - \\beta } \\right) = 1$$ and $$\\,\\cos \\left( {\\alpha + \\beta } \\right) = 1\/e$$ where $$\\alpha ,\\,\\beta \\in \\left[ { - \\pi ,\\pi } \\right].$$\nParis of $$\\alpha ,\\,\\beta$$ which satisfy both the equations is\/are\nA\n0\nB\n1\nC\n2\nD\n4\n4\n\nIIT-JEE 2004 Screening\n\nGiven both $$\\theta$$ and $$\\phi$$ are acute angles and $$\\sin \\,\\theta = {1 \\over 2},\\,$$ $$\\cos \\,\\phi = {1 \\over 3},$$ then the value of $$\\theta + \\phi$$ belongs to\nA\n$$\\left( {{\\pi \\over 3},\\left. {{\\pi \\over 2}} \\right]} \\right.$$\nB\n$$\\left( {{\\pi \\over 2},{{2\\pi } \\over 3}} \\right)$$\nC\n$$\\left( {{{2\\pi } \\over 3},\\left. {{{5\\pi } \\over 6}} \\right]} \\right.$$\nD\n$$\\left( {{{5\\pi } \\over 6},\\pi } \\right]$$\n\nJoint Entrance Examination\n\nJEE Main JEE Advanced WB JEE\n\nGATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN\n\nNEET\n\nClass 12","date":"2022-05-28 06:59:21","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 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\": 0.8607729077339172, \"perplexity\": 8886.77999911094}, \"config\": {\"markdown_headings\": false, \"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-2022-21\/segments\/1652663013003.96\/warc\/CC-MAIN-20220528062047-20220528092047-00456.warc.gz\"}"} | null | null |
/*
* To change this license header, choose License Headers in Project Properties.
* To change this template file, choose Tools | Templates
* and open the template in the editor.
*/
package com.yousoft.stram.persistence.dao.service;
import com.yousoft.stram.domain.ColorsVehicule;
import com.yousoft.stram.domain.Vehicule;
import com.yousoft.stram.exception.StramException;
import org.springframework.stereotype.Service;
/**
*
* @author jguinart
*/
public interface ColorsVehiculeService {
int insertColorsVehicule(Vehicule vehicule) throws StramException;
ColorsVehicule getColorsVehiculeByName(Vehicule vehicule) throws StramException;
}
| {
"redpajama_set_name": "RedPajamaGithub"
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Q: Detect img src and then change it with javascript This is my code:
var i;
var pic = document.getElementById('image');
var picSrc = pic.src;
var fullSrc = picSrc.split('h.jpg')[0] + '.jpg';
pic.src = fullSrc;
document.getElementById('next').onmousedown = function () {
i = 0;
// it works up to here
pic.addEventListener("DOMAttrModified", function(event) {
if (i == 0 && event.attrName == "src") {
pic = document.getElementById('image');
i = 1; // this is to prevent endless loop
picSrc = pic.src;
fullSrc = picSrc.split('h.jpg')[0] + '.jpg';
pic.src = fullSrc;
}
});
return true;
};
It should work on imgur's horizontal layout albums, and replace the low-res images with full-res ones, one image at a time (currently displayed image).
On click of the "next" button, a new image is displayed. However, the script does not load the next full-res image. It only works with the first image loaded.
A: You're messing up your scope completely, invalidating the entire code after the first run. This should also pop up more than enough errors in your console. Reshuffle assignments to the right spot:
document.getElementById('next').onmousedown = function () {
var i;
var pic = document.getElementById('image');
var picSrc = pic.src;
var fullSrc = picSrc.split('h.jpg')[0] + '.jpg';
pic.src = fullSrc;
pic.addEventListener("DOMAttrModified", function(event) {
if (i == 0 && event.attrName == "src") {
pic = document.getElementById('image');
i = 1; // this is to prevent endless loop
picSrc = pic.src;
fullSrc = picSrc.split('h.jpg')[0] + '.jpg';
pic.src = fullSrc;
}
});
return true;
};
Depending on how the page works specifically (I can't see without having a real world use case) you might have to reassign the entire mousedown event as well.
A: On every mousedown you are adding a new DOMAttrModified event listener.
Try arranging your code to something like following:
var pic = document.getElementById('image');
var i;
pic.addEventListener("DOMAttrModified", function(event) {
if (i == 0 && event.attrName == "src") {
//pic = document.getElementById('image');
i = 1; // this is to prevent endless loop
picSrc = pic.src;
fullSrc = picSrc.split('h.jpg')[0] + '.jpg';
pic.src = fullSrc;
i = 0;
});
document.getElementById('next').onmousedown = function () {
var picSrc = pic.src;
var fullSrc = picSrc.split('h.jpg')[0] + '.jpg';
pic.src = fullSrc;
});
You should also try using the addEventListener instead of onmousedown
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,881 |
Q: Will IIS always use virtual directory over a real directory? If I setup a virtual folder under a website, but there is also a real folder with the same name then can I trust that IIS will always use the virtual path rather than the real path?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 457 |
\section{Introduction}
On a connected non-compact Riemannian manifold $M$ without boundary,
consider the heat diffusion governed by $\LL^Vf:=\Delta f + b\cdot
\nabla f -V f$ where $f\in C_0^\infty(M)$ (the space of all real
infinitely differentiable functions with compact support), where
$\Delta, \nabla$ are respectively the Laplace-Beltrami operator and
the gradient on $M$. Here the vector field $b$ is locally
Lipschitzian and represents the macroscopic velocity of the heat
diffusion, $V: M\to \rr^+$ is a locally bounded potential killing
the heat.
Let $u(t,x)dx$ be the heat distribution at time $t$. It satisfies
the well known Fokker-Planck equation in the distribution sense
\bequ \label{FP}
\partial_t u = (\LL^V)^* u(t,x)= \Delta u - div(ub) -
V u, \ u(0,\cdot)\ \text{given}. \nequ
A $L^1(M,dx)$-solution to (\ref{FP}) means that $t\to
u(t)=u(t,\cdot)$ is continuous from $\rr^+$ to $L^1(M,dx)$ and
$$
\<u(t)-u(0), f\> = \int_0^t \<u(s), \LL^V f\> ds, \ \forall t\ge0, \
f\in C_0^\infty(M),
$$
where $\<f,g\>=\int_Mf(x)g(x)dx$.
The study on this subject has a long history when $\LL^V
=\Delta$:
\bdes
\item{a)} the subject was opened by S. T. Yau \cite{Yau1, Yau2}.
Once if $M$ is complete and $1<p<+\infty$, every nonnegative
subharmonic functions in $L^p(M,dx)$ are constant (\cite{Yau1}),
and the $L^p$-uniqueness of the above Fokker-Planck equation holds
(due to Strichwarz \cite{Str1}). In \cite{WZ2} it is proved that the
$L^p$-Liouville property for nonnegative subharmonic functions
implies the $L^p$-uniqueness of the above Fokker-Planck equation for
general $\LL^V$ instead of $\Delta$.
\item{b)} For the $L^\infty$-Liouville property, Yau \cite{Yau1}
proved that every bounded harmonic function is constant if $M$ has
nonnegative Ricci curvature. The last curvature condition is shown
to be sharp, since there are infinitely many bounded harmonic
functions on a simply connected manifold with sectional curvature
identically $-1$. The final result in this opposite direction was
obtained by Sullivan \cite{Su} and Anderson \cite{An}: on a complete
$M$ with (strongly) negative sectional curvature they identified
the Martin boundary of $M$ as the sphere at infinity $S(\infty)$.
See Anderson-Schoen \cite{AS}, Schoen-Yau \cite{SY} for development
of this subject.
\item{c)} For the $L^\infty$-uniqueness of (\ref{FP}) with
$\LL^V=\Delta$, Davies \cite{Da} proved that it is equivalent to the
stochastic completeness of $M$ (i.e., the Brownian motion on $M$
does not explode). Grigor'yan \cite{Gri1} found sharp volume growth
condition for the stochastic completeness of $M$.
\item{d)} The question of $L^1$-uniqueness for (\ref{FP}) is much more delicate.
Azencott \cite{Az} and P. Li and Schoen \cite{LS} found several
counter-examples for which the $L^1$-uniqueness of (\ref{FP}) fails.
P. Li \cite{Li} found the following sharp sufficient condition for
the $L^1$-uniqueness of (\ref{FP}) (with $\LL^V=\Delta$) on a
complete Riemannian manifold : \bequ \label{PLi} Ric_x \ge -C( 1+
d(x,o)^2) \nequ where $Ric_x$ is the Ricci curvature at $x$, $C>0$
is some constant, $o$ is some fixed point and $d(x,o)$ is the
Riemannian distance. Under that condition he proved that every
nonnegative $L^1(M,dx)$-subharmonic function is constant.
\ndes
Recently the second named author and Y. P. Zhang \cite{WZ2}
introduced the $L^\infty$-uniqueness of $\LL^V$ and prove that it is
equivalent to the $L^1$-uniqueness of (\ref{FP}) and also to the
$L^1$-uniqueness of the resolvent equation:
\bequ \label{resolvent} \text{if }\ u\in L^1(M,dx)\ \text{ verifies
}\ [(\LL^V)^* -1] u=0, \ \text{ then }\ u=0. \nequ
Furthermore when $M=\rr^d$ and $V=0$, necessary and sufficient
conditions are found for the $L^1$-uniqueness of (\ref{FP}) in the
one-dimensional case ($d=1$), and sharp sufficient conditions are
obtained in the multi-dimensional case. Our main purpose of this
work is to generalize the results of \cite{WZ2}. However this is not
just a generalization, indeed the new difficulty is comparable to
that in the classical passage from the Laplacian $\Delta$ to the
Schr\"odinger operator $-\Delta +V$.
The $L^2$-uniqueness for (\ref{FP}) might seem to be the most
important and natural. This is true from the point of view of
quantum mechanics when $b=0$ (in such case it is also equivalent to
the $L^2$-uniqueness of the associated Schr\"odinger equation or the
essential self-adjointness of $-\Delta +V$). But from the point of
view of heat diffusion, the $L^1$-uniqueness is physically
meaningful and it is then important: indeed in the heat diffusion
interpretation, $u(t,x)\ge 0$ is the energy ($=$ heat) density and
the $L^1$-norm $\int_M |u(t,x)| dx$ is the total energy in the
system at time $t$; the quantities $\int u^2(t,x) dx$ or $\int
|\nabla u(t,x)|^2 dx$, though called energy in mathematical
language, are not energy in the physics of heat diffusion.
Let us explain where comes the non-uniqueness of solutions to the
Fokker-Planck equation (\ref{FP}) from two points of view.
{\bf 1) Mathematically.} When $M$ is not complete, one can impose
different boundary conditions on the ``boundary" $\partial M:= \bar M \backslash
M$ (which may vary and depend on different topologies) to obtain different solutions, such as Dirichlet boundary and
Neumann boundary etc. Even if $M$ is complete, integrability or
growth conditions will be required to assure the uniqueness of
solution.
{\bf 2) Physically.} The non-uniqueness comes from the
interchange of heat between $M$ and its ``boundary". For example
the $L^\infty$-uniqueness of (\ref{FP}) with $\LL^V=\Delta$ is
equivalent to the non-explosion of the Brownian Motion on $M$ (i.e. $M$
is stochastically complete) by \cite{Da},
which means that the heat from the interior of $M$ can not reach
the boundary $\partial$ (the one-point compactification of $M$).
This intuitive idea is realized on a connected open domain $M$ of
$\rr^d$ for $\Delta -V$ and for the Nelson's diffusions
$\Delta - \nabla \phi\cdot \nabla$ by the second named author in
\cite{Wu98} and \cite{Wu99}.
There is another way of interchange of heat between $M$ and its
``boundary": the heat at the boundary can enter into the interior
of $M$. Indeed for the one-dimensional Sturm-Liouville operator
without killing potential (i.e., $V=0$) on an open interval $M$ of $\rr$, the second named author
with Y. Zhang \cite{WZ1, WZ2} proved that the $L^1$-uniqueness of
the associated Fokker-Planck equation is equivalent to say that
the boundary is {\it no entrance } boundary in the classification
of Feller, which exactly means in the probabilistic interpretation
that the heat at the boundary can not enter into the interior
of $M$. This is very intuitive: if the heat at the ``boundary" can
enter into the interior of $M$, new energy can be inserted from the
``boundary" into $M$ without being perceived by the local operator
$\LL^V$, and then destroys the $L^1$-uniqueness of (\ref{FP}).
The goal of this work is to realize the last physical intuition for
general $\LL^V$. All results in this work are inspired by
probabilistic (=physical) ideas, but for a larger audience all
crucial proofs will be analytic.
This paper is organized as follows. In the next section, we
introduce some preliminaries and present characterizations and
applications of the $L^\infty$-uniqueness of $\LL^V$ to the
$L^1$-Liouville property. Section 3 is devoted to the study of one
dimensional Sturm-Liouville operators $\LL^V=a(x)\frac
{d^2}{dx^2}+b(x)\frac d{dx} -V$. We shall furnish a necessary and
sufficient condition for the $L^\infty$-uniqueness of $\LL^V$ by
means of a new notion of {\it no entrance boundary}. A comparison
principle is derived and several examples are presented. In Section
4, we establish a sharp sufficient condition for the
$L^\infty$-uniqueness of $\LL^V$ on Riemannian manifolds by means of
comparison with a one-dimensional model. Several examples are
presented.
\section{$L^\infty$-uniqueness of pre-generator and $L^1$-Liouville property}
Throughout this paper we assume that vector filed $b$ is locally
Lipschitzian and the killing potential $V$ is nonnegative and
locally bounded (measurable of course).
\subsection{Background on $L^\infty$-uniqueness of pre-generator}
Given the operator $\LL^V$ acting on $C_0^\infty(M)$, let
$(X_t)_{0\le t\le \sigma}$ be the (stochastic) diffusion generated
$\LL = \Delta + b\cdot\nabla$, defined on $(\Omega, \FF,
(\pp_x)_{x\in M})$, where $\sigma$ is the explosion time (see
Ikeda-Watanabe \cite{IW}). Then by Feynman-Kac formula,
\bequ \label{21a} P_t^V g(x) = \ee^x 1_{t<\sigma}g(X_t)
\exp\left(-\int_0^t V(X_s) ds\right) \nequ is one semigroup
generated by $\LL^V$, i.e., for all $t\ge0$,
$$
P_t^V f - f =\int_0^t P_s^V (\LL^V f) ds, \ \forall f\in
C_0^\infty(M).
$$
But $(P_t^V)$ is not strongly continuous on $L^\infty(M, dx)$ w.r.t.
the norm $\|\cdot\|_\infty$ (indeed Lotz's theorem says that the
generator of every strongly continuous (or $C_0-$) semigroup of
operators on $(L^\infty, \|\cdot\|_\infty)$ is bounded). So it is
impossible to define the uniqueness of $C_0-$semigroup on $L^\infty$
generated by $\LL^V$, in the norm $\|\cdot\|_\infty$. That's why we
introduce in \cite{WZ2} the topology $\CC(L^\infty, L^1)$ on
$L^\infty$ of uniform convergence over compact subsets of $L^1$. It
is proved in \cite{WZ2} that a semigroup of bounded operators on
$L^\infty$ is strongly continuous on $L^\infty$ with respect to
$\CC(L^\infty, L^1)$ if and only if $(P_t)=(Q_t^*)$, where $(Q_t)$
is a $C_0$-semigroup on $L^1$ (w.r.t. the $L^1$-norm). Now the
$L^\infty$-uniqueness of $\LL^V$ can be defined as
\begin{defn}{\rm (\cite{WZ2})}
We call that $\LL^V$ is $L^\infty$-unique, if the closure
$\overline{\LL^V}$ of $(\LL^V, C_0^\infty(M))$ is the generator of
$(P_t^V)$ on $(L^\infty, \CC(L^\infty, L^1))$, in the graph topology
induced by $\CC(L^\infty, L^1)$.
\end{defn}
Let
\bequ \label{radius} \lambda_0:= \lim_{t\to\infty} \frac 1t \log
\|P_t^V1\|_\infty = \inf_{t>0} \frac 1t \log \sup_{x\in M}
P_t^V1(x),\nequ i.e., $e^{\lambda_0 t}$ is the spectral radius of
$P_t^V$ in $L^\infty(M,dx)$, which is always in the spectrum of
$P_t^V$ in $L^\infty(M,dx)$. Since $V\ge0$, we have always
$\lambda_0\le0$. Recall
\bthm \label{thm21} {\rm (\cite[a particular case of Theorem
2.1]{WZ2})} The following properties are equivalent:
\bdes
\item{(i)} $\LL^V$ is $L^\infty$-unique;
\item{(ii)} for some or equivalently for all $\lambda>\lambda_0$, if $ u\in L^1(M,dx)$ verifies
$[(\LL^V)^* -\lambda] u=0$, then $u=0$;
\item{(iii)} the Fokker-Planck equation (\ref{FP}) has a unique
$L^1(M,dx)$-solution;
\item{(iv)} $(P_t^V)$ given by (\ref{21a}) is the unique
$C_0$-semigroup on $(L^\infty, \CC(L^\infty, L^1))$ such that its
generator extends $\LL^V$. \ndes
\nthm
By the theory for elliptic partial differential equations (PDE),
$$P_t^Vf(x)= \int_M p_t^V(x,y) f(y)dy$$
and it is known that if $0\le u(0)=u(0,\cdot)\in L^1(M,dx)$,
$$
u(t,y):= (P_t^V)^* u(0) (y) = \int_M u(0,x) p_t^V(x,y) dx
$$
is the minimal nonnegative solution to (\ref{FP}).
\subsection{$L^1$-Liouville property}
At first we should understand the meaning of harmonic functions
related with $\LL^V$. When $\LL^V=\Delta$, a harmonic function $h$
(i.e., $\Delta h=0$) is a solution independent of $t$ to (\ref{FP})
(i.e., the equilibrium distribution of heat). For $\LL=\Delta
+b\cdot \nabla$, the equilibrium distribution $h$ of heat satisfies
Kolmogorov's equation
$$
\LL^* h=\Delta h - div(hb)=0.
$$
However in presence of the killing potential $V\ge0$, usually
equilibrium distribution $h$ is zero. So some further interpretation
is required. Since $p_t^V(x,y)>0, \ dy-a.e.$ for every $x\in M$, the
dimension of \bequ\label{II} \II:=\{h\in L^1(M,dx); \ (P_t^V)^* h=
e^{\lambda_0 t}h, \forall t\ge0 \} \nequ is at most one
(Perron-Frobenius theorem), and if its dimension is one, then it is
generated by some strictly positive $h_0$ such that $\int_M h_0
dx=1$ (by the theory of positive operators \cite{MN91}).
\bdef A function $h\in L^1_{loc}(M,dx)$ (the space of real locally
$dx$-integrable functions on $M$) is said to be
$(\LL^V-\lambda)^*$-harmonic where $\lambda\in \rr$, if
$$
\<h, (\LL^V -\lambda ) f\>=0, \ \forall f\in C_0^\infty(M)
$$
(recall that $\<f,g\>=\int_M fgdx$). It is said to be
$(\LL^V-\lambda)^*$-subharmonic, if
$$
\<h, (\LL^V -\lambda ) f\>\ge0, \ \forall 0\le f\in C_0^\infty(M).
$$
\ndef
We now state our result about the $L^1$-Liouville property.
\bthm\label{thm22} Assume that $\LL^V$ defined on $C_0^\infty(M)$ is
$L^\infty$-unique. Let $\lambda\in\rr$. Then for $h\in L^1(M,dx)$,
it is $(\LL^V-\lambda)^*$-harmonic if and only if
$$
(P_t^V)^* h= e^{\lambda t} h, \ \forall t\ge0.
$$
In particular we have the following alternatives : \bdes \item{(a)}
If $\lambda>\lambda_0$ or $\lambda=\lambda_0$ but $dim(\II)=0$, then
every $(\LL^V-\lambda)^*$-harmonic function $h$ in $L^1(M,dx)$ is
zero.
\item{(b)} If $\lambda=\lambda_0$ and $dim(\II)=1$, then every
$(\LL^V-\lambda)^*$-harmonic function $h$ in $L^1(M,dx)$ is $c h_0$
where $h_0$ is the strictly positive element in $\II$ such that
$\int_M h_0 dx=1$ and $c$ is a constant.
\ndes \nthm
The results above without the $L^\infty$-uniqueness of $\LL^V$ are
in general false, see Li-Schoen's Example \ref{exa41}.
\bprf The sufficient part is obvious by differentiating on $t=0$
(and holds true even without the $L^\infty$-uniqueness of $\LL^V$). Let us prove the
necessity. Consider the generator $\LL^V_{(\infty)}$ of $(P_t^V)$
in $L^\infty(M,dx)$. For every $f$ belonging the domain of
definition $\dd(\LL^V_{(\infty)})$, there is a nest $(f_i)$ in
$C_0^\infty(M)$ such that
$$
f_i\to f, \ \LL^V f_i\to \LL^V_{(\infty)}f
$$
in the topology $\CC(L^\infty, L^1)$ by the assumed
$L^{\infty}$-uniqueness. Thus we obtain for all $f\in
\dd(\LL^V_{(\infty)})$,
$$
\<h, (\LL^V_{(\infty)} - \lambda) f\>=0
$$
which implies (since $P_t^V f\in \dd(\LL^V_{(\infty)})$ for all
$t\ge0$)
$$
\frac d{dt}\<e^{-\lambda t} (P_t^V)^*h, f\>=\frac d{dt} \<h,
e^{-\lambda t} P_t^V f\>=\<h, (\LL^V_{(\infty)} - \lambda)
e^{-\lambda t} P_t^V f\>=0, \ \forall t\ge0
$$
where it follows that $\<e^{-\lambda t} (P_t^V)^*h, f\>=\<h,f\>$.
Since $\dd(\LL^V_{(\infty)})$ is dense in $L^\infty(M,dx)$ with
respect to $\CC(L^\infty, L^1))$ (\cite{WZ2}), we get $e^{-\lambda
t} (P_t^V)^*h=h$ for all $t\ge0$.
When $\lambda>\lambda_0$, the Liouville property in (a) is
equivalent to the $L^\infty$-uniqueness of $\LL^V$ (\cite[Theorem
0.2 or Theorem 2.1]{WZ2}). If $\lambda=\lambda_0$, the last part of (a) and (b) follow easily
from the previous equivalence.
\nprf
\bexam\label{exa21} {\rm $\LL^V=\Delta$. Assume that $\Delta$ is
$L^\infty$-unique. By Theorem \ref{thm22}, we have
\bdes \item{(i)} If $M$ is not stochastically complete, then every
integrable harmonic function $h$ is zero. Indeed by Theorem
\ref{thm22} we have $P_t ^*h=h$ where $(P_t)$ is the Brownian motion
(or heat) semigroup. Then $P_t^*|h|\ge |h|$. Since $P_t1<1$
everywhere on $M$, we get that if $h\ne 0$ in $L^1(M,dx)$,
$$
\<1, |h|\> > \<P_t 1, |h|\> =\<1, P_t^*|h|\> \ge \<1, |h|\>
$$
which is a contradiction.
\item{(ii)} If $M$ is stochastically complete and the volume of
$M$ is infinite, then $dim(\II)=0$ and consequently every integrable
harmonic function is zero.
Indeed, if in contrary $dim(\II)=1$, i.e., $\II$ is spanned by some
nonnegative non-zero function $h_0\in L^1(M,dx)$, since
$\lambda_0=0$ by the stochastic completeness of $M$, $h_0dx$ is an
invariant probability measure of the Brownian motion semigroup
$(P_t)$, which implies that the kernel $R_1:=\int_0^\infty e^{-t}
P_tdt$ is positively recurrent (\cite[Proposition 10.1.1]{MT93}).
But for such Markov kernel $R_1$, it has no other nonnegative
invariant measure than $ch_0 dx$ (\cite[Theorem 10.0.1]{MT93}) for
some constant $c>0$. However $dx$ is an invariant measure of $R_1$,
which is infinite. This contradiction yields that $dim(\II)=0$.
\item{(iii)} If $M$ is stochastically complete and the volume of
$M$ is finite, then $dim(\II)=1$ and $\II$ coincides with $\rr$ and
consequently every integrable harmonic function is constant. The
argument is as in (ii). See Example \ref{exa41} for a stochastically
complete and finite volume manifold for which $\Delta$ is not
$L^\infty$-unique and the $L^1$-Liouville property is violated.
\ndes
} \nexam
The argument in the example above leads to
\bcor\label{cor21} Let $\LL^V=\LL=\Delta +b \cdot\nabla$, i.e.,
$V=0$. Assume that $\LL$ is $L^\infty(M,dx)$-unique. Then
\bdes \item{(a)} If the diffusion $(X_t)_{0\le t<\sigma}$ generated
by $\LL$ is explosive, i.e., $\pp_x(\sigma<+\infty)>0$ for some (or
equivalently for all) $x\in M$, then every $\LL^*$-harmonic function
$h$ in $L^1(M,dx)$ is zero.
\item{(b)} If the diffusion $(X_t)_{0\le t<\sigma}$ generated by
$\LL$ is not explosive, i.e., $\pp_x(\sigma<+\infty)=0$ for all
$x\in M$, then either there is no non-zero $\LL^*$-harmonic and
integrable function, or there is one positive $dx$-integrable
$\LL^*$-harmonic function $h_0$ such that for every non-zero
$\LL^*$-harmonic function $h\in L^1_{loc}(M,dx)$, if $h\ge 0$ or
$h\in L^1(M,dx)$, then $h$ is a constant multiple of $h_0$. \ndes
\ncor
In summary if $\LL^V$ is $L^\infty$-unique, we have the
$L^1$-Liouville property stated in Theorem \ref{thm22} and Corollary
\ref{cor21}. Then the main task remained to us is to check the
$L^\infty$-uniqueness of $\LL^V$.
From now on in this paper, $L^{\infty}$ will be endowed with the
topology $\CC (L^{\infty}, L^1)$, and the $L^\infty$-uniqueness of
operators and $C_0$-semigroups etc. on $L^{\infty}$ are always
w.r.t. $\CC (L^{\infty}, L^1)$.
\section{$L^\infty$-uniqueness of Sturm-Liouville operator}
Consider the following Sturm-Liouville operator:
\begin{equation}\label{generator}
\LL^Vf(x) = a(x)f^{\prime\prime} + b(x)f^{\prime}-V(x)f,
\quad\forall f\in C_0^{\infty}(x_0,y_0)
\end{equation}
$(-\infty\leq x_0 <y_0\leq
+\infty)$. Assume that the coefficients $a,\ b,\ V$ of $\LL^V$ in
(\ref{generator}) satisfy
\begin{equation}\label{ass1}
\ a(x),\ b(x)\in L^{\infty}_{loc}(x_0, y_0)
\end{equation}
\begin{equation}\label{ass2}
a(x)>0 \ dx{\rm-a.e.} \ ;\ \frac 1{a(x)},\ V(x) \in
L^{\infty}_{loc}(x_0, y_0); \ V(x)\geq0;
\end{equation}
where $L^{\infty}_{loc}(x_0, y_0)$ (resp. $L^{1}_{loc}(x_0, y_0)$ )
denotes the space of real measurable functions which are essentially
bounded (resp. integrable) w.r.t. the Lebesgue measure $dx$ on any
compact sub-interval of $(x_0, y_0)$. Fix a point $c\in (x_0, y_0)$
and let
\begin{equation}\label{feller1}
s'(x)=\exp\left(-\int^x_c\frac{b(t)}{a(t)}dt\right), \ m'(x)
=\dps\frac{1}{a(x)}\exp\left(\int^x_c\frac{b(t)}{a(t)}dt\right).
\end{equation}
Their primitives $s$ and $m$ are respectively the scale and
speed functions of Feller. Below $m$ will also denote the measure
$m'(x)dx$. It is easy to see that
$$
\<\LL^Vf,\ g\>_m=\<f,\ \LL^Vg\>_m, \ \forall f, g\in
C^\infty_0(x_0, y_0)
$$
where $\dps \<f,\ g\>_m:=\dps\int^{y_0}_{x_0}f(x)g(x)m'(x)dx.$ For
$f\in C^\infty_0(x_0, y_0)$, we can write $\LL^V$ in the following
form of Feller,
$$
\LL^Vf= \frac {d}{dm} \frac d{ds}f-Vf.
$$
Now regard $\LL^V$ as an operator on $L^p(m):=L^p((x_0,y_0),m)$,
$p\in [1,+\infty]$, with domain of definition $C^\infty_0(x_0,
y_0)$. Recall that $ L^\infty(m)$ is endowed always with the
topology $\CC(L^\infty(m), L^1(m))$. Again let $(X_t)_{0\le
t<\sigma}$ be the diffusion in $(x_0,y_0)$ generated by $\LL$ with
the explosion time $\sigma$ (cf. \cite{IM}) and define $P_t^V$ by
the Feynman-Kac formula as in (\ref{21a}). $P_t^V$ is
$m$-symmetric, and its generator $\LL^V_{(p)}$ in
$L^p(m)=L^p((x_0,y_0), m)$ extends $\LL^V$ (defined on
$C_0^\infty(x_0,y_0)$). The problem resides again in the uniqueness.
$\LL^V$ is said {\it $L^p(m)$-unique} ($1\le p\le +\infty$), if
its closure in $L^p(m)$ coincides with $\LL^V_{(p)}$. That
$L^p(m)$-uniqueness is equivalent to the uniqueness of
solution $t\to u(t)$ (continuous from $\rr^+\to L^q(m)$) to the following integral version of
Fokker Planck equation
$$
\<u(t)-u(0), \LL^V f\>_m =\int_0^t \<u(s), \LL^V f\>_m ds, \ \forall
f\in C_0^\infty(x_0,y_0),\ \forall t\ge0
$$
for every $u(0)\in L^q(m)$ given, where $q$ is the conjugate number
of $p\in [1,+\infty]$, i.e., $\dps q=\frac{p}{p-1}$. ({\it In other
words the $L^p$-uniqueness of $\LL^V$ is equivalent to the
$L^q$-uniqueness of the associated Fokker-Planck equation.}) It is
also equivalent to : for any $h\in L^q(m)$,
\bequ\label{Lq} (\<h, (\LL^V-1)f\>_m=0, \ \forall f\in
C_0^\infty(x_0,y_0))\implies h=0. \nequ See \cite{WZ2} for numerous
other characterizations.
The study of $L^2$-uniqueness of the Sturm-Liouville operators was
born with the {\it limit point--limit cycle} theory of Weil (see
\cite{RS}). In a series of pioneering works (here we mention only
\cite{fe1, fe2}) W. Feller investigated thoroughly the different
sub-Markov generator-extensions of $\LL^V$.
The recent study is concentrated on the case where $V=0$. Wielens
\cite{[Wi]} obtained the characterization of $L^2$-uniqueness (or
equivalently the essential self-adjointness) of $\LL$. Furthermore,
Eberle \cite{[Eb]} and Djellout \cite{Dj97} have completely
characterized the $L^p$-uniqueness of $\LL$ for $1<p<\infty$. The
$L^1$-uniqueness, the $L^\infty$-uniqueness are characterized in
\cite{Wu99} and \cite{WZ2}, respectively.
In presence of the killing potential, the problem of uniqueness
becomes much more difficult, just because it is hard to obtain {a
priori} estimates about solutions of the second order ordinary
differential equation with a potential. This can be seen for an
example in the theory of Weil: $\Delta - c/x^2$ $(c>0)$ acting on
$C_0^\infty(0,+\infty)$ is $L^2((0,+\infty),dx)$-unique iff $c\ge
3/4$ (see Reed-Simon \cite{RS}). This simple example (but profound
characterization) excludes any easy integral test criteria such as
those in no killing case.
Our purpose is to find an explicit characterization of the
$L^{\infty}$-uniqueness of $(\LL^V, C^\infty_0(x_0, y_0))$.
\subsection{Main result}
The main result of this section is
\begin{thm}\label{thm1}
$(\LL^V, C^\infty_0(x_0, y_0))$ is unique in $L^{\infty}(m)$ iff
for some or equivalently all $\delta>0$
\begin{equation}\label{1.1}
\int_{c}^{y_0}\sum_{n\geq0}I_n^{V+\delta}(y)m'(y)dy=+\infty,
\end{equation}
\begin{equation}\label{1.2}
\int_{x_0}^{c}\sum_{n\geq0}J_n^{V+\delta}(y)m'(y)dy=+\infty,
\end{equation}
where for all $V\ge 0$,
$$
\aligned I_0^V(y)&=1,\ \ I_n^V(y)=\int_{c}^{y}
s'(r)dr\int_{c}^{r}m'(t)
V(t)I^V_{n-1}(t)dt, \ y\ge c;\\
J_0^V(y)&=1,\ \
J_n^V(y)=\int_{y}^{c}s'(r)dr\int_{r}^{c}m'(t)V(t)J^V_{n-1}(t)dt, \
y\le c.
\endaligned
$$
\end{thm}
\bdef \label{def31}We say that $y_0$ (resp. $x_0$) is {\it no
entrance} boundary for $\LL^V$ if (\ref{1.1}) (resp. (\ref{1.2}))
holds for some or equivalently for all $\delta>0$. \ndef
In other words the $L^\infty$-uniqueness of $\LL^V$ is equivalent to
say that $x_0, y_0$ are no entrance boundary in the sense of
Definition \ref{def31}. In the presence of the killing potential
$V\ge0$, our definition of no entrance boundary is different from
the classical one of Feller (see Ito-Mckean \cite{IM}), so it is a
new notion. The comparison is given in Corollary \ref{cor32} and
Remarks \ref{rem33}.
\brmk\label{thm1-rmk1} {\rm Denote by $I_n^V(x)$ (resp. $J_n^V(x)$)
by $I_n^V(c;x)$ (resp. $J_n^V(x;c)$). One can prove that (\ref{1.1})
and (\ref{1.2}) do not depend on $c\in (x_0,y_0)$. Its proof is
given later.
} \nrmk
\subsection{Proof of Theorem \ref{thm1}}
Throughout this section, the dual operator $(\LL^V)^*$ is taken
w.r.t. $m$, NOT w.r.t. $dx$ unlike in other places of the paper.
We begin with a series of technical lemmas similar to {\bf Lemma
4.5, Lemma 4.6, Lemma 4.7 } of \cite{WZ2}, so we omit their proofs.
\begin{lem}\label{domain}
Let $u,v\in L^1_{loc}((x_0,y_0),m)$ such that
$$
\<u, \LL^V f\>_m = \<v, f\>_m,\ \forall f\in C_0^\infty(x_0,y_0).
$$
Then
\bdes \item{(i)} $u$ has a $C^1$-smooth $dx$-version ${\tilde u}$
such that ${\tilde u}'$ is absolutely continuous;
\item{(ii)} $g:=a \tilde u'' +b \tilde u'-V\tilde u=(1/m')
(\tilde u'/s')'-V\tilde u\in L^1_{loc}(m)$. \ndes In that case
$v=g$.
\end{lem}
\begin{lem}\label{diff} Suppose that $h$ is $C^1(x_0,y_0)$ such that $h'$ is absolutely
continuous and $(h'/s')'=Vm'h$.
Assume that $c_1\in (x_0, y_0)$, $h(c_1)>0$ and $h'(c_1)>0$ (resp.
$h'(c_1)<0$). Then $h'(y)>0$ (resp. $h'(y)<0$) for $\forall y\in
(c_1, y_0)$ (resp. $\forall y\in (x_0, c_1)$).
\end{lem}
\begin{lem}\label{solution} {\rm (essentially due to Feller \cite{fe1})}
Assume that $V\ge\delta>0$, $dx-a.e.$, then there exist two strictly
positive $C^1$-functions $h_k, k=1, 2$ on $(x_0, y_0)$ such that
(1) For $k=1, 2$, $h_k'$ is absolutely continuous, and $(h'_k/s'
)'=m'V h_k$, a.e.;
(2) $h_1'>0$ and $h_2'<0$ over $(x_0, y_0)$.
\end{lem}
Our key observation is
\bprop\label{prop31} Let $h$ be any $C^1$-function on $(x_0,y_0)$
such that $h'$ is absolutely continuous and $(h'/s')'=m'Vh,
dx-a.e.$.
\bdes
\item{(a)} If $h(c)>0, h'(c)>0$ and $\int_c^{y_0}V(t)m'(t)dt>0$, then there is a positive constant
$C$ such that
\bequ\label{prop31a} h(c)\sum_{n=0}^\infty I^V_n(x)\le h(x) \le C
\sum_{n=0}^\infty I^V_n(x), \ \forall x\ge c. \nequ
\item{(b)} If $h(c)>0, h'(c)<0$ and $\int_{x_0}^cV(t)m'(t)dt>0$, then there is a positive constant $C$
such that
\bequ\label{prop31b} h(c)\sum_{n=0}^\infty J^V_n(x)\le h(x) \le C
\sum_{n=0}^\infty J^V_n(x), \ \forall x\le c. \nequ
\ndes
\nprop
\bprf (a) By Lemma \ref{diff}, $h'(r)>0$ for $r\in [c, y_0)$.
Notice that
\begin{align*}
h(x)=&h(c)+\int^x_{c}h'(r)dr \\
=&h(c)+
\int^x_{c}\left\{\frac{h'(c)}{s'(c)}s'(r)
+s'(r)\int^r_{c}m'(t)V(t)h(t)dt \right\}dr\\
>&h(c)+\int^x_{c}s'(r)dr\int^r_{c}m'(t)V(t)h(t)dt.
\end{align*}
Thus using the above inequality recursively, we easily obtain:
$$h(x)\ge h(c)\sum_{n=0}^{+\infty}I^V_n(x),\ \forall x\ge c$$
which is the first inequality in (\ref{prop31a}).
For the second inequality in (\ref{prop31a}), letting $K(x):\
=\int_c^{x}(h'(c)/s'(c))s'(r)dr$ and fixing $c_0\in (c, y_0)$ such
that $\int_c^{c_0} V(t)m'(t)dt>0$, we can choose suitable positive
constants $C_1, \ C_2,\ C_3$ such that for all $x\ge c$,
$$
\aligned
K(x) & \le
C_2+C_1\int_{c_0}^{x}s'(r)dr\\
& \le
C_2+C_3\int_{c}^{x}s'(r)dr\int_{c}^{r}m'(t)V(t)dt=
C_2+C_3I^V_1(x).
\endaligned
$$
Setting $C_4=h(c)+C_2$, we get for all $x\ge c$:
\begin{align*}
h(x)=&h(c)+
\int^x_{c}\left\{\frac{h'(c)}{s'(c)}s'(r)
+s'(r)\int^r_{c}m'(t)V(t)h(t)dt \right\}dr\\
\leq & C_4+C_3I_1(x)+\int_{c}^{x}
s'(r)dr\int_{c}^{r}m'(t)V(t)h(t)dt.
\end{align*} Using it inductively we obtain for all $x\ge c$,
\begin{align*}
h(x)\le & C_4+ (C_3+C_4)I_1(x)+C_3I_2(x)\\
& \ \ \ +\int_{c}^{x}s'(r_1)dr_1\int_c^{r_1} m'(t_1)V(t_1)dt_1
\int^{t_1}_{c}s'(r_2)dr_2\int^{r_2}_{c}m'(t_2)V(t_2)h(t_2)dt_2\\
& \cdots\cdots\cdots\cdots\cdots\cdots\cdots\\
\leq & C_4+(C_3+C_4)\sum_{n=1}^{+\infty}I^V_n(x).
\end{align*}
where the second inequality in (\ref{prop31a}) follows.
(b) Similar to part(a). \nprf
Let us now to the
\bprf[Proof of Remarks \ref{thm1-rmk1}] We prove here the no
entrance property of $y_0$ does not depend on $c$. Denote $I_n^V(x)$
by $I_n^V(c;x)$ to emphasize the role of $c$. Let $x_0<c<c_1<y_0$.
By Feller's lemma \ref{solution}, there is a strictly increasing
positive $C^1$-function $h=h_1$ on $(x_0,y_0)$ such that $h'$ is
absolutely continuous and
$$
(h'/s')' = (V+\delta)h m'
$$
a.e. on $(x_0,y_0)$. Hence $h'(x)>0$ over $(x_0,y_0)$. By
Proposition \ref{prop31}, there is a constant $C>0$ such that for
all $x\ge c_1$,
$$
h(c) \sum_{n=0}^\infty I_n^{V+\delta}(c;x) \le h(x) \le C
\sum_{n=0}^\infty I_n^{V+\delta}(c_1;x).
$$
That completes the proof.
\nprf
\begin{proof}[Proof of Theorem \ref{thm1}] Since the constant
$\lambda_0$ defined in (\ref{radius}) is non-positive, according to
Theorem \ref{thm21} (or more precisely \cite[Theorem 2.1]{WZ2}), the
$L^\infty(m)$-uniqueness of $\LL^V$ is equivalent to : for some or
equivalently for all $\delta>0$, if $h\in L^1(m)$ such that
$$
\<h, (\LL^V-\delta)f\>_m=\<h, \LL^{V+\delta}f\>_m=0,\ \forall f\in
C_0^\infty(x_0,y_0)
$$
then $h=0$. By Lemma \ref{domain}, for such $h$, we may assume that
$h\in C^1(x_0,y_0)$ and $h'$ is absolutely continuous and
\bequ\label{equation}(h'/s' )'=m' (V+\delta) h.\nequ
{\bf Part ``if":} Assume (\ref{1.1}) and (\ref{1.2}) hold for some
$\delta>0$. Suppose in contrary that $0\ne h\in L^1(m)$ is a
solution of (\ref{equation}). We can assume that $h>0$ on some
interval $[x_1, y_1]\subset (x_0, y_0)$ where $x_1<y_1$. Notice that
$h'\not\equiv 0$ on $(x_1, y_1)$ by (\ref{equation}).
\noindent {\bf Case (i):} $h'(c_1)>0$ for some $c_1\in (x_1, y_1)$.
We
obtain from Proposition \ref{prop31}(a):
$$\int_{c_1}^{y_0}h(y)m'(y)dy\ge h(c_1)\int_{c_1}^{y_0}\sum_{n=0}^{+\infty}I^{V+\delta}_n(y)m'(y)dy=+\infty; $$
which is a contradiction with the assumption that $h\in L^1(m)$.
\noindent {\bf Case (ii)}: $h'(c_1)<0$ for some $c_1\in (x_1,
y_1)$. By Proposition \ref{prop31}(b), we have
$$\int_{x_0}^{c_1}m'(y)h(y)dy\ge h(c_1)\int_{x_0}^{c_1}\sum_{n=0}^{+\infty}J^{V+\delta}_n(y)m'(y)dy=+\infty.$$
{\bf Part ``only if":} Let us prove that (\ref{1.2}) holds for all
$\delta>0$. Indeed assume in contrary that for some $\delta>0$,
$$
\int^c_{x_0}m'(y)\sum_{n=0}^{+\infty} J_n^{V+\delta}(y)dy<+\infty.
$$
In particular $\int_{x_0}^c m'(y)dy<\infty.$ Consider a solution $h$
of (\ref{equation}) such that $h>0$ and $h'<0$ over $(x_0, y_0)$,
whose existence is assured by Feller's Lemma \ref{solution}. We
shall prove that $h\in L^1(m)$.
{\bf (1) Integrability near $y_0$:} Let $c\in (x_0,y_0)$. For $y\in
(c,y_0)$ we have
$$ 0\geq h'(y)/s'(y)= h'(c)/s'(c)+\int_c^ym'(t)h(t)(\delta+V(t))dt$$
which implies that $\dps \delta\int_c^{y_0}m'(t)h(t)dt\leq
-h'(c)/s'(c)<+\infty$.
{\bf (2) Integrability near $x_0$:} By Proposition \ref{prop31}(b),
$$
\int_{x_0}^c m'(t)h(t)dt\le C \int_{x_0}^c \sum_{n=0}^{+\infty}
J_n^{V+\delta}(y) m'(y)dy <+\infty.
$$
That completes the proof of the necessity of (\ref{1.2}). For the
necessity of (\ref{1.1}) for all $\delta>0$, the proof is similar :
The only difference is to use a positive and increasing solution $h$
of (\ref{equation}) (whose existence is guaranteed by Lemma
\ref{solution}).
\end{proof}
\subsection{Several corollaries}
\begin{lem}\label{cor31-lem}
Assume that $V(x)=0,\ \forall x\in(x_0, y_0)$. The point $y_0$
is no entrance boundary, i.e.
(\ref{1.1}) holds iff
\begin{equation}\label{entr1}
\int_{c}^{y_0}m'(y)dy\int_{c}^{y}s'(x)dx=+\infty;
\end{equation}
and $x_0$ is no entrance boundary, i.e. (\ref{1.2}) holds iff
\begin{equation}\label{entr2}
\int_{x_0}^{c}m'(y)dy\int_{y}^{c}s'(x)dx=+\infty.
\end{equation}
\end{lem}
\begin{proof} We prove here only the equivalence between (\ref{1.1})
and (\ref{entr1}).
$(\ref{entr1})\implies (\ref{1.1}).$ Let $I_n=I_n^{V+1}$ with $V=0$.
This implication is obvious because for some $c_1\in (c,y_0)$,
$$
\int_c^{y_0} I_1(y) m'(y)dy \ge \int_c^{y_0} m'(y)dy \int_{c_1}^y
s'(x)dx \cdot \int_c^{c_1} m'(y) dy=+\infty.
$$
$(\ref{1.1})\implies (\ref{entr1}).$ If
$m(y_0):=m([c,y_0))=+\infty$, then both (\ref{1.1}) and
(\ref{entr1}) hold true. Assume then $m(y_0)<+\infty$. We have
$$\aligned
\int_{c}^{y_0}I_n(y)m'(y)dy&\leq
m(y_0)\int_{c}^{y_0}m'(t_1)dt_1\int_{c}^{t_1}s'(r_1)dr_1\cdots\int_{c}^{r_{n-1}}m'(t_n)dt_n\int_{c}^{t_n}s'(r_n)dr_n\\
&\le m(y_0)\frac{1}{n!}\left(\int_{c}^{y_0}m'(t)dt\int_{c}^{t}s'(r)dr\right)^n,\\
\endaligned$$
hence $$ \int_{c}^{y_0}\sum_{n\geq0}I_n(y)m'(y)dy\leq
m(y_0)\exp{\left(\int_{c}^{y_0}m'(y)dy\int_{c}^{y}s'(x)dx\right)}.
$$ Then (\ref{entr1}) follows immediately from (\ref {1.1}).
\end{proof}
From the lemma above we get immediately from Theorem \ref{thm1}
\begin{cor}{\rm (\cite[Theorem 4.1]{WZ2}\label{cor31})}
$(\LL, C^\infty_0(x_0, y_0))$ is unique in $L^{\infty}(m)$ if and
only if
\begin{equation}\label{wu1}
\int_{c}^{y_0}m'(y)dy\int_{c}^{y}s'(x)dx=+\infty;
\end{equation} and
\begin{equation}\label{wu2}
\int_{x_0}^{c}m'(y)dy\int_{y}^{c}s'(x)dx=+\infty;
\end{equation} hold.
\end{cor}
\bcor\label{cor32} If
\bequ\label{cor32a} \int_c^{y_0} (1+V(t))m'(t) dt \int_c^t
s'(r)dr<+\infty \nequ then $y_0$ is entrance boundary for $\LL^V$.
Similarly if
\bequ\label{cor32b} \int_{x_0}^c (1+V(t))m'(t) dt \int_{t}^c
s'(r)dr<+\infty \nequ then $x_0$ is entrance boundary for $\LL^V$.
\ncor
\bprf This is obtained by the same proof as that of Lemma
\ref{cor31-lem}. \nprf
\brmk\label{rem33} {\rm In the theory of Feller (see \cite{IM}),
(\ref{cor32a}) and (\ref{cor32b}) are used for the definition of
entrance boundary of $y_0$ and $x_0$ for $\LL^V$. So our definition
of entrance boundary for $\LL^V$ is equivalent to his one if $V=0$
by Corollary \ref{cor31}, but strictly weaker in the presence of a
zero potential $V\ge 0$. For example when $c\in (0,2)$, $0$ is
entrance boundary for $d^2/dx^2 - c/x^2$ on $(0,+\infty)$ in our
sense, but it is not in the sense of Feller's (\ref{cor32b}), see
Example \ref{exam1}.
} \nrmk
We now turn to
\begin{thm}\label{thm1-cor2} (comparison principle) Let
$$
\LL_kf(x)= a_k(x) f''(x) + b_k(x) f'-V_k(x)f, \ \forall f\in
C_0^{\infty}(x_0, y_0)
$$
where $(a_k,\ b_k,\ V_k),\ k=1, 2$ satisfy (\ref{ass1}) and
(\ref{ass2}). Assume that for some $c\in (x_0,y_0)$,
\begin{equation}\label{con}
a_1(x)\ge a_2(x), V_2(x)\geq V_1(x), \end{equation} \bdes
\item{(a).} If \bequ\label{comp2}
\frac {b_1(x)}{a_1(x)} \le \frac{ b_2(x)}{a_2(x)}, \ x\ge c,\\
\nequ and $y_0$ is no entrance boundary for $\LL_1$, so it is for
$\LL_2$.
\item{(b).} If \bequ\label{comp3}
\frac {b_1(x)}{a_1(x)} \ge \frac{ b_2(x)}{a_2(x)},\ x\le c,\nequ
and $x_0$ is no entrance boundary for $\LL_1$, so it is for $\LL_2$.
\ndes
\end{thm}
The conditions above are guided by the intuitive picture of no
entrance boundary. Assume $a_1=a_2$ and $y_0$ is no entrance
boundary for $\LL_1$. Condition $b_2\ge b_1$ means that the heat in
the second system described by $\LL_2$ goes more rapidly to the
boundary $y_0$ than in the first system, and condition $V_2\ge V_1$
means that the heat in the second system is killed more rapidly than
in the first. Then the heat from the boundary $y_0$ goes more
difficultly into the interior in the second system than in the first
one.
\begin{proof} We prove here only the implication for $y_0$.
Let $I_{n,k} (k=1, 2; n\in \nn)$ denote
``$I_n^{V+1}$" with respect to (w.r.t.) $\LL_k \ (k=1, 2).$ By
conditions (\ref{con}) and (\ref{comp2}), we have
\begin{align}
& m_1'(t)(1+V_1(t))\leq m_2'(t)(1+V_2(t)); \\
& \exp\left\{\int_r^{y}\frac{b_1(u)}{a_1(u)}du\right\}\leq
\exp\left\{\int_r^{y}\frac{b_2(u)}{a_2(u)}du\right\}, \ y>r.
\end{align}
Letting $B_{n,k}:=\int_c^{y_0} m_k'(y)I_{n,k}(y)dy$, we have
\begin{align*}
B_{n,k}=&\iint\limits_{c\le r_1\le t_1\le y_0} \frac{1}{a_k(t_1)}
e^{\int_{r_1}^{t_1}\frac{b_k(u)}{a_k(u)}du}dr_1\,dt_1\iint\limits_{c\le
r_2\le t_2\le r_1}
\frac{1}{a_k(t_2)}e^{\int_{r_2}^{t_2}\frac{b_k(u)}{a_k(u)}du}\left(V_k(t_2)+1\right)dr_2\,dt_2\\
&\cdots\iint\limits_{c\le r_n\le t_n\le r_{n-1}}
\frac{1}{a_k(t_n)}e^{\int_{r_n}^{t_n}\frac{b_k(u)}{a_k(u)}du}\left(V_k(t_n)+1\right)dr_ndt_n\int_0^{r_n}
(V_k(t_{n+1})+1)m_k'(t_{n+1}) dt_{n+1}.
\end{align*}
Thus $\forall n$, $B_{n,1}\leq B_{n,2}$. Hence the conclusion
follows.
\end{proof}
\subsection{Dirichlet or Neumann boundary problem}
Consider the Sturm-Liouville operator $\LL^V$ on $[x_0,y_0)$ where
$x_0\in \rr$ and $x_0<y_0\le +\infty$, where $a,b,V$ satisfy always
(\ref{ass1}) and (\ref{ass2}) on $[x_0,y_0)$ (instead of
$(x_0,y_0)$). Consider
$$
\DD_D:=\{f\in C_0^\infty[x_0,y_0);\ f(x_0)=0\}
$$
and
$$
\DD_N:=\{f\in C_0^\infty[x_0,y_0);\ f'(x_0)=0\}.
$$
Denote by $\LL^V_D$ (resp. $\LL^V_N$) the operator with domain of
definition $\DD_D$ (resp. $\DD_N$), which corresponds to the
Dirichlet boundary (resp. Neumann) boundary condition at $x_0$. One
can define $\LL^V_D$ and $\LL^V_N$ similarly on $(x_0,y_0]$ where
$-\infty\le x_0<y_0<+\infty$.
With exactly the same proof we have
\bthm \label{thmA1} $\LL^V_D$ (or $\LL^V_N$) is
$L^\infty([x_0,y_0),m)$-unique iff $y_0$ is no entrance boundary.
$\LL^V_D$ (or $\LL^V_N$) is $L^\infty((x_0,y_0],m)$-unique iff $x_0$
is no entrance boundary. \nthm
\subsection{About $L^p(m)$-uniqueness of $\LL^V$}
\bprop\label{prop3.14} (\cite{WYZ}) $\LL^V$ is $L^1(m)$-unique iff
$$
\aligned &I_1^{V+1}(y_0)=\int_c^{y_0} s'(r) dr \int_c^r m'(t) (1+
V(t))
dt=+\infty;\\
&J_1^{V+1}(x_0)=\int^c_{x_0} s'(r) dr \int_r^c m'(t) (1+ V(t))
dt=+\infty.
\endaligned
$$ \nprop
With exactly the same proof as that of Theorem \ref{thm1}, we have
by Proposition \ref{prop31},
\bprop\label{prop3.15} Let $p\in (1,+\infty)$. $\LL^V$ is
$L^p(m)$-unique iff for some or all $\delta>0$,
\bequ\label{prop3.15a} \int_c^{y_0}\left(\sum_{n=0}^\infty
I^{V+\delta}_n(y) \right)^q m'(y) dy =+\infty;\nequ
\bequ\label{prop3.15b} \int_{x_0}^c\left(\sum_{n=0}^\infty
J^{V+\delta}_n(y) \right)^q m'(y) dy =+\infty. \nequ \nprop
\bdef\label{def32} {\rm If the condition (\ref{prop3.15a}) (resp.
(\ref{prop3.15b})) is verified, $y_0$ (resp. $x_0$) will be called
{\it $L^q(m)$-no entrance boundary} for $\LL^V$. } \ndef
So the no entrance boundary in Definition \ref{def31} is $L^1(m)$-no
entrance boundary. If $V=0$, a much easier criterion is available :
\bprop\label{prop3.16} {\rm (due to Eberle \cite{[Eb]} and
Djellout \cite{Dj97})} Assume that $V=0$. $\LL$ is $L^p(m)$-unique
iff
\bequ\label{prop3.16a}
\int_{c}^{y_0}\left(\int_{c}^{y}s'(x)dx\right)^qm'(y)dy=+\infty;
\nequ
and
\bequ\label{prop3.16b}
\int_{x_0}^{c}\left(\int_{y}^{c}s'(x)dx\right)^qm'(y)dy=+\infty.
\nequ
\nprop
\subsection{Several examples}
For applications of the comparison principle in Theorem
\ref{thm1-cor2}, we should have some standard examples.
\bexam\label{exam1} {\bf (combination of Weil's example and Bessel's
diffusion)} {\rm Let
$$
(x_0,\ y_0)=(0,+\infty),\ \LL^V
f=f''+\frac{\gamma}{x}f'-\frac{c}{x^2},\ c\geq0,\gamma\in \rr.
$$
When $\gamma=0$, this is Weil's example mentioned before, and when
$c=0$, it is the Bessel's process with dimension $\gamma+1$. For
this example $s'(x)=x^{-\gamma},\ m'(x)=x^\gamma$ and $V(x)=c/x^2$.
$+\infty$ is no entrance boundary for $\LL f=
f''+\frac{\gamma}{x}f'$, so for $\LL^V$ by the comparison principle
in Theorem \ref{thm1-cor2}. Furthermore condition (\ref{prop3.16a})
is verified for $y_0=+\infty$, so does (\ref{prop3.15a}).
One decreasing solution for
$(\LL^V)^*h=0$ is given by
$$
h_c(x)= x^{\alpha},\
\alpha=\frac{-(\gamma-1)-\sqrt{(\gamma-1)^2+4c}}{2}.
$$
We have \bdes
\item{(a).} $(\LL,\ C_0^\infty(0,+\infty))$ is $L^1-$unique if and
only if $c>0$ or $c=0$ and $\gamma\geq1$, by Proposition
\ref{prop3.14}.
\item{(b)} If $c=0$ and $p\in (1, +\infty]$, $(\LL,\ C_0^\infty(0,+\infty))$ is
$L^p(m)-$unique if and only if $\gamma\leq-1$ or $\gamma\ge 2p-1$ by
Proposition \ref{prop3.16}.
\item{(c).} Let $p\in(1,+\infty]$ and $c>0$. $(\LL,\ C_0^\infty(0,+\infty))$ is
$L^p(m)-$unique if and only if $\alpha q+\gamma\leq -1 $ (where
$\dps \alpha=\frac{-(\gamma-1)-\sqrt{(\gamma-1)^2+4c}}{2},\
\frac{1}{p}+\frac{1}{q}=1$), or equivalently
$$
c\ge c_{cr}(q,\gamma):= \frac{(\gamma+1)^2}{q^2} -
\frac{\gamma^2-1}{q}.
$$
When $p=2, \gamma=0$, we find Weil's critical value $3/4$.
\item{(d)} If $\gamma=0$ and $p\in(1,+\infty]$, $(\LL,\
C_0^\infty(0,+\infty))$ is $L^p(dx)-$unique if and only if
$c\geq\frac{1}{q^2}+\frac{1}{q}$. This is a particular case of (c).
\ndes
\bprf[Proof of part (c)] If $\alpha q+\gamma\leq -1 $, $\int_0^1
h_c^q (x) m'(x)dx=+\infty$. By Proposition \ref{prop31},
$\sum_{n=0}^\infty J_n^V\notin L^q((0,1], m'(x)dx)$, hence
$\sum_{n=0}^\infty J_n^{V+1}\notin L^q((0,1], m'(x)dx)$ (for
$J_n^{V+1}\ge J_n^V$). Thus by Theorem \ref{thm1} and Proposition
\ref{prop3.15}, $\LL^V$ is $L^p(m)$-unique.
If $\alpha q+\gamma > -1$, $\int_0^1 h_{c(1+\vep)}^{q}
(x) m'(x)dx<+\infty$ for some small $\vep>0$. By Proposition
\ref{prop31}, $\sum_{n=0}^\infty J_n^{(1+\vep)V}\in L^{q}((0,1],
m'(x)dx)$. But for $x\in (0,1]$, as $V+\delta =\frac c{x^2} + \delta
\le \frac {c(1+\vep)}{x^2}$ for $\delta\in (0,\vep)$, we have
$J_n^{V+\delta}\le J_n^{(1+\vep)V}$. Therefore $\sum_{n=0}^\infty
J_n^{V+\delta}\in L^q((0,1], m)$, $\LL^V$ is not $L^p(m)$-unique by
Theorem \ref{thm1} and Proposition \ref{prop3.15}. \nprf } \nexam
\bexam{\rm Let $(x_0,\ y_0)=(0,+\infty)$, $c\geq0,\gamma,\kappa\in
\rr, \kappa\ne 0$ and
$$\LL^V f=x^{\kappa}\left(f''+\frac{\gamma}{x}f'-\frac{c}{x^2}\right),\ f\in C_0^\infty(0,+\infty).$$
We have $s'(x)=x^{-\gamma}$, $m'(x)=x^{-\kappa+\gamma}$. Let $h_c$
be given as in the previous example, which is again
$(\LL^V)^*$-harmonic function. By the same proof as above, we have :
$0$ is {\it $L^q(m)$-no entrance boundary} iff $\alpha q +\gamma
-\kappa\le -1$, where $\dps
\alpha=\frac{-(\gamma-1)-\sqrt{(\gamma-1)^2+4c}}{2}$ and $q\in
[1,+\infty)$. }\nexam
\bexam\label{exa33}{\rm Let $(x_0,y_0)=\rr$, $a(x)=1$ and
$b(x)=\gamma (|x|^\alpha)'$ for $|x|>1$ and continuous on $\rr$, and
$V(x)=c |x|^\beta$ for $|x|>1$ and continuous and nonnegative on
$\rr$. Here $\gamma \in \rr$ and $\alpha, \beta, c\ge0$. In this
example $m'(x)= e^{\gamma |x|^\alpha}$, $s'(x)=e^{-\gamma
|x|^\alpha}$ for $|x|>1$ (for simplicity we have forgotten a
constant factor in $m'$ and $s'$, which plays no role in our
history). The operator $\LL^V$ is given by
$$
\LL^V f = f'' + \gamma \alpha {\rm sgn}(x) |x|^{\alpha-1} f' -
c|x|^\beta, \ |x|>1.
$$
{\bf 1).} For all $1<p<\infty$, $\LL^V$ is $L^p(m)$-unique by
applying Proposition \ref{prop3.16} to the case $V=0$ and then
Proposition \ref{prop3.15}.
\vskip10pt\noindent {\bf 2).} $\LL^V$ is $L^1(m)$-unique iff
$\gamma\le 0$ or ($\gamma>0$ and $1_{c>0}\beta\ge \alpha-2$), by
Proposition \ref{prop3.14}.
\vskip10pt\noindent {\bf 3)} $\LL$ is $L^\infty(m)$-unique iff
$\gamma\ge 0$ or ``$\gamma<0$ and $\alpha \le 2$" (by \cite[Example
4.10]{WZ2}). In such case $\LL^V$ is $L^\infty(m)$-unique by the
comparison principle in Theorem \ref{thm1-cor2}.
Let $\gamma<0$ and $\alpha>2$ below. Then $\LL$ is not
$L^\infty(m)$-unique, and our purpose is to find the critical
potential $V=c|x|^\beta$ so that $\LL^V$ is $L^\infty(m)$-unique or
equivalently $+\infty$ is no entrance boundary.
{\bf Claim : }{\it Let $\gamma<0$ and $\alpha>2$ and
$\beta=\alpha-2$. Set $c_{cr}=|\gamma| \alpha (\alpha-2)$. If
$c>c_{cr}$, $\LL^V$ is $L^\infty(m)$-unique; if $c<c_{cr}$, $\LL^V$
is not $L^\infty(m)$-unique. }
By the symmetry we have only to regard if $+\infty$ is no entrance
boundary. For two positive functions $f,g$, we write $f\sim g$ (at
$+\infty$), if $\lim_{x\to +\infty} f(x)/g(x)=1$; and $f\propto g$
(at $+\infty$), if there are two positive constants $C_1,C_2$ such
that $C_1 f(x)\le g(x)\le C_2 f(x)$ for all $x$ large enough (say
$x\gg 1$).
To prove the claim, consider $h=s(\log s)^{\kappa}$, where $s(1)=e$
and $\kappa>0$. We have $\LL h =h \tilde V$ where
$$
\tilde V = \frac{s'^2}{s^2}\left(\frac \kappa {\log s} +
\frac{\kappa(\kappa-1)}{(\log s)^2 }\right)\sim |\gamma|\kappa
\alpha^2 x^{\alpha -2}
$$
by using $\dps s(x)\sim \frac{e^{|\gamma| x^\alpha}}{|\gamma| \alpha
x^{\alpha-1}}$. Note that $\dps s(\log s)^{\kappa} m'\propto
\frac{1}{x^{\alpha-1-\alpha\kappa}}$. Then $h\in L^1([1,+\infty),
m)$ iff $\kappa< \kappa_0=(\alpha-2)/\alpha$. For $\kappa=\kappa_0$,
$\tilde V\sim c_{cr} x^{\alpha -2}$. Now one can conclude the claim
by means of Proposition \ref{prop31} and Theorem \ref{thm1}.
} \nexam
\section{Riemannian manifold case: comparison with one-dimensional
case}
In this section, let $(M, g)$ be a connected oriented
non-compact
Riemannian manifold of dimension $d\geq $2 with metric $g$ without
boundary, but not necessarily complete. Throughout this section we
denote by $dx$ the volume element, given in local coordinates by $
dx|_U=\sqrt{G}dx_1dx_2\cdots dx_d, \mbox{ where }\ G=det(g_{ij})$.
Let $TM$ be the tangent bundle on $M$. Let $L_{loc}^p(M)$ ($p\in
[1,+\infty]$) be the space of all real measurable functions $f$
such that $f1_K\in L^p(M):=L^p(M,dx)$ for every compact subset $K$
of $M$. Let $H^{1, 2}(M)$ (resp. $H^{1, 2}_{loc}(M)$) be the space
of those functions $f\in L^2(M)$ (resp. $f\in L^2_{loc}(M)$) such
that $|\nabla f|\in L^2(M)$ (resp. $|\nabla f|\in L^2_{loc}(M)$)
where the gradient $\nabla f$ is taken in the distribution sense,
and $|\cdot|$ is the Riemannian metric.
Let us consider the following operator:
\begin{equation}\label{manifoldgenerator}
\LL^Vf(x)=\Delta f(x)+ b (x)\cdot\nabla f(x)
-V(x)f(x),\ f\in C_0^\infty(M)
\end{equation}
where $\Delta, \nabla$ are respectively the Laplace-Beltrami
operator and the gradient on $M$, and $b $ is a locally Lipschitzian
vector field, $0\le V\in L^\infty_{loc}(M)$ (assumed throughout this
section). We write $\LL$ instead of $\LL^V$ if $V=0$.
Now regard $\LL^V$ as an operator
on $L^\infty(M)$ which is endowed with the topology $\CC \left(L^{\infty}(M),\ L^1(M)\right)$, with domain of definition
$C_0^{\infty}\left(M\right)$. Our purpose is to find some sharp
sufficient condition for the $L^\infty$-uniqueness of $\left(\LL^V,\
C_0^{\infty}\left(M\right)\right)$.
\vskip10pt
{\bf{Assumption (A)}}
\bdes
\item{\bf(1) } $\rho:\ M\longrightarrow[x_0, y_0)$ is surjective, where $0\leq
x_0<y_0\leq +\infty$, such that $\rho^{-1}\left([x_0,\ l]\right)$
is compact subset for all $l\in[x_0, y_0)$, and there is some $c\in
[x_0,y_0)$ such that $\rho$ is $C^2$-smooth and $|\nabla \rho|>0$ on
$[\rho>c]$;
\item{\bf (2)} there exist $\alpha (r),\ \beta(r),\ q(r) \in L_{loc}^{\infty}\left([x_0,
y_0),dr\right)$, $q(r)\geq0, \ \alpha>0,\ 1/\alpha (r)\in
L^\infty_{loc}([x_0,y_0),dr)$ and $c\in[x_0,\ y_0)$ such that
$dx$-a.e. on $[\rho>c]$, \ndes
\bequ \label{22ass2} |\nabla \rho(x)|^2\leq \alpha(\rho(x)); \nequ
\begin{equation}\label{2ass1}
\LL\rho (x)\geq \frac{\beta(\rho(x))}{\alpha(\rho\left(x\right))}
|\nabla \rho(x)|^2;
\end{equation}
\begin{equation}
\label{2ass2} V(x)\geq q(\rho (x)).
\end{equation}
\begin{thm}\label{mainthm}
Under the assumption {\bf (A)}, if $y_0$ is no entrance boundary for
$\LL^{1, q}=\alpha(r)\frac{d^2}{dr^2}+\beta(r)\frac{d}{dr}-q(r)$,
then $\left(\LL^V, C_0^\infty(M)\right)$ is
$L^{\infty}(M,dx)$-unique.
\end{thm}
\brmk{\rm Our assumption {\bf (A)} is inspired from the comparison
theorems in the theory of stochastic differential equations
(\cite[Chap. VI, Sections 4 and 5]{IW}). Assume that
$|\nabla\rho|^2=\alpha(\rho)$. Let $(X_t)_{0\le t<\sigma}$ be the
diffusion generated by $\LL$ and $\eta_t$ by
$\alpha(r)\frac{d^2}{dr^2}+\beta(r)\frac{d}{dr}$ with
$\rho(X_0)=\eta_0>c$. If (\ref{22ass2}) holds, one can realize $X_t$
and $\eta_t$ on the same probability space so that $\rho(X_t)\ge
\eta_t$ before returning to $c$. In other words $\rho(X_t)$ goes to
$y_0$ (i.e., $X_t$ goes to infinity) more rapidly than $\eta_t$.
Then under the assumption {\bf (A)}, if $y_0$ is no entrance
boundary for $\LL^{1,q}$, the heat from ``boundary" of $M$ is again
more difficult to enter into $M$ : it should be ``no entrance
boundary". The result above justifies this intuition. }\nrmk
Let us begin with a Kato type inequality.
\begin{lem}\label{kato}
If $u\in L^1(M, dx)$ satisfies
\begin{equation}\label{2equation}
\int_M u(\LL^V-1)fdx=0, \quad\forall f \in C_0^{\infty}(M)
\end{equation}
Then $u\in C^1(M)$ (more precisely one version of $u$ is
$C^1$-smooth) and
\begin{equation}\label{kato2}
-\int_M \nabla |u|\cdot \nabla fdx+\int_M b \cdot \nabla f\cdot
|u|dx\geq \int_M(V+1)f|u|dx
\end{equation}
for all positive, compactly supported functions $f\in
H_{loc}^{1, 2}(M)$.\\
\end{lem}
\begin{proof} By \cite[Theorem 1]{BR1}, we know $u\in H_{loc}^{1,
2}(M)\cap L_{loc}^{\infty}(M).$ Using $\Delta u = div(ub )+(V+1)u$
and Sobolev's embedding theorems recursively, $u\in C^1(M)$. The
remained proof can follow word-by-word Eberle \cite[Theorem 2.5 step
2]{[Eb]} (in ``$V=0$" case), so omitted.
\end{proof}
\blem\label{lem43} If $u\in H^{1,2}_{loc}(M)$ satisfies
$$
\<u, \LL^V f\>=0, \ \forall f\in C_0^\infty(M)
$$
and $u=0$ $dx-a.e.$ outside of some compact subset $K$ of $M$, then
$u=0$. \nlem
This is contained in the folklore of elliptic PDE, so we omit its
proof.
\begin{proof}[\it Proof of Theorem \ref{mainthm}] According to Theorem \ref{thm21},
we have only to show that the equation
\begin{equation} \label{mainthm1}
\int_M u(\LL^V-1)fdx=0 , \quad\forall f \in C_0^{\infty}(M)
\end{equation}
has no non-trivial
$L^1(M,dx)$ solution. Assume by absurd that there is some non-zero
$u\in L^1(M)$ satisfying (\ref{mainthm1}). By Lemma \ref{kato},
$u\in C^1(M)$ and (\ref{kato2}) holds.
For all $r_1,r_2$ such that $x_0\leq c<r_1<r_2<y_0$, put $h(r):=\min\{r_2-r_1,
(r_2-|r|)^+\}$ and $f:=h(\rho(x))$. Plugging such $f$ into (\ref{kato2}) we obtain:
\bequ \label{222}
\int_{\{r_1\leq \rho(x)\leq r_2\}}\nabla |u| \cdot \nabla
\rho dx-\int_{\{r_1\leq \rho(x)\leq r_2\}}b \cdot \nabla \rho\cdot
|u|dx\geq\int_M(V+1)|u|h(\rho) dx.
\nequ
Since $\nabla |u|\cdot \nabla \rho=div(|u|\nabla
\rho)-|u|\Delta\rho$, we have
\begin{align*}
\int_{\{r_1\leq \rho(x)\leq r_2\}}&\nabla |u|\cdot \nabla
\rho dx=\int_{\{r_1\leq \rho(x)\leq r_2\}}[div(|u|\nabla
\rho)-|u|\Delta\rho ]dx\\
&\stackrel{(i)}{=}\int_{\{\rho=r_2\}}|u|\cdot|\nabla
\rho|d\sigma_M-\int_{\{\rho=r_1\}}|u|\cdot|\nabla
\rho|d\sigma_M-\int_{\{r_1\leq \rho(x)\leq r_2\}}|u|\Delta\rho dx
\end{align*}
where $(i)$ follows from the Divergence Theorem, $\sigma_M$ is the
$(d-1)$-dimensional surface measure on the $C^2$-smooth $\{\rho=
r\}$ induced by the volume measure $dx$. Using the preceding
equality, we get:
\begin{align*}
\int_{\{\rho=r_2\}}|u|\cdot|&\nabla
\rho|d\sigma_M-\int_{\{\rho=r_1\}}|u|\cdot|\nabla
\rho|d\sigma_M
-\int_{\{r_1\leq \rho(x)\leq r_2\}}|u|\cdot|\nabla
\rho|^2\frac{\beta(\rho)}{\alpha(\rho)}dx\\
&\stackrel{(i)}{\geq}\int_{\{\rho=r_2\}}|u|\cdot|\nabla
\rho|d\sigma_M-\int_{\{\rho=r_1\}}|u|\cdot|\nabla
\rho|d\sigma_M-\int_{\{r_1\leq \rho(x)\leq r_2\}}|u|\LL\rho dx\\
&=\int_{\{r_1\leq \rho(x)\leq r_2\}}\nabla |u|\cdot \nabla
\rho dx+\int_{\{r_1\leq \rho(x)\leq r_2\}}|u|\Delta\rho dx-\int_{\{r_1\leq \rho(x)\leq r_2\}}|u|\LL\rho
dx\\
&\stackrel{(ii)}{\geq} \int_M (V+1)|u|h(\rho)dx\ge
\int_M\frac{V+1}{\alpha(\rho)}|u|\cdot|\nabla
\rho|^2h(\rho)dx\\
&\stackrel{(iii)}{\geq}\int_M\frac{q(\rho)+1}{\alpha(\rho)}|u|\cdot|\nabla
\rho|^2h(\rho)dx
\end{align*}
where $(i)$ follows from condition (\ref{2ass1}), $(ii)$ follows
from (\ref{222}) and (\ref{22ass2}), $(iii)$ follows from
(\ref{2ass2}).
Now let
$G(r)=\int_{\{\rho(x)\leq r\}}|\nabla \rho|^2\cdot |u| dx$. By the
Co-area formula (Federer v\cite[Theorem 3.2.12]{Federer}): for
$r_2>r_1>c$, \bequ\label{co} \int_{\{\rho(x)\in
[r_1,r_2]\}}\mid\nabla \rho(x)\mid f(x)dx=\int_{r_1}^{r_2}
dr\int_{\{\rho(x)=r\}}f(x)d\sigma_M, \nequ $G$ is absolutely
continuous on $r\in (c,y_0)$ and
$$G'(r) = \int_{\{\rho(x)= r\}} |\nabla \rho|\cdot|u|
d\sigma_M,\ dr-a.e. \ r>c. $$ From now on we fix $G'(r)$ as the
right hand side above. By the Co-area formula we also have
$$\int_{\{r_1\leq \rho(x)\leq
r_2\}}|u|\cdot|\nabla
\rho|^2\frac{\beta(\rho)}{\alpha(\rho)}dx=\int_{r_1}^{r_2}G'(r)\frac{\beta(r)}{\alpha(r)}dr
$$ the previous inequality
is read as :
\begin{equation}\label{2aaa}
G'(r_2)-G'(r_1)-\int_{r_1}^{r_2}G'(r)\frac{\beta(r)}{\alpha(r)}dr\ge
\int_c^{y_0}\frac{q(r)+1}{\alpha(r)}G'(r)h(r)dr
\end{equation}
for $c<r_1<r_2.$ Since
\begin{align*}
\int_c^{y_0}\frac{q(r)+1}{\alpha(r)}G'(r)h(r)dr &=
\int_{r_1}^{r_2}\frac{q(r)+1}{\alpha(r)}(r_2-r)G'(r)dr+\int_{c}^{r_1}(r_2-r_1)G'(r)\frac{q(r)+1}{\alpha(r)}dr\\
& =
\int_{r_1}^{r_2}\frac{q(t)+1}{\alpha(t)}G'(t)\int_t^{r_2}dsdt+(r_2-r_1)\int_c^{r_1}G'(r)\frac{q(t)+1}{\alpha(t)}dt\\
&=\int_{r_1}^{r_2}ds\int_{r_1}^{s}\frac{q(t)+1}{\alpha(t)}G'(t)dt+\int_{r_1}^{r_2}ds\int_c^{r_1}G'(t)\frac{q(t)+1}{\alpha(t)}dt\\
& = \int_{r_1}^{r_2}ds\int_{c}^{s}\frac{q(t)+1}{\alpha(t)}G'(t)dt.
\end{align*}
Substituting this into (\ref{2aaa}), we obtain
$$
-\int_{r_1}^{r_2}G'(r)\frac{\beta(r)}{\alpha(r)}dr+G'(r_2)-G'(r_1)\geq
\int_{r_1}^{r_2}ds\int_{c}^{s}\frac{q(t)+1}{\alpha(t)}G'(t)dt
$$
for $c<r_1<r_2$. It can re-written as a differential equality :
\begin{equation}\label{felform}
\LL^{-}G:=G''(r)-\frac{\beta(r)}{\alpha(r)}G'(r)=\int_c^{r}\frac{q(t)+1}{\alpha(t)}G'(t)dt
+H'
\end{equation}
in distribution on $(c, y_0)$, where $H:(c,y_0)\to\rr$ is
nondecreasing and right continuous. Then $G'$ admits a right
continuous version. We shall assume $G'$ is itself right continuous
below.
Consider $\tilde
m'(r)=\exp\left(\int_c^r\frac{\beta(t)}{\alpha(t)}dt\right)$,
$s'(r)=\exp\left(-\int_c^r\frac{\beta(t)}{\alpha(t)}dt\right)$,
which are respectively the derivative of the scale and speed
function associated with $\LL^-$. Then $m'(r):=\frac {\tilde
m'(r)}{\alpha(r)}, s'(r)$ are respectively the derivative of the
speed and scale function associated with $\LL^{1,q}$. Hence we can
write (\ref{felform}) in the Feller's form,
$$
(G'/\tilde m ')'\ge s'\int_c^{\cdot}\frac{q(t)+1}{\alpha(t)}G'(t)dt.
$$
Let us prove now that $\exists r_0\in (c,y_0),\ G ' (r_0)>0.$
Indeed, if in contrary $G'(r)=0, \forall r>c$, then $u=0, dx-a.e.$
on $[\rho>c]$, i.e., outside of the compact set
$K=\rho^{-1}[x_0,c]$. Thus $u=0$ on $M$ by Lemma \ref{lem43}, a
contradiction with our assumption.
The above inequality in distribution implies that for
$dr-a.e.\ r>r_0, r_0\in(x_0, y_0)$
\begin{align*}
\frac{dG}{d\tilde m }(r) & \geq
\frac{dG}{d\tilde m }(r_0)+\int_{r_0}^rs'(u)du\int_{r_0}^{u}\frac{q(t)+1}{\alpha(t)}G'(t)dt\\
& =\frac{dG}{d\tilde m
}(r_0)+\int_{r_0}^rs'(u)du\int_{r_0}^{u}(q(t)+1) m '(t)
\frac{dG}{d\tilde m }(t)dt.
\end{align*}
Using the above inequality by induction as in the proof of
Proposition \ref{prop31} we get
$$
\frac{G'}{\tilde m '}(y)\geq C\sum_{n=0}^{+\infty}I_n^{q+1}(y)
$$
where $I_0^{q+1}=1, \ I_n^{q+1}(y)=\int_{r_0}^ys'(r)dr\int_{r_0}^r
(q(t)+1) m '(t) I_{n-1}^{q+1}(t)dt,\ C=\frac{dG}{d\tilde m
}(r_0)>0.$
Using Co-area formula and our assumption, we obtain:
\begin{align*}
\int_{\{r_0\leq\rho\}}|u|dx&\geq\int_{\{r_0\leq\rho\}}\frac{|u|\cdot|\nabla \rho|^2}{\alpha(\rho)}
dx\\
&=\int_{r_0}^{y_0}\frac{G'(r)}{\alpha(r)}dr= \int_{r_0}^{y_0} m '(r) \frac{dG}{d\tilde m } (r)dr\\
&\geq C\sum_{n=0}^{+\infty}\int_{r_0}^{y_0} m '(r) I_n^{q+1}(r)dr.
\end{align*}
Thus $\int_{\{r_0\leq\rho\}}|u|dx=\infty$ by our assumption that
$y_0$ is no entrance boundary for $\LL^{1,q}$. This is in
contradiction with the assumption that $u\in L^1(M, dx)$.
\end{proof}
With the same proof we have the two sides' version of Theorem
\ref{mainthm} :
\begin{thm}\label{twosides}
We suppose \bdes
\item{\bf(1) } $\rho:\ M\longrightarrow(x_0, y_0)$ is surjective, where $-\infty\leq
x_0<y_0\leq +\infty$, such that $\rho^{-1}\left([x_1,\ x_2]\right)$
is compact subset for all $x_1<x_2$ in $(x_0, y_0)$, and there are
$c_1<c_2$ in $(x_0,y_0)$ such that $\rho$ is $C^2$-smooth, $|\nabla
\rho|>0$ on $[\rho<c_1]\cup [\rho>c_2]$ ;
\item{\bf (2)} there exist $\alpha (r),\ \beta(r),\ q(r) \in L_{loc}^{\infty}\left((x_0,
y_0), dr\right)$, $q(r)\geq0, \ \alpha>0,\ 1/\alpha (r)\in
L^\infty_{loc}(x_0,y_0)$ and $c_1<c_2$ in $(x_0,\ y_0)$ such that
$dx-a.e. $ on $[\rho<c_1]\cup [\rho>c_2]$,
\ndes \bequ\label{cor4.16} |\nabla \rho|^2\leq
\alpha(\rho),\ \text{$dx-a.e. $ on } \ [\rho<c_1]\cup
[\rho>c_2];\nequ
\bequ\label{cor4.17} \LL \rho \geq|\nabla \rho|^2\frac{\beta(\rho)}{\alpha(\rho)},\
\text{$dx-a.e. $ on } \
[\rho>c_2];\nequ \bequ\label{cor4.18} \LL \rho \leq|\nabla
\rho|^2\frac{\beta(\rho)}{\alpha(\rho)},\
\text{$dx-a.e. $ on } \
[\rho<c_1];\nequ
\bequ\label{cor4.19}V(x)\geq q(\rho (x)),\ \text{$dx-a.e. $ on } \ [\rho<c_1]\cup
[\rho>c_2];\nequ If $x_0,y_0$ are no entrance boundaries for
$\LL^{1, q}=\alpha(r)\frac{d^2}{dr^2}+\beta(r)\frac{d}{dr}-q(r)$,
then $\left(\LL^V, C_0^\infty(M)\right)$ is $L^{\infty}(M,
dx)$-unique.
\end{thm}
\bcor Suppose that M is a Cartan-Hadamard manifold with dimension
$d\ge 2$ (i.e. complete, simply connected with non-positive
sectional curvature). Let $d(x)$ be the distance from some fixed
point $o$ to $x$. Then
\bdes
\item{(a)} $\Delta$ is $L^\infty(M,dx)$-unique. In particular the
$L^1$-Liouville property holds : every $dx$-integrable
($\Delta$-)harmonic function is constant.
\item{(b)} Assume that $b $ verifies $b (x)\cdot \nabla d(x)\ge
-L [1+d^2(x)], \ x\ne o$ for some constant $L>0$. Then $\LL^V=\Delta
+b \cdot \nabla-V$ is $L^\infty(M,dx)$-unique. In particular the
$L^1$-Liouville property in Theorem \ref{thm22} and Corollary
\ref{cor21} holds true.
\item{(c)} Assume that $b $ verifies $b (x)\cdot \nabla d(x)\ge
-L [1+d^\alpha(x)]$ for some constants $L>0$ and $\alpha>2$. If
$V(x)\ge cd(x)^{\alpha-2}$ with $c>L \alpha (\alpha-2)$, then
$\LL^V=\Delta +b \cdot \nabla-V$ is $L^\infty(M,dx)$-unique. In
particular the $L^1$-Liouville property in Theorem \ref{thm22}
holds.
\ndes
\ncor
\bprf {\bf (a)} The $L^\infty$-uniqueness of $\Delta$ is a particular case of part
(b). Then the $L^1$-Liouville theorem follows from Example
\ref{exa21}.
{\bf (b)} Recall that on the Cartan-Hadamard manifold, the exponential map $\exp : T_oM\to M$
is a diffeomorphism. The distance function $d(x)$ is
$C^\infty$-smooth on $M\backslash \{o\}$, and the Laplacian
comparison theorem says that (\cite{Chee, SY})
$$
\Delta d(x)\ge \frac{d-1}{d(x)}, \ x\ne o.
$$
Hence the assumption {\bf (A)} holds with $\rho(x)=d(x)$,
$[x_0,y_0)=\rr^+$, $\alpha(r)=1$, $q(r)=0$ and $\beta(r)=-(L+1) r^2$
(for some point $c\in \rr^+$ large enough in {\bf (A)}). By Example
\ref{exa33}, $+\infty$ is no entrance boundary for $\LL^{1,q}$, thus
$\LL^V$ is $L^\infty$-unique by Theorem \ref{mainthm}.
{\bf (c)} Take $\rho(x)=d(x)$ as above and $\LL^{1,q}$ as follows : $[x_0,y_0)=\rr^+$,
$\alpha(r)=1$, $q(r)=c r^\alpha$ and $\beta(r)=-(L+\vep)
r^\alpha$, where $\vep>0$ is small enough so that $c>(L+\vep)\alpha(\alpha-2)$.
The assumption {\bf (A)} is satisfied. Again by Example
\ref{exa33}, $+\infty$ is no entrance boundary for $\LL^{1,q}$, therefore $\LL^V$ is
$L^\infty$-unique by Theorem \ref{mainthm}.
\nprf
\bexam\label{exa41}{\bf (the first example of \cite{LS})} {\rm Let $M$ be a
compact surface with arbitrary genus. Assume the metric on $M$
around some point $o\in M$ is flat. Hence locally around $o$ we can
write the metric in polar coordinates as
$$ds^2_0=dr^2+r^2d\theta ^2$$
we choose the new metric to be
$$
ds^2=\varrho^2ds_0^2.
$$
Choose $\varrho$ to be arbitrary outside a neighborhood of $o$ (say
$r\ge\delta$), and for $r\in(0,\delta]$,
\begin{equation}
\varrho(\theta,r)=\varrho(r)= r^{-1}(-\log r)^{-1}\left(\log(-\log
r)\right)^{-\alpha},
\end{equation}where $0<\delta<e^{-2}$ and $0<\alpha\leq1$ (It is assumed in \cite{LS} that $\frac12<\alpha\le 1$).
$(M\backslash \{o\}, ds^2)$ is (metrically) complete, stochastically
complete and its volume is finite. The Green's function on $(M,
ds_0^2)$ (with the pole at $o$) $G(o,x)=f(x)$ is a positive harmonic
on $M\setminus\{o\}$ w.r.t. $ds_0^2$, then w.r.t. $ds^2$. Let
$\Delta, \nabla, |\cdot|$ be respectively the Laplacian operator,
the gradient and the Riemannian norm in the metric $ds^2$. Note
$\Delta r=\frac{1}{r\varrho(r)^2},\ |\nabla
r|=\frac{1}{\varrho(r)}$. Consider the Sturm-Liouville operator
$\dps \LL^1:
=\frac{1}{\varrho^2(r)}\frac{d^2}{d^2r}+\frac{1}{r\varrho^2(r)}\frac{d}{dr}$,
the derivative of speed function and that of scale function of $\LL$
are respectively
$m'(r)=\varrho^2(r)\exp\left(\int_1^r\frac{1}{t}dt\right)=\varrho^2(r)r,\
s'(r)=\exp\left(-\int_1^r\frac{1}{t}dt\right)=\frac{1}{r}.$ \bdes
\item{(i). } Let $\alpha\in (0, \frac12]$. Since \begin{align*}
\int_0^\delta m'(r)dr\int_r^\delta s'(t)dt&=\int_0^\delta \varrho^2(r)r\log(\frac{1}{r})dr\\
&=\int_0^\delta \frac{1}{r\log(\frac{1}{r})(\log\log(\frac{1}{r}))^{2\alpha}}dr\\
&=+\infty,
\end{align*}
it follows that $0$ is no entrance boundary for $\LL^1$. By Theorem
\ref{mainthm}, $\left(\Delta, C_0^\infty(M\setminus\{o\})\right)$ is
$L^{\infty}(M\backslash \{o\})$-unique. Then the $L^1$-Liouville
property holds true on $(M\backslash \{o\}, ds^2)$ by Corollary
\ref{cor21}.
\item{(ii). }If $\alpha\in (\frac12,1]$. The Green function
$f(x)=G(o,x)$ with respect to the old metric $ds_0^2$ is
$\Delta$-harmonic and $dx$-integrable as observed in \cite{LS}.
$\Delta$ can not be $L^\infty$-unique on $M\backslash \{o\}$ by
Example
\ref{exa21} (or Corollary \ref{cor21}). On the other hand, we have
\begin{align*}
\int_0^1m'(r)dr\int_r^1s'(t)dt&=\int_0^1\varrho^2(r)r\log(\frac{1}{r})dr<+\infty.
\end{align*}
That shows the sharpness of Theorem \ref{mainthm}.
\ndes
}
\nexam
\bexam {\rm
Let $D$ be the unit open ball centered at the origin of $\rr^d(d\ge 2)$. Let
$\LL^V:=\Delta -V(x)$ be defined on $C_0^\infty(D)$. Let
$\rho(x)=|x|$ ($|x|$ denotes the Euclidian metric). For this
example, $|\nabla\rho(x)|=1$ and $\Delta \rho(x)=\frac{d-1}{r}\ge 0
$ where $r=r(x)=|x|$. If $V(x)\ge \frac{c}{\left(1-|x|\right)^2}$,
where $c$ is a constant such that $c\ge 2$. The assumption {\bf (A)}
is satisfied for $\LL^{1,q}:=\frac{d^2}{dr^2}-\frac{c}{(1-r)^2}$. By
Example \ref{exam1}, $1$ is no entrance boundary for $\LL^{1,q}$,
then $(\LL^V,C_0^\infty(D))$ is $L^\infty$-unique by Theorem
\ref{mainthm}.
} \nexam
\bexam{\rm
Let us consider $\LL^V:=\Delta-V$ defined on $C_0^\infty(D)$ where $D:=\rr^d\setminus{0}(d\ge 2).$
Let $\rho(x)=r(x):=|x|$, then $|\nabla \rho(x)|=1,\Delta
\rho(x)=\frac{d-1}{r(x)}$. If $V(x)\ge \frac{c}{r^2(x)}$ for $x$ close to $0$ (say $|x|<\delta$), where the constant $c$ satisfies
$$
c\ge d^2 - (d-1)^2+1=2d
$$
since $0$ and $+\infty$ is no entrance boundary
for $\dps \LL^{1,q}:=\frac{d^2}{dr^2}+\frac{d-1}{r}\frac{d}{dr}-\frac{c}{r^2} 1_{(0,\delta)}(r)$ by Example
\ref{exam1},
$(\LL^V,C_0^\infty(D))$
is $L^\infty$-unique by Theorem \ref{mainthm}.
In contrary if
$V=c/|x|^2$ with $0\le c<2d$, as $\LL^{1,q}$ is not
$L^\infty(m)$-unique (again by Example
\ref{exam1}), there is some $h\in L^1((0,+\infty), m)\bigcap C^1(0,+\infty)$ such
that $h'$ is absolutely continuous and
$\frac{d^2h}{dr^2}+\frac{d-1}{r}\frac{dh}{dr}-\frac{ch}{r^2}=h$ (such $h$ is indeed $C^\infty$-smooth).
Let $\tilde h(x)=h(r(x))$, we see readily that $\tilde h\in L^1(D,dx)$ and
$\Delta \tilde h-V\tilde h=\tilde h$ over $D=\rr^d\backslash
\{0\}$.
Thus $\Delta-V$ defined on $C_0^\infty(D)$ is not
$L^\infty(D,dx)$-unique.
} \nexam
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,279 |
Mrs. Elton in America by Diana Birchall
Last week I read Mrs. Elton in America, published in 2008 by Sourcebooks, which comprises all three of the Mrs. Elton stories by Diana Birchall. It's a fascinating notion - taking one of the most hated characters in Austen and attempting to redeem her. This began with In Defense of Mrs. Elton, the second story in the volume, which is a fairly straight forward retelling of Emma from Mrs. Elton's perspective. The entire text of this story can be found online at jasna.org along with a set of quirky illustrations by Juliet McMaster which are sadly absent from the Sourcebooks edition (like this one of Mr. Elton). Mrs. Elton is displayed as a misunderstood woman, aware of her own social awkward ness and inclined to put her foot in her mouth when sincerely attempting to endear herself to her new neighbors. It is a pretty story and the depiction of the relationship between Emma and Mrs. Elton, which is followed many years beyond the end of the original novel, is highly believable.
The first story in the volume (the second one composed by Ms. Birchall) is entitled The Courtship of Mrs. Elton and is available online at jasnasaz.org. I understand that this has been adapted into a short play. It is my favorite of the stories in this collection. Miss Augusta Hawkins is a young lady in Bath for yet another season and is absolutely determined that this year will find her married. She meets Mr. Elton almost immediately upon her (and his) arrival and they are instantly taken with each other:
There was nothing new in this social round to Augusta, with her eight seasons' experience of the place; but it did often occur to her, in the course of her multifarious activities, that, of all the people she had met during them, none had ever been more attractive to her than this Mr. Elton. He was very handsome, and decidedly agreeable; that he liked her was beyond question; and the Miss Milmans had swiftly found out, and swiftly related to her, that he was installed in an excellent and modern vicarage in one of the very finest towns of England, as well as being possessed of a comfortable competence of his own. Augusta had lived enough years in the world to know that she could hardly do better; that this might, indeed, be her last and best chance; and though she did not call herself desperate, she had already made up her mind, before she set eyes upon him for the second time, that, if he were ever to ask her to marry him, she would accept.
Surrounded by company so vulgar that she seems (comparatively) the embodiment of refinement, Miss Hawkins fastidiously encourages Mr. Elton in a manner that must be gratifying, considering his recent rejection. So satisfied are both parties by the success of the courtship that one may honestly remark (without the bitterness that tinges Frank Churchill's words), "Happy couple! How well they suit one another." This story inspires much more sympathy in me for Mrs. Elton than the other two. It is a thoroughly sweet tale and I will certainly reread it many times in the future.
The final (and longest) story in this book is Mrs. Elton in America. I have been puzzling over what to say about it for a week and am still at a loss. The story sees the transformation of Mrs. Elton from the character created by Jane Austen into an entirely new and unrecognizable creature. Roughly picking up where In Defense of Mrs. Elton leaves off, the Eltons, having over spent, retrench to the United States as missionaries. Here they survive horrific hardships as they head West in the ubiquitous covered wagon, where Mr. Elton is sent to convert the Comanches. By the end of the book Mrs. Elton has been fully democratized, turning into a sensible, no nonsense woman. I admit to being transfixed by the story as I read it but cannot actually say I liked it. It is a curious read but one too far outside the scope of Austen for me to feel comfortable with it.
Mrs. Elton in America leaves me wanting more books of redemption for Austen's less likable characters. As far as I know, no one has yet to defend Elizabeth Elliot, Lucy Steele, or Isabella Thorpe, all of whom seem ripe for such treatment. Aunt Norris would also be a challenging but fascinating character to defend.
Labels: Art, Emma, Reviews, Sequels, Short Stories
ibmiller November 30, 2009 at 12:06 PM
Hmmm. I don't think I could stomach defenses of Lucy Steele or Aunt Norris - they are two character who evoke blind, violent rage from me.
But minor characters getting a focus is indeed a very interesting idea - I've mostly found it in the internet side of fandom. Though you can see Marsha Altman's Darcys and Bingleys series as doing something like this for Caroline Bingley (though as the series progresses, I think it starts to do what Mrs. Elton in America does and stop really being connected with Austen - that and her prose style isn't that great to begin with, and only gets worse).
Alexa Adams November 30, 2009 at 12:58 PM
I think Monica Fairview did a better job with Caroline Bingley than Marsha Altman. I really enjoyed Jane Odiwe's treatment of Lydia Bennet, though it was a bit far fetched, and she has indicated that she is writing more novels along that line (though she wont reveal which character is her focus). I was also intrigued (though far from convinced) by Judith Brockelhurst's redemption of Maria Bertram.
I figure it would be really easy to put Lucy Steele's back story together and in such a way that her actions appear at least somewhat justified. Aunt Norris would be much harder. I have been thinking about her a lot lately and in a much more sympathetic manner than I have be accustomed to, ever since reading "Three Sisters" in Jane Greensmith's Intimations of Austen. Mrs. Elton certainly is an easier character, being comic rather than sinister, to sympathize with.
Meredith November 30, 2009 at 6:40 PM
Excellent review, Alexa! I think I would find the first two more interesting than the third. I need to get my own copy of this!!
Actually there is a book out there about Elizabeth Elliot and its called Mercy Embrace by Laura Hile. I believe it is a trilogy and she only has the first book out so far.
I don't think Mrs. Norris could ever be redeemed in my eyes, she is just pure meanness and hatred towards Fanny, however, I can see your point about Lucy Steele and I am curious to see if she will ever get her own book...
Alexa Adams December 1, 2009 at 10:09 AM
I just ordered Mercy's Embrace and consulted your Persuasion list to make sure I haven't missed any gems. Can you tell me anything about Sir Willy?
Perhaps I should have never brought up Mrs. Norris but now that I have I feel like I must defend the notion. Mrs. Norris makes me cringe in Mansfield Park, but what was she like as a young woman? Obviously she has been forced to make herself valuable to her community by inserting herself into the household at Mansfield. Constantly living in subservience to an insipid sister is sure to pay a toll on anyone. She doesn't even have her own children to occupy her interest. In fact, the more I think of it, Mrs. Norris is remarkably like Mrs. Elton - imagine if the latter was mistress of a rectory near Maple Grove.
ibmiller December 1, 2009 at 5:04 PM
See, I understand about both Lucy and Mrs. Norris - they could certainly have motives. But they are bullies - and cruel ones at that - and their preferred targets are the kindest and most defenseless characters they know. My own reaction to them is more than cringing or annoyance - I almost literally see red when I read their parts. I do tend to overreact (in life and literature), but rarely do I have an almost physical hatred of fictional characters (I think the villain of Gaudy Night by Dorothy Sayers is the only other character that inspires such a feeling of rage at sadistic mental torturing of helpless targets).
I've not heard of Monica Fairview - and I think Altman's work is much more enjoyably read on fanfiction.net than paid for, even considering the gorgeous printing job Sourcebooks did.
Alexa Adams December 1, 2009 at 5:51 PM
You haven't come across The Other Mr. Darcy? It just came out this fall. Here's my review: http://alexaadams.blogspot.com/2009/10/other-mr-darcy-by-monica-fairview.html
I love people who feel passionately about the characters in the stories they love. Mrs. Norris is certainly warrants red vision. She is certainly a bully, almost a text book example of the play ground tyrant. It just feels like, after strumming up sympathy for Mrs. Elton, anything is possible.
Later Days at Highbury by Joan Austen-Leigh
All About the Brontes Challenge 2010
Regina Jeffers & Wayward Love
Intimations of Austen by Jane Greensmith
Book Cover!!
My Personal Austen Rankings
Another "What If?" Post - Why must we follow the D...
Willoughby's Return by Jane Odiwe
Emma Mashup
My Favorite Austen Book Covers
For the love of Austen! Let's boycott the monsters. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,048 |
module FileFixtures
def fixture_file_path(filename)
Rails.root.join *%w[spec fixtures files], filename
end
end
RSpec.configure do |config|
config.include FileFixtures
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,402 |
\section{Appendix}
\subsection{More Implementation Details}
\subsubsection{Network Details}
For MobileNet-V2 encoder, we increase the final resolution of the encoder to $1/16$ by adding a dilation to the last stage and removing a stride from the first convolution of this stage. For ResNet-50 and SwinB encoders, we remove the last stage directly. The encoder features are flattened into sequences before LSTT. In LSTT, the input channel dimension is 256, and the head number is set to 8 for all the attention modules. To increase the receptive field of LSTT, we insert a depth-wise convolution layer with a kernel size of 5 between two layers of each feed-forward module. In our default setting of the short-term memory $\mathbf{n}$, only the previous ($t-1$) frame is considered, which is similar to the local matching strategy~\cite{feelvos,cfbi}. After LSTT, all the output features of LSTT blocks are reshaped into 2D shapes and will serve as the decoder input. Then, the FPN decoder progressively increases the feature resolution from $1/16$ to $1/4$ and decreases the channel dimension from 256 to 128 before the final output layer, which is used for identification decoding.
\noindent\textbf{Patch-wise Identity Bank:} Since the spatial size of LSTT features is only 1/16 of the input video, we can not directly assign identities to the pixels of high-resolution input mask to construct a low-resolution identification embedding. To overcome this problem, we further propose a strategy named patch-wise identity bank. In detail, we first separate the input mask into non-overlapping patches of 16$\times$16 pixels. The original identity bank with $M$ identities is also expanded to a patch-wise identity bank, in which each identity has 16$\times$16 sub-identity vectors corresponding to 16$\times$16 positions in a patch. Hence, the pixels of an object region with different patch positions will have different sub-identity vectors under an assigned identity. By summing all the assigned sub-identities in each patch, we can directly construct a low-resolution identification embedding while keeping the shape information inside each patch.
\subsubsection{Training Details}
All the videos are firstly down-sampled to 480p resolution, and the cropped window size is 465$\times$ 465. For optimization, we adopt the AdamW~\cite{adamw} optimizer and the sequential training strategy~\cite{cfbi}, whose sequence length is set to 5. The loss function is a 0.5:0.5 combination of bootstrapped cross-entropy loss and soft Jaccard loss~\cite{nowozin2014optimal}. For stabilizing the training, the statistics of BN~\cite{bn} modules and the first two stages in the encoder are frozen, and Exponential Moving Average (EMA)~\cite{polyak1992acceleration} is used. Besides, we apply stochastic depth~\cite{huang2016deep} to the self-attention and the feed-forward modules in LSTT.
The batch size is 16 and distributed on 4 Tesla V100 GPUs. For pre-training, we use an initial learning rate of $4\times10^{-4}$ and a weight decay of $0.03$ for 100,000 steps. For main training, the initial learning rate is set to $2\times10^{-4}$ and the weight decay is $0.07$. In addition, the training steps are 100,000 for YouTube-VOS or 50,000 for DAVIS. To relieve over-fitting, the initial learning rate of encoders is reduced to a 0.1 scale of other network parts. All the learning rates gradually decay to $2\times10^{-5}$ in a polynomial manner~\cite{cfbi}.
\input{Figures/id_bank}
\input{Figures/long_short_term}
\subsection{Visualization of Identity Bank}
In AOT, the identity bank is randomly initialized, and all the $M$ identification vectors are learned by being randomly assigned to objects during the training phase. Intuitively, all the identification vectors should be equidistant away from each other in the feature space because their roles are equivalent. To validate our hypothesis, we visualize the similarity between every two identification vectors in Fig.~\ref{fig:id_bank}.
In our default setting, $M=10$ (Fig.~\ref{fig:id_bank_10}), all the vectors are far away from each other, and the similarities remain almost the same. This phenomenon is consistent with our above hypothesis. In other words, the reliability and effectiveness of our identification mechanism are further verified.
In the ablation study, using more identities leads to worse results. To analyze the reason, we also visualize the learned identity banks with more vectors. Fig.~\ref{fig:id_bank_15},~\ref{fig:id_bank_20}, and~\ref{fig:id_bank_30} demonstrate that maintaining equidistant between every two vectors becomes more difficult when the identity bank contains more vectors, especially when $M=30$. There are two possible reasons for this phenomenon: (1) No training video contains enough objects to be assigned so many identities, and thus the network cannot learn to associate all the identities simultaneously; (2) the used space with only 256 dimensions is difficult for keeping more than 10 objects to be equidistant.
\subsection{Illustration of Long Short-term Attention}
To facilitate understanding our long-term and short-term attention modules, we illustrate their processes in Fig.~\ref{fig:long_short_term}. Since the temporal smoothness between the current frame and long-term memory frames is difficult to guarantee, the long-term attention employs a non-local manner to match all the locations in the long-term memory. In contrast, short-term attention only focuses on a nearby spatial-temporal neighborhood of each current-frame location.
\input{Figures/attention}
\subsection{Visualization of Hierarchical Matching and Propagation}
In our AOT, we propose to construct a hierarchical framework, \emph{i.e.}, LSTT, for multi-object matching and propagation, and the ablation study indicates that using more LSTT layers (or blocks) results in better VOS performance. To further validate the effectiveness of LSTT and analyze the behavior of each LSTT layer, we visualize long-term and short-term attention maps in each layer during inference, as shown in Fig.~\ref{fig:attention} and~\ref{fig:occlusion}.
At the bottom of Fig.~\ref{fig:attention}, the attention maps become more accurate and sharper as the index of layers increases. In the first layer, \emph{i.e.}, $l=1$, the current features have not aggregated the multi-object mask information from memory frames, and the long-term attention map is very vague and contains a lot of wrong matches among the objects and the background. Nevertheless, as the layer index increases, the mask information of all the objects is gradually aggregated so that the long-term attention becomes more and more accurate. Similarly, the quality, especially the boundary of objects, of the short-term attention improves as the layer index increases. Notably, the short-term attention performs well even in the first layer, $l=1$, which is different from the long-term attention. The reason is that the neighborhood matching of short-term attention is easier than the long-term matching of long-term attention. However, long-term attention is still necessary because short-term attention will fail in some cases, such as occlusion (Fig.~\ref{fig:occlusion}).
In short, the visual analysis further proves the necessity and effectiveness of our hierarchical LSTT. The hierarchical matching is not simply a combination of multiple matching processes. Critically, the multi-object information will be gradually aggregated and associated as the LSTT structure goes deeper, leading to more accurate attention-based matching.
\input{Figures/occlusion}
\subsection{Compare with More Methods}
We compare our AOT with more VOS methods in Table~\ref{tab:full_comparisons} and~\ref{tab:full_davis2016}. To compare with latest real-time methods~\cite{realtimevos1,realtimevos2}, we introduce the real-time AOT variant, \emph{i.e.}, AOT-T.
Even on the single-object DAVIS 2016~\cite{davis2016}, AOT-S (89.4\%, 40.0FPS) can achieve comparable speed with SAT~\cite{realtimevos1} (83.1\%, 39FPS) and comparable performance with CFBI~\cite{cfbi} (89.4\%, 6.3FPS). On the multi-object DAVIS 2017~\cite{davis2017}, AOT-T (79.9\%, 51.4FPS) significantly outperforms SAT (72.3\%, 19.5FPS) and GC (71.4\%, 12.5FPS). Particularly, on the large-scale multi-object YouTube-VOS~\cite{youtubevos}, AOT-T/S (80.2\%/82.6\%) achieves superior performance compared to previous real-time methods, SAT (63.6\%) and GC (73.2\%), while still maintaining a real-time speed (41.0FPS/27.1FPS).
\input{Tables/full_comparisons}
\input{Tables/full_davis2016}
\subsection{Additional Qualitative Results}
We supply more qualitative results under multi-object scenarios on the large-scale YouTube-VOS~\cite{youtubevos} and the small-scale DAVIS 2017~\cite{davis2017} in Fig.~\ref{fig:youtubevos} and~\ref{fig:davis2017}, respectively. As demonstrated, our AOT-L is robust to many challenging VOS cases, including similar objects, occlusion, fast motion, and motion blur, etc.
\subsection{Border Impact and Future Works}
The proposed AOT framework greatly simplifies the process of multi-object VOS and achieves a significant performance of effectiveness, robustness, and efficiency. Some AOT variants can achieve promising results while keeping real-time speed. In other words, AOT may promote the applications of VOS in real-time video systems, such as video conference, self-driving car, augmented reality, etc.
Nevertheless, the speed and accuracy of AOT can still be further improved. There is still a very large accuracy gap between the real-time AOT-T and the state-of-the-art SwinB-AOT-L. Moreover, AOT uses only a lightweight encoder and decoder. How to design stronger yet efficient encoders and decoders for VOS is still an open question.
As to related areas of VOS, the simple yet effective identification mechanism should also be promising for many tasks requiring multi-object matching, such as interactive video object segmentation~\cite{oh2019fast,miao2020memory}, video instance segmentation~\cite{vis,bertasius2020classifying,vist}, and multi-object tracking~\cite{milan2016mot16,pointtrack,wang2019towards}. Besides, the hierarchical LSTT may serve as a new solution for processing video representations in these tasks.
\input{Figures/youtubevos}
\input{Figures/davis2017}
\section{Conclusion}
This paper proposes a novel and efficient approach for video object segmentation by associating objects with transformers and achieves superior performance on three popular benchmarks. A simple yet effective identification mechanism is proposed to associate, match, and decode all the objects uniformly under multi-object scenarios. For the first time, processing multiple objects in VOS can be efficient as processing a single object by using the identification mechanism. In addition, a long short-term transformer is designed for constructing hierarchical object matching and propagation for VOS. The hierarchical structure allows us to flexibly balance AOT between real-time speed and state-of-the-art performance by adjusting the LSTT number.
\zongxin{We hope the identification mechanism will help ease the future study of multi-object VOS and related tasks (\emph{e.g.}, video instance segmentation, interactive VOS, and multi-object tracking), and AOT will serve as a solid baseline.}
\section{Experimental Results}\label{sec:experiments}
We evaluate AOT on popular multi-object benchmarks, YouTube-VOS~\cite{youtubevos} and DAVIS 2017~\cite{davis2017}, and single-object benchmark, DAVIS 2016~\cite{davis2016}. For YouTube-VOS experiments, we train our models on the YouTube-VOS 2019 training split. For DAVIS, we train on the DAVIS-2017 training split. When evaluating YouTube-VOS, we use the default 6FPS videos, and all the videos are restricted to be smaller than $1.3\times480p$ resolution. As to DAVIS, the default 480p 24FPS videos are used.
The evaluation metric is the $\mathcal{J}$ score, calculated as the average Intersect over Union (IoU) score between the prediction and the ground truth mask, and the $\mathcal{F}$ score, calculated as an average boundary similarity measure between the boundary of the prediction and the ground truth, and their mean value, denoted as $\mathcal{J}$\&$\mathcal{F}$. We evaluate all the results on official evaluation servers or with official tools.
\subsection{Compare with the State-of-the-art Methods}
\noindent \textbf{YouTube-VOS}~\cite{youtubevos} is the latest large-scale benchmark for multi-object video segmentation and is about 37 times larger than DAVIS 2017 (120 videos). Specifically, YouTube-VOS contains 3471 videos in the training split with 65 categories and 474/507 videos in the validation 2018/2019 split with additional 26 unseen categories. The unseen categories do not exist in the training split to evaluate algorithms' generalization ability.
As shown in Table~\ref{tab:youtubevos}, AOT variants achieve superior performance on YouTube-VOS compared to the previous state-of-the-art methods. With our identification mechanism, AOT-S (82.6\% $\mathcal{J}\&\mathcal{F}$) is comparable with CFBI+~\cite{cfbip} (82.8\%) while running about $7\times$ faster (27.1 \emph{vs}~ 4.0FPS). By using more LSTT blocks, AOT-B improves the performance to 83.5\%. Moreover, AOT-L further improves both the seen and unseen scores by utilizing the memory reading strategy, and our R50-AOT-L (\textbf{84.1\%/84.1\%}) significantly outperforms the previous methods on the validation 2018/2019 split while maintaining an efficient speed (14.9FPS).
\input{Figures/comparisons}
\input{Tables/davis2016}
\noindent \textbf{DAVIS 2017}~\cite{davis2017} is a multi-object extension of DAVIS 2016. The validation split of DAVIS 2017 consists of 30 videos with 59 objects, and the training split contains 60 videos with 138 objects. Moreover, the testing split contains 30 more challenging videos with 89 objects in total.
Table~\ref{tab:davis} shows that our R50-AOT-L (\textbf{Y}) surpasses all the competitors on both the DAVIS-2017 validation (\textbf{84.9\%}) and testing (\textbf{79.6\%}) splits and maintains an efficient speed (18.0FPS). Notably, such a multi-object speed is the same as our single-object speed on DAVIS 2016. For the first time, processing multiple objects can be as efficient as processing a single object over the AOT framework. We also evaluate our method without training with YouTube-VOS, and AOT-S (79.2\%) performs much better than KMN~\cite{KMN} (76.0\%) by +3.2\%.
\noindent \textbf{DAVIS 2016}~\cite{davis2016} is a single-object benchmark containing 20 videos in the validation split. Although our AOT aims at improving multi-object video segmentation, we also achieve a new state-of-the-art performance on DAVIS 2016 (R50-AOT-L (\textbf{Y}), \textbf{91.1\%}). Under single-object scenarios, the multi-object superiority of AOT is limited, but R50-AOT-L still maintains an about $2\times$ efficiency compared to KMN (18.0 \emph{vs}~ 8.3FPS). Furthermore, our smaller variant, AOT-B (89.9\%), achieves comparable performance with CFBI+ (89.9\%) while running 5$\times$ faster (29.6 \emph{vs}~ 5.9FPS).
Apart from the above results, replacing the AOT encoder from commonly used ResNet50 to SwinB can further boost our performance to higher level (Table~\ref{tab:youtubevos}, \ref{tab:davis}, and \ref{tab:davis2016}).
\zongxin{\noindent\textbf{Qualitative results:} Fig.~\ref{fig:comparisons} visualizes some qualitative results in comparison with CFBI~\cite{cfbi}, which only associates each object with its relative background. As demonstrated, CFBI is easier to confuse multiple highly similar objects. In contrast, our AOT tracks and segments all the targets accurately by associating all the objects uniformly. However, AOT fails to segment some tiny objects (\textit{ski poles} and \textit{watch}) since we do not make special designs for tiny objects.}
\input{Tables/ablation}
\subsection{Ablation Study}
In this section, we analyze the main components and hyper-parameters of AOT and evaluate their impact on the VOS performance in Table~\ref{tab:ablation}.
\textbf{Identity number:} The number of the identification vectors, $M$, have to be larger than the object number in videos. Thus, we set $M$ to 10 in default to be consistent with the maximum object number in the benchmarks~\cite{youtubevos,davis2017}. As seen in Table~\ref{tab:id_number}, $M$ larger than 10 leads to worse performance since (1) no training video contains so many objects; (2) embedding more than 10 objects into the space with only 256 dimensions is difficult.
\textbf{Local window size:} Table~\ref{tab:window_size} shows that larger local window size, $\lambda$, results in better performance. Without the local attention, $\lambda=0$, the performance of AOT significantly drops from 80.3\% to 74.3\%, which demonstrates the necessity of the local attention.
\textbf{Local frame number:} In Table~\ref{tab:local_frame}, we also try to employ more previous frames in the local attention, but using only the $t-1$ frame (80.3\%) performs better than using 2/3 frames (80.0\%/79.1\%). A possible reason is that the longer the temporal interval, the more intense the motion between frames, so it is easier to introduce more errors in the local matching when using an earlier previous frame.
\textbf{LSTT block number:} As shown in Table~\ref{tab:lstt_number}, the AOT performance increases by using more LSTT blocks. Notably, the AOT with only one LSTT block (77.9\%) reaches a fast real-time speed (41.0FPS) on YouTube-VOS, although the performance is -2.4\% worse than AOT-S (80.3\%). By adjusting the LSTT block number, we can flexibly balance the accuracy and speed of AOT.
\textbf{Position embedding:} In our default setting, we apply fixed sine spatial positional embedding to the self-attention following~\cite{detr}, and our local attention is equipped with learned relative positional embedding~\cite{relativepos}. The ablation study is shown in Table~\ref{tab:pos_emb}, where removing the sine embedding decreases the performance to 80.1\% slightly. In contrast, the relative embedding is more important than the sine embedding. Without the relative embedding, the performance drops to 79.7\%, which means the motion relationship between adjacent frames is helpful for local attention. We also tried to apply learned positional embedding to self-attention modules, but no positive effect was observed.
\section{Implementation Details}\label{sec:implementation}
\noindent\textbf{Network Details:} For sufficiently validating the effectiveness of our identification mechanism and LSTT, we mainly use light-weight backbone encoder, MobileNet-V2~\cite{sandler2018mobilenetv2}, and decoder, FPN~\cite{fpn} with Group Normalization~\cite{gn}. The spatial neighborhood size $\lambda$ is set to 15, and the number of identification vectors, $M$, is set to 10, which is consistent with the maximum object number in the benchmarks~\cite{youtubevos,davis2017}. AOT performs well with PaddlePaddle~\cite{paddlepaddle} and PyTorch~\cite{pytorch}.
More details can be found in the supplementary material.
\noindent\textbf{Architecture Variants:} We build several AOT variant networks with different LSTT layer number $L$ or long-term memory size $\mathbf{m}$. The hyper-parameters of these variants are: \textbf{(1) AOT-Tiny}: $L=1$, $\mathbf{m}=\{1\}$; \textbf{(2) AOT-Small}: $L=2$, $\mathbf{m}=\{1\}$; \textbf{(3) AOT-Base}: $L=3$, $\mathbf{m}=\{1\}$; \textbf{(4) AOT-Large}: $L=3$, $\mathbf{m}=\{1,1+\delta,1+2\delta,1+3\delta,...\}$. In the experiments, we also equip AOT-L with ResNet50 (R50)~\cite{resnet} or Swin-B~\cite{swin}.
AOT-S is a small model with only 2 layers of LSTT block. Compared to AOT-S, AOT-T utilizes only 1 layer of LSTT, and AOT-B/L uses 3 layers. In AOT-T/S/B, only the first frame is considered into long-term memory, which is similar to \cite{feelvos,cfbi}, leading to a smooth efficiency. In AOT-L, the predicted frames are stored into long-term memory per $\delta$ frames, following the memory reading strategy~\cite{spacetime}. We set $\delta$ to 2/5 for training/testing.
\noindent\textbf{Training Details:} Following~\cite{rgmp,spacetime,EGMN,KMN}, the training stage is divided into two phases: (1) pre-training on sythetic video sequence generated from static image datasets~\cite{voc,coco,cheng2014global,shi2015hierarchical,semantic} by randomly applying multiple image augmentations~\cite{rgmp}. (2) main training on the VOS benchmarks~\cite{youtubevos,davis2017} by randomly applying video augmentations~\cite{cfbi}.
More details are in the supplementary material.
\section{Introduction}\label{sec:introduction}
Video Object Segmentation (VOS) is a fundamental task in video understanding with many potential applications, including augmented reality~\cite{ngan2011video} and self-driving cars~\cite{zhang2016instance}. The goal of semi-supervised VOS, the main task in this paper, is to track and segment object(s) across an entire video sequence based on the object mask(s) given at the first frame.
Thanks to the recent advance of deep neural networks, many deep learning based VOS algorithms have been proposed recently and achieved promising performance. STM~\cite{spacetime} and its following works~\cite{KMN,EGMN} leverage a memory network to store and read the target features of predicted past frames and apply a non-local attention mechanism to match the target in the current frame. FEELVOS~\cite{feelvos} and CFBI~\cite{cfbi,cfbip} utilize global and local matching mechanisms to match target pixels or patches from both the first and the previous frames to the current frame.
Even though the above methods have achieved significant progress, the above methods learn to decode scene features that contain a single positive object. Thus under a multi-object scenario, they have to match each object independently and ensemble all the single-object predictions into a multi-object segmentation, as shown in Fig.~\ref{fig:post_ensemble}. Such a post-ensemble manner eases network architectures' design since the networks are not required to adapt the parameters or structures for different object numbers. However, {modeling multiple objects independently, instead of uniformly, is inefficient in exploring multi-object contextual information to learn a more robust feature representation for VOS.} In addition, processing multiple objects separately yet in parallel requires multiple times the amount of GPU memory and computation for processing a single object. This problem restricts the training and application of VOS under multi-object scenarios, especially when computing resources are limited.
To solve the problem, Fig.~\ref{fig:aot} demonstrates a feasible approach to associate and decode multiple objects uniformly in an end-to-end framework. Hence, we propose an Associating Objects with Transformers (AOT) approach to match and decode multiple targets uniformly. First, an identification mechanism is proposed to assign each target a unique identity and embed multiple targets into the same feature space. Hence, the network can learn the association or correlation among all the targets. Moreover, the multi-object segmentation can be directly decoded by utilizing assigned identity information. Second, a Long Short-Term Transformer (LSTT) is designed for constructing hierarchical object matching and propagation. Each LSTT block utilizes a long-term attention for matching with the first frame's embedding and a short-term attention for matching with several nearby frames' embeddings. Compared to the methods~\cite{spacetime,KMN} utilizing only one attention layer, we found hierarchical attention structures are more effective in associating multiple objects.
We conduct extensive experiments on two popular multi-object benchmarks for VOS, \emph{i.e.}, YouTube-VOS~\cite{youtubevos} and DAVIS 2017~\cite{davis2017}, to validate the effectiveness and efficiency of the proposed AOT. {Even using the light-weight Mobilenet-V2~\cite{sandler2018mobilenetv2} as the backbone encoder}, the AOT variant networks achieve superior performance on the validation 2018 \& 2019 splits of the large-scale YouTube-VOS (ours, $\mathcal{J}\&\mathcal{F}$ \textbf{82.6$\sim$ 84.5}\% \& \textbf{82.2$\sim$ 84.5}\%) while keeping more than $\mathbf{2\times}$ faster multi-object run-time (\textbf{27.1$\sim$ 9.3}FPS) compared to the state-of-the-art competitors (\emph{e.g.}, CFBI~\cite{cfbi}, 81.4\% \& 81.0\%, 3.4FPS). We also achieve new state-of-the-art performance on both the DAVIS-2017 validation (\textbf{85.4}\%) and testing (\textbf{81.2}\%) splits. Moreover, AOT is effective under single-object scenarios as well and outperforms previous methods on DAVIS 2016~\cite{davis2016} (\textbf{92.0}\%), a popular single-object benchmark. Besides, our smallest variant, AOT-T, can maintain real-time multi-object speed on all above benchmarks (\textbf{51.4}FPS on 480p videos). Particularly, AOT ranked $\mathbf{1^{st}}$ in the Track 1 (Video Object Segmentation) of the 3rd Large-scale Video Object Segmentation Challenge.
\input{Figures/better}
Overall, our contributions are summarized as follows:
\begin{itemize}
\vspace{-0.5em}
\item We propose an identification mechanism to associate and decode multiple targets uniformly for VOS. For the first time, multi-object training and inference can be efficient as single-object ones, as demonstrated in Fig.~\ref{fig:complexity}.
\vspace{-0.5em}
\item Based on the identification mechanism, we design a new efficient VOS framework, \emph{i.e.}, Long Short-Term Transformer (LSTT), for constructing hierarchical multi-object matching and propagation. LSTT achieves superior performance on VOS benchmarks~\cite{youtubevos,davis2017,davis2016} while maintaining better efficiency than previous state-of-the-art methods. {To the best of our knowledge, LSTT is the first hierarchical framework for object matching and propagation by applying transformers~\cite{transformer} to VOS.}
\end{itemize}
\section{Associating Objects with Transformers}
In this section, we introduce our identification mechanism proposed for efficient multi-object VOS. Then, we design a new VOS framework, \emph{i.e.}, long short-term transformer, based on the identification mechanism for constructing hierarchical multi-object matching and propagation.
\subsection{Identification Mechanism for Multi-object Association}
Many recent VOS methods~\cite{spacetime,EGMN,KMN} utilized attention mechanisms and achieved promising results.
To formulate, we define $Q \in \mathbb{R}^{HW \times C}$, $K \in \mathbb{R}^{THW \times C}$, and $V \in \mathbb{R}^{THW \times C}$ as the query embedding of the current frame, the key embedding of the memory frames, and the value embedding of the memory frames respectively, where $T$, $H$, $W$, $C$ denote the temporal, height, width, and channel dimensions. The formula of a common attention-based matching and propagation is,
\begin{equation}
\label{equ:att}
Att(Q,K,V)=Corr(Q,K)V=softmax(\frac{QK^{tr}}{\sqrt{C}})V,
\end{equation}
where a matching map is calculated by the correlation function $Corr$, and then the value embedding, $V$, will be propagated into each location of the current frame.
In the common single-object propagation~\cite{spacetime}, the binary mask information in memory frames is embedded into $V$ with an additional memory encoder network and thus can also be propagated to the current frame by using Eq.~\ref{equ:att}. A convolutional decoder network following the propagated feature will decode the aggregated feature and predict the single-object probability logit of the current frame.
The main problem of propagating and decoding multi-object mask information in an end-to-end network is how to adapt the network to different target numbers. To overcome this problem, we propose an identification mechanism consisting of identification embedding and decoding based on attention mechanisms.
First, an \textbf{Identification Embedding} mechanism is proposed to embed the masks of multiple different targets into the same feature space for propagation. As seen in Fig.~\ref{fig:id}, we initialize an identity bank, $D \in \mathbb{R}^{M \times C}$, where $M$ identification vectors with $C$ dimensions are stored. For embedding multiple different target masks, each target will be randomly assigned a different identification vector. Assuming $N$ ($N<M$) targets are in the video scenery, the formula of embedding the targets' one-hot mask, $Y \in \{0,1\}^{THW \times N}$, into a identification embedding, $E \in \mathbb{R}^{THW \times C}$, by randomly assigning identification vector from the bank $D$ is,
\begin{equation}
\label{equ:id}
E=ID(Y,D)=YPD,
\end{equation}
where $P \in \{0,1\}^{N \times M}$ is a random permutation matrix, satisfying that $P^{tr}P$ is equal to a $M \times M$ unit matrix, for randomly selecting $N$ identification embeddings. After the $ID$ assignment, different target has different identification embedding, and thus we can propagate all the target identification information from memory frames to the current frame by attaching the identification embedding $E$ with the attention value $V$, \emph{i.e.},
\begin{equation}
V'=AttID(Q,K,V,Y|D)=Att(Q,K,V+ID(Y,D))=Att(Q,K,V+E),
\end{equation}
where $V' \in \mathbb{R}^{HW \times C}$ aggregates all the multiple targets' embeddings from the propagation.
For \textbf{Identification Decoding}, \emph{i.e.}, predicting all the targets' probabilities from the aggregated feature $V'$, we firstly predict the probability logit for every identity in the bank $D$ by employing a convolutional decoding network $F^{\mathcal D}$, and then select the assigned ones and calculate the probabilities, \emph{i.e.},
\begin{equation}
Y'=softmax(PF^{\mathcal D}(V'))=softmax(PL^D),
\end{equation}
where $L^D \in \mathbb{R}^{HW \times M}$ is all the $M$ identities' probability logits, $P$ is the same as the selecting matrix used in the identity assignment (Eq.~\ref{equ:id}), and $Y' \in [0,1]^{HW \times N}$ is the probability prediction of all the $N$ targets.
For training, common multi-class segmentation losses, such as cross-entropy loss, can be used to optimize the multi-object $Y'$ regarding the ground-truth labels. The identity bank $D$ is trainable and randomly initialized at the training beginning. To ensure that all the identification vectors have the same opportunity to compete with each other, we randomly reinitialize the identification selecting matrix $P$ in each video sample and each optimization iteration.
\subsection{Long Short-Term Transformer for Hierarchical Matching and Propagation}
Previous methods~\cite{spacetime,KMN} always utilize only one layer of attention (Eq.~\ref{equ:att}) to aggregate single-object information. In our identification-based multi-object pipeline, we found that a single attention layer cannot fully model multi-object association, which naturally should be more complicated than single-object processes. Thus, we consider constructing hierarchical matching and propagation by using a series of attention layers. Recently, transformer blocks~\cite{transformer} have been demonstrated to be stable and promising in constructing hierarchical attention structures in visual tasks~\cite{detr,vit}. Based on transformer blocks, we carefully design a Long Short-Term Transformer (LSTT) block for multi-object VOS.
Following the common transformer blocks~\cite{transformer,devlin2018bert}, LSTT firstly employs a self-attention layer, which is responsible for learning the association or correlation among the targets within the current frame. Then, LSTT additionally introduces a long-term attention, for aggregating targets' information from long-term memory frames and a short-term attention, for learning temporal smoothness from nearby short-term frames. The final module is based on a common 2-layer feed-forward MLP with GELU~\cite{gelu} non-linearity in between. Fig.~\ref{fig:lstt} shows the structure of an LSTT block. Notably, all these attention modules are implemented in the form of the multi-head attention~\cite{transformer}, \emph{i.e.}, multiple attention modules followed by concatenation and a linear projection. Nevertheless, we only introduce their single-head formulas below for the sake of simplicity.
\noindent\textbf{Long-Term Attention} is responsible for aggregating targets' information from past memory frames, which contains the reference frame and stored predicted frames, to the current frame. Since the time intervals between the current frame and past frames are variable and can be long-term, the temporal smoothness is difficult to guarantee. Thus, the long-term attention employs non-local attention like Eq.~\ref{equ:att}. Let $X_l^{t} \in \mathbb{R}^{HW \times C}$ denotes the input feature embedding at time $t$ and in block $l$, where $l \in \{1,...,L\}$ is the block index of LSTT, the formula of the long-term attention is,
\begin{equation}
AttLT(X_l^{t},X_l^{\mathbf{m}},Y^{\mathbf{m}}) = AttID( X_l^{t} W^K_l, X_l^{\mathbf{m}} W^K_l, X_l^{\mathbf{m}} W^V_l,Y^{\mathbf{m}}|D),
\end{equation}
where $X_l^{\mathbf{m}}=Concat(X_l^{m_1},...,X_l^{m_T})$ and $Y^{\mathbf{m}}=Concat(Y^{m_1},...,Y^{m_T})$ are the input feature embeddings and target masks of memory frames with indices $\mathbf{m}=\{m_1,...,m_T\}$. Besides, $W^K_l \in \mathbb{R}^{C \times C_k}$ and $W^V_l\in \mathbb{R}^{C \times C_v}$ are trainable parameters of the space projections for matching and propagation, respectively. Instead of using different projections for $X_l^{t}$ and $X_l^{\mathbf{m}}$, we found the training of LSTT is more stable with a siamese-like matching, \emph{i.e.}, matching between the features within the same embedding space ($l$-th features with the same projection of $W^K_l$).
\noindent\textbf{Short-Term Attention} is employed for aggregating information in a spatial-temporal neighbourhood for each current-frame location. Intuitively, the image changes across several contiguous video frames are always smooth and continuous. Thus, the target matching and propagation in contiguous frames can be restricted in a small spatial-temporal neighborhood, leading to better efficiency than non-local processes. Considering $n$ neighbouring frames with indices $\mathbf{n}=\{t-1,...,t-n\}$ are in the spatial-temporal neighbourhood, the features and masks of these frames are $X_l^{\mathbf{n}}=Concat(X_l^{t-1},...,X_l^{t-n})$ and $Y^{\mathbf{n}}=Concat(Y^{t-1},...,Y^{t-n})$, and then the formula of the short-term attention at each spatial location $p$ is,
\begin{equation}
AttST(X_l^{t},X_l^{\mathbf{n}},Y^{\mathbf{n}}|p) = AttLT(X_{l,p}^{t},X_{l,\mathcal{N}(p)}^{\mathbf{n}},Y_{l,\mathcal{N}(p)}^{\mathbf{n}}),
\end{equation}
where $X_{l,p}^{t} \in \mathbb{R}^{1 \times C} $ is the feature of $X_{l}^{t}$ at location $p$, $\mathcal{N}(p)$ is a $\lambda \times \lambda$ spatial neighbourhood centered at location $p$, and thus $X_{l,\mathcal{N}(p)}^{\mathbf{n}}$ and $Y_{l,\mathcal{N}(p)}^{\mathbf{n}}$ are the features and masks of the spatial-temporal neighbourhood, respectively, with a shape of $n\lambda^2\times C$ or $n\lambda^2\times N$.
When extracting features of the first frame $t=1$, there is no memory frames or previous frames, and hence we use $X_l^{1}$ to replace $X_l^{\mathbf{m}}$ and $X_l^{\mathbf{n}}$. In other words, the long-term attention and the short-term attention are changed into self-attentions without adjusting the network structures and parameters.
\section{Related Work}
\noindent\textbf{Semi-supervised Video Object Segmentation.}
Given one or more annotated frames (the first frame in general), semi-supervised VOS methods propagate the manual labeling to the entire video sequence. Traditional methods often solve an optimization problem with an energy defined over a graph structure~\cite{tradition1,tradition3,tradition2}. In recent years, VOS methods have been mainly developed based on deep neural networks (DNN), leading to better results.
Early DNN methods rely on fine-tuning the networks at test time to make segmentation networks focus on a specific object. Among them, OSVOS~\cite{osvos} and MoNet~\cite{xiao2018monet} fine-tune pre-trained networks on the first-frame ground-truth at test time. OnAVOS~\cite{onavos} extends the first-frame fine-tuning by introducing an online adaptation mechanism. Following these approaches, MaskTrack~\cite{masktrack} and PReM~\cite{premvos}
utilize optical flow to help propagate the segmentation mask from one frame to the next. Despite achieving promising results, the test-time fine-tuning restricts the network efficiency.
Recent works aim to achieve a better run-time and avoid using online fine-tuning. OSMN~\cite{osmn} employs one convolutional network to extract object embedding and another one to guide segmentation predictions. PML~\cite{pml} learns pixel-wise embedding with a nearest neighbor classifier, and VideoMatch~\cite{videomatch} uses a soft matching layer that maps the pixels of the current frame to the first frame in a learned embedding space. Following PML and VideoMatch, FEELVOS~\cite{feelvos} and CFBI~\cite{cfbi,cfbip} extend the pixel-level matching mechanism by additionally matching between the current frame and the previous frame. RGMP~\cite{rgmp} also gathers guidance information from both the first frame and the previous frame but uses a siamese encoder with two shared streams. STM~\cite{spacetime} and its following works (\emph{e.g.}, EGMN~\cite{EGMN} and KMN~\cite{KMN}) leverage a memory network to embed past-frame predictions into memory and apply a non-local attention mechanism on the memory to decode the segmentation of the current frame. SST~\cite{sstvos} utilizes attention mechanisms in a different way, \emph{i.e.}, transformer blocks~\cite{transformer} are used to extract pixel-level affinity maps and spatial-temporal features. The features are target-agnostic, instead of target-aware like our LSTT, since the mask information in past frames is not propagated and aggregated in the blocks. Instead of using matching mechanisms, LWL~\cite{LWLVOS} proposes to use an online few-shot learner to learn to decode object segmentation.
The above methods learn to decode features with a single positive object and thus have to match and segment each target separately under multi-object scenarios, consuming multiple times computing resources of single-object cases. The problem restricts the application and development of the VOS with multiple targets. Hence, we propose our AOT to associate and decode multiple targets uniformly and simultaneously, as efficiently as processing a single object.
\noindent\textbf{Visual Transformers.} Transformers~\cite{transformer} was proposed to build hierarchical attention-based networks for machine translation. Similar to Non-local Neural Networks~\cite{nonlocal}, transformer blocks compute correlation with all the input elements and aggregate their information by using attention mechanisms~\cite{att}. Compared to RNNs, transformer networks model global correlation or attention in parallel, leading to better memory efficiency, and thus have been widely used in natural language processing (NLP) tasks~\cite{devlin2018bert,radford2019language,synnaeve2019end}. Recently, transformer blocks were introduced
to many computer vision tasks, such as image classification~\cite{vit,vaswani2021scaling,swin}, object detection~\cite{detr}/segmentation~\cite{vistr}, and image generation~\cite{parmar2018image}, and have shown promising performance compared to CNN-based networks.
Many VOS methods~\cite{lin2019agss,spacetime,EGMN,KMN} have utilized attention mechanisms to match the object features and propagate the segmentation mask from past frames to the current frames. Nevertheless, these methods consider only one positive target in the attention processes, and how to build hierarchical attention-based propagation has been rarely studied. In this paper, we carefully design a long short-term transformer block, which can effectively construct multi-object matching and propagation within hierarchical structures for VOS.
\section{Revisit Previous Solutions for Video Object Segmentation}
In VOS, many common video scenarios have multiple targets or objects required for tracking and segmenting. Benefit from deep networks, current state-of-the-art VOS methods~\cite{spacetime,cfbi} have achieved promising performance. Nevertheless, these methods focus on matching and decoding a single object. Under a multi-object scenario, they thus have to match each object independently and ensemble all the single-object predictions into a multi-object prediction, as demonstrated in Fig.~\ref{fig:post_ensemble}. Let $F^{\mathcal{N}}$ denotes a VOS network for predicting single-object segmentation, and $A$ is an ensemble function such as $softmax$ or the soft aggregation~\cite{spacetime}, the formula of such a post-ensemble manner for processing $N$ objects is like,
\begin{equation}
Y'=A(F^{\mathcal{N}}(I^t,I^{\mathbf{m}},Y^{\mathbf{m}}_1),...,F^{\mathcal{N}}(I^t,I^{\mathbf{m}},Y^{\mathbf{m}}_N)),
\end{equation}
where $I^t$ and $I^\mathbf{m}$ denote the image of the current frame and memory frames respectively, and $\{Y^{\mathbf{m}}_1,..., Y^{\mathbf{m}}_N\}$ are the memory masks (containing the given reference mask and past predicted masks) of all the $N$ objects. This manner extends networks designed for single-object VOS into multi-object applications, so there is no need to adapt the network for different object numbers.
Although the above post-ensemble manner is prevalent and straightforward in the VOS field, processing multiple objects separately yet in parallel requires multiple times the amount of GPU memory and computation for matching a single object and decoding the segmentation. This problem restricts the training and application of VOS under multi-object scenarios when computing resources are limited.
To make the multi-object training and inference as efficient as single-object ones, {an expected solution should be capable of associating and decoding multiple objects uniformly instead of individually. To achieve such an objective, we propose an identification mechanism to embed the masks of any number (required to be smaller than a pre-defined large number) of targets into the same high-dimensional space. Based on the identification mechanism, a novel and efficient framework, \emph{i.e.}, Associating Objects with Transformers (AOT), is designed for propagating all the object embeddings uniformly and hierarchically, from memory frames to the current frame.}
As shown in Fig.~\ref{fig:aot}, our AOT associates and segments multiple objects within an end-to-end framework. For the first time, processing multiple objects can be as efficient as processing a single object (Fig.~\ref{fig:complexity}). Compared to previous methods, our training under multi-object scenarios is also more efficient since AOT can associate multiple object regions and learn contrastive feature embeddings among them uniformly.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,554 |
John Etter Clark (March 29, 1915 – June 3, 1956) was a provincial politician, teacher and farmer from Alberta, Canada. He was a member of the Legislative Assembly of Alberta from 1952 until committing one of the deadliest mass murders in Alberta history and killing himself.
Early life
John Etter Clark was born in Stettler, Alberta in 1915. He became a part-time school teacher and a farmer. Clark inherited the farm founded by his father. He married Margaret Dinwoodie in 1947 and they had four children.
Political career
Clark ran for a seat in the Alberta Legislature, representing the Stettler district, in the 1952 Alberta general election as a Social Credit candidate. The four-way race was hotly contested, and Clark won on the second vote count to hold the district for his party.
Clark ran for a second term in the 1955 Alberta general election. He won a sizable majority to defeat two other candidates and hold his seat.
Mass murder and subsequent suicide
On June 3, 1956 Pete Parrott, a neighbour residing on a farm leased from Clark next to his farm in Erskine, Alberta, stopped over for a social visit. He found six bodies and one wounded person, each shot at least once through the head with .22 calibre bullets, and one shot multiple times. The wounded victim was taken to a local hospital and died shortly after.
The Royal Canadian Mounted Police descended on the scene with 14 special field agents. Clark had fled and was not among the dead, who included his wife, son, three daughters, a hired farmhand and a visitor. The murder weapon was a single-shot .22 calibre rifle that Clark had borrowed from his uncle. He was expected to travel to Saskatchewan on June 1, 1956, to help manage the Social Credit campaign in the 1956 Saskatchewan general election, but failed to show up without explanation.
Police found Clark's body lying on the edge of a dugout approximately 600 yards (550 m) from the farmhouse where the murders took place. It had wounds from a single self-inflicted bullet through the head and the murder weapon lying at its feet. It was found adorned in night attire as if Clark had been preparing to go to bed. Thirty-two RCMP Officers who travelled the range on horseback with a team of tracking dogs conducted the search. A team of three Mounties on a Royal Canadian Air Force Otter conducted an aerial search. The mounties spotted Clark's body from the air a few hours after the search began.
Clark had previously been hospitalized for a month and a half after a nervous breakdown in 1954, and another during the legislature's 1956 spring session.
It, along with the Cook Family murders in 1959, were the deadliest mass murders in Alberta's history, until Phu Lang killed nine in Edmonton in December 2014.
References
External links
Legislative Assembly of Alberta Members Listing
1915 births
1956 suicides
20th-century criminals
Alberta Social Credit Party MLAs
Canadian farmers
Canadian mass murderers
Canadian murderers of children
Canadian schoolteachers
Familicides
Murder–suicides in Canada
People from the County of Stettler No. 6
Suicides by firearm in Alberta
Mass murder in Alberta | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 252 |
Ranjit Studios, also known as Ranjit Movietone, was an Indian film production company with studio facilities located in Mumbai, Maharashtra, India. It produced films between 1929 and mid-1970s. The studio was founded by Chandulal Shah along with Gohar Kayoum Mamajiwala. It was one of the three largest studios in Bollywood of its time, besides Kohinoor Film Company and Imperial Film Company.
The company began production of silent films in 1929 under the banner Ranjit Film Company and by 1932 had made 39 pictures, most of them social dramas. The company changed its name to Ranjit Movietone in 1932 and during the 1930s produced numerous successful talkies at the rate of about six a year. At this time, the studio employed around 300 actors, technicians and other employees.
Ranjit productions were mostly filmed in the Hindi, Punjabi and Gujarati languages.
The company ended at some time in the late 1960s.
References
External links
Chandulal Shah biography
Ranjit Films on the IMDb
Ranjit Studios on the IMDb
Hindi cinema
Film studios in Mumbai
1929 establishments in India
Mass media companies established in 1929
Companies disestablished in the 1960s
1960s disestablishments in India | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,711 |
ROB SHIRAKBARI (also known as ROB SHROCK) is the Music Director for pop/soul icon, DIONNE WARWICK, and for years Rob was the keyboardist, arranger and music director for legendary composer, BURT BACHARACH. He is currently the producer, arranger and songwriting partner for UK artist, RUMER.
Rob has recorded and performed with such legendary artists as Duffy, Rumer, Adele, Elvis Costello, Leann Rimes, Sir Elton John, Stevie Wonder, Sheryl Crow, Whitney Houston, Chrissie Hynde, Ray Charles, Faith Hill, Sir Cliff Richard, Wynonna, Aretha Franklin, Isaac Hayes, Alexandra Burke, Chris Botti, Rufus Wainwright, Joe McElderry, Garth Brooks, Jamie Cullum, Michael Bublé, Peabo Bryson, 'N Sync, Clay Aiken, Rubben Studdard, Kristy Lee Cook, Boy George, Caro Emerald, Ivan Lins, Gladys Knight, Frank Sinatra, Queen Latifah, David Foster, Gloria Estefan, Barry Manilow, Cyndi Lauper, Ben Folds, Olivia Newton-John, Patti LaBelle, Smokey Robinson, Jeffrey Osborne, David Sanborn, Trijntje Oosterhuis, Johnny Mathis, Natalie Cole, George Duke, Ronald Isley and a host of others.
In 2000, Rob served as one of three Musical Directors for the 72nd Academy Awards© (along with Don Was and Burt Bacharach). Rob wrote the arrangements and conducted the pit orchestra for the four-hour awards show. As well, Rob played keyboards in the on-camera band that featured Ray Charles, Garth Brooks, Queen Latifah, Faith Hill, Dionne Warwick and Isaac Hayes in a tribute to Oscar-winning songs.
Rob also sings, plays guitar and keyboards and is the bandleader for AM/FM, the best 70s Classic Rock band in the USA.
As a writer, Rob is a long-time contributor to Keyboard and Electronic Musician magazines (among others), writing feature articles and product reviews. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,913 |
Q: Committing a running docker container (with pause), then run it, what's the images internal state Let's assume I have a Python program running inside a docker container
import time
counter = 0
while True:
counter += 1
print counter
time.sleep(1)
What happens if I do a commit on that running container, and then use that new image to run a new container?
The docs state that a running container is paused (cgroups freezer), and after commiting it gets unpaused. In what state is the image in? SIGKILL? I assume that the python program won't be running anymore when doing a docker start on that image, correct?
I'm asking because I have a couple of Java servers (Atlassian) running in the container, so I wonder if I'm doing daily backups via commit on that container, and I then "restore" (docker run ... backup/20160118) one of the backups, in what state will the servers be in?
A: Docker commit only commits the filesystem changes of a container, so any file that has been added, removed, or modified on the filesystem since the container was started.
Note that any volume (--volume or VOLUME inside the dockerfile) is not part of the container's filesystem, so won't be committed.
In-memory state: "Checkpoint and Restore"
Committing a container, including it's current (in-memory) state, is a lot more complex. This process is called "checkpoint and restore". You can find more information about this on https://criu.org. There's currently a pull request to add basic support for checkpoint and restore to Docker; https://github.com/docker/docker/pull/13602, but that feature does not yet support "migrating" such containers to a different machine.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,686 |
\section{Introduction}
Throughout this paper, $X$ and $Y$ denote separable infinite-dimensional Fr\'{e}chet spaces, and $L(X,Y)$ denotes the space of continuous linear operators from $X$ to $Y$. When $Y=X$, we let $L(X)=L(X,X)$. A sequence $(T_n)$ in $L(X,Y)$ is said to be {\em hypercyclic } provided there is some $x$ in $X$ for which the set $\{ T_nx: \ n\ge 0\}$ is dense in $Y$. Such vector $x$ is called a {\em hypercyclic vector} for $(T_n)$.
An operator $T$ in $L(X)$ is said to be hypercyclic whenever its sequence of iterates $(T^n)$ is a hypercyclic sequence.
An important question about a hypercyclic operator is whether it supports any infinite-dimensional {\em closed} subspace in which every non-zero vector is hypercyclic. Such a subspace is called a \emph{hypercyclic subspace}. This notion is interesting --see \cite[Chapter 8]{BayartMatheron} or \cite[Chapter 10]{Grosse}-- because some hypercyclic operators do not support hypercyclic subspaces while some other operators do. Indeed, Montes \cite{Montes} showed that no scalar multiple of the backward shift on $\ell^p$ ($1\le p<\infty$) supports a hypercyclic subspace. On the other hand, the collective work of Bernal and Montes \cite{Bernal}, Petersson \cite{Petersson}, Shkarin \cite{ShkarinD} and Menet \cite{Menet1} shows that each non-scalar convolution operator on the space $H(\mathbb{C})$ of entire functions has a hypercyclic subspace. Convolution operators on $H(\mathbb C)$ are those that commute with the operator $D$ of complex differentiation, and are precisely those of the form
\[
\phi (D): H(\mathbb C)\to H(\mathbb C ), \ f\mapsto \sum_{n\ge 0} a_n D^nf,
\]
where $\phi (z)=\sum_{n\ge 0} a_n z^n \in H(\mathbb C )$ is of exponential type \cite{Godefroy}.
\subsection{Two sufficient criteria for the existence of hypercyclic subspaces} \label{Ss:Hcs}
Several criteria for the existence or the non-existence of hypercyclic subspaces are known; we will be
particularly interested in the next two theorems. We first recall the following.
\begin{definition}(Le\'{o}n and M\"{u}ller \cite{Leon})\label{def C fre}
A sequence $(T_n)$ in $L(X,Y)$ satisfies \emph{Condition} (C) along a given strictly increasing sequence $(n_k)$ of positive integers provided
\begin{enumerate}
\item[(1)]\ For each $x$ in some dense subset of $X$ we have $T_{n_k}x\to 0$, and
\item[(2)]\ For each continuous seminorm $p$ on $X$, $\bigcup_k T_{n_k}(p^{-1}[0,1))$ is dense in $Y$.
\end{enumerate}
\end{definition}
First formulated on the Banach space setting, Condition (C) ensures by a standard Baire Category argument that the sequence $(T_n)$ has a residual set of hypercyclic vectors whenever $Y$ is separable \cite{Menet1}.
\begin{theorem}[{\bf Criterion ${\bf M_0}$ }\cite{Menet1}]\label{thm M0}
Let $(T_n)$ be a sequence in $L(X,Y)$ satisfying Condition~$(C)$ along a sequence $(n_k)$. Suppose that $X$ supports a continuous norm and that there exists an infinite-dimensional closed subspace $M_0$ of $X$ such that \[T_{n_k}x\rightarrow 0 \ \ \mbox{ for all $x\in M_0$.}\] Then $(T_n)$ has a hypercyclic subspace.
\end{theorem}
\begin{definition} Let $(M_n)$ be a sequence of infinite-dimensional closed subspaces of $X$ with $M_n\supseteq M_{n+1}$ for all $n$.
A sequence $(T_n)$ in $L(X, Y)$ is said to be {\em equicontinuous along $(M_n)$} provided for each continuous seminorm $q$ on $Y$ there exists a continuous seminorm $p$ of $X$ so that for each $n\in\mathbb{N}$
\[
q(T_nx) \le p(x) \ \ \mbox{ for each $x\in M_n$.}
\]
\end{definition}
\begin{theorem}[ {\bf Criterion ${\bf (M_k)}$} \cite{Menet1}]\label{thm Mk}
Let $(T_n)$ be a sequence in $L(X,Y)$ satisfying Condition $(C)$ along a sequence $(n_k)$. Suppose that $X$ supports a continuous norm and that
$(T_{n_k})$ is equicontinuous along some non-increasing sequence $(M_k)$ of closed, infinite-dimensional subspaces of $X$.
Then $(T_n)$ has a hypercyclic subspace.
\end{theorem}
The first versions of Criterion $M_0$ and Criterion $(M_k)$ appeared under the stronger assumption of satisfying the Hypercyclicity Criterion instead of Condition $(C)$.
Criterion $M_0$ first appeared in 1996
for the case of operators on Banach spaces and is due to Montes~\cite{Montes}; this was generalized to operators on Fr\'{e}chet spaces with a continuous norm by Bonet, Mart\'inez and Peris~\cite{Bonet} and independently by Petersson~\cite{Petersson}, as well as to sequences of operators on Banach spaces by Le\'{o}n and M\"{u}ller~\cite{Leon}. The present version on sequences of operators on Fr\'{e}chet spaces with a continuous norm is due to Menet~\cite{Menet1}.
Criterion $(M_k)$ first appeared implicitly for the case of operators on Hilbert and Banach spaces (and where the sequence $(M_k)$ is constant) in the works of Le\'{o}n and Montes \cite{LeonMontes} and by Gonz\'{a}lez et al \cite{Gonzalez}, respectively; this was extended in 2006 by Le\'on and M\"{u}ller \cite{Leon} to sequences of operators on Banach spaces and more recently by Menet \cite{Menet1} to the present version on Fr\'{e}chet spaces with a continuous norm. Both Criterion $M_0$ and Criterion $(M_k)$ have generalizations to Fr\'{e}chet spaces without continuous norm as well \cite{Menet1bis}.
\vspace{.1in}
\subsection{Two themes about hypercyclic subspaces we consider}
This paper explores a link between Criterion $M_0$ and Criterion $(M_k)$ and its consequences within two themes: subspaces of $\mathcal{U}$-frequently hypercyclic vectors
and subspaces of common hypercyclic vectors.
The first theme is motivated by recent work of Bonilla and Grosse-Erdmann~\cite{Bonilla2} that initiated the study of {\em frequently hypercyclic subspaces,} that is, of hypercyclic subspaces consisting entirely (but the origin) of frequently hypercyclic vectors.
The notions of frequently hypercyclic vectors and of $\mathcal{U}$-frequently hypercyclic vectors are due to Bayart and Grivaux~\cite{Bayart04} and to Shkarin~\cite{Shkarin0}, respectively. For a set $A$ of non-negative integers, the quantities
\[\underline{\text{dens}}(A):=\liminf_N\frac{\#(A\cap [0,N])}{N+1} \mbox{ and } \overline{\text{dens}}(A):=\limsup_N\frac{\#(A\cap [0,N])}{N+1} \]
are respectively called the lower density and the upper density of $A$.
\begin{definition}\label{D:fhc} Let $(T_n)$ be a sequence in $L(X,Y)$. A vector $x\in X$ is said to be {\em frequently hypercyclic} (respectively, $\mathcal{U}$-\emph{frequently hypercyclic}) for $(T_n)$ provided for each non-empty open subset $U$ of $Y$, the set $\{n\ge 0: T_n x\in U\}$ has positive lower density (respectively, positive upper density).
\end{definition}
Bonilla and Grosse-Erdmann \cite{Bonilla3, Bonilla2} showed that each non-scalar convolution operator $\phi (D)$ on $H(\mathbb C )$ is frequently hypercyclic and that it has a frequently hypercyclic subspace whenever $\phi\in H(\mathbb{C})$ is transcendental, and asked whether the derivative operator $D$ has a frequently hypercyclic subspace as well \cite[Problem 2]{Bonilla2}. We will consider here the following related question.
\begin{question} \label{Q:P(D)ufhcs}
Does the derivative operator have a $\mathcal{U}$-frequently hypercyclic subspace on $H(\mathbb{C})$? What about $P(D)$, where
$P$ is a non-constant polynomial?
\end{question}
The next question is motivated by recent work of Bayart and Ruzsa \cite{Bayart3}, who completely characterized frequent hypercyclicity and $\mathcal{U}$-frequent hypercyclicity among unilateral and bilateral weighted shift operators on $\ell^p$ and $\ell^p(\mathbb Z )$ $(1\le p<\infty )$, respectively, as well as on $c_0$ and $c_0(\mathbb{Z})$, respectively. Hence it is natural to ask:
\begin{question} \label{Q:B_wUfhcs}
Which weighted shifts support a $\mathcal{U}$-frequently hypercyclic subspace?
\end{question}
\vspace{.05in}
The second theme we consider is common hypercyclic subspaces. Recall the following.
\begin{definition}
A vector $x\in X$ is called a \emph{common hypercyclic vector} for a given family $\mathcal{F}=\{ (T_{n,\lambda})_{n\ge 0} \}_{\lambda\in \Lambda}$ of sequences in $L(X,Y)$ provided $x$ is a hypercyclic vector for each $(T_{n,\lambda})_{n\ge 0}$ in $\mathcal{F}$.
We also say that the common hypercyclic vectors of a given family $\mathcal{F}=\{ T_{\lambda} \}_{\lambda\in \Lambda}$ of operators on $X$ are the common hypercyclic vectors of the family of corresponding sequences of iterates $\{ (T_{\lambda}^n)_{n\ge 0} \}_{\lambda\in \Lambda}$.
A closed, infinite-dimensional subspace consisting entirely (but the origin) of common hypercyclic vectors for $\mathcal{F}$ is called a {\em common hypercyclic subspace} for $\mathcal{F}$. A dense, linear subspace consisting entirely (but the origin) of common hypercyclic vectors for $\mathcal{F}$ is called a {\em common hypercyclic manifold} for $\mathcal{F}$.
\end{definition}
If the family $\mathcal{F}=\{ T_{\lambda} \}_{\lambda\in \Lambda}$ of hypercyclic operators is countable, then it has a residual set of common hypercyclic vectors
and a common hypercyclic manifold \cite{Grivaux2003}. If $\mathcal{F}$ is uncountable, however, it may fail to have a single common hypercyclic vector, even if each operator $T_\lambda$ has a hypercyclic subspace \cite{Aron}. The first example of an uncountable family with a common hypercyclic vector was given by Abakumov and Gordon \cite{Abakumov}, who showed that $\{ \lambda B \}_{\lambda >1}$ has a common hypercyclic vector, where $B$ is the unweighted backward shift on $\ell^2$. The importance of common hypercyclic vectors within linear dynamics is showcased in \cite[Chapter 7]{BayartMatheron} and \cite[Chapter 11]{Grosse}.
In a remarkable paper, Costakis and Sambarino \cite{Costakis2} considered unilateral backward shifts $B_{w_\lambda}$ $(\lambda >1)$ on $\ell^2$ with weight sequence $w_\lambda =(1+\frac{\lambda}{n})_{n\ge 1}$ and
showed that the set of common hypercyclic vectors for the family
\[
\{ B_{w_\lambda } \}_{\lambda >1 }
\]
is residual on $\ell^2$. Each such $B_{w_\lambda}$ has a hypercyclic subspace. Hence the following
\begin{question}{ ({\bf Costakis and Sambarino \cite[Problem 3]{Costakis2}})} \label{Q:Bwchcs}
Does the family $\{ B_{w_\lambda} \}_{\lambda >1}$ support a common hypercyclic subspace?
\end{question}
We already mentioned that any given non-scalar convolution operator $\phi(D)$ on $H(\mathbb C )$ has a hypercyclic subspace. But the family of its (non-zero) scalar multiples
\[
\{ \lambda\, \phi(D) \}_{|\lambda |>0}
\]
has a residual set of common hypercyclic vectors, as established by Costakis and Mavroudis~\cite{Costakis} for the case $\phi$ is a non-constant polynomial, by Bernal~\cite{Bernal2009} for the case $\phi$ is of growth order no larger than $\frac{1}{2}$, and by Shkarin \cite{Shkarin} for the general transcendental case. Hence it is natural to ask.
\begin{question} \label{Q:P(D)common}
Does the family of non-zero scalar multiples of a non-scalar convolution operator on $H(\mathbb C )$ support a common hypercyclic subspace?
\end{question}
We mention that Bayart~\cite{Bayart} extended Criterion~$M_0$ to one that shows the existence of common hypercyclic subspaces (see Theorem~\ref{CritM0}), but this extension has proven difficult to apply in many natural examples.
\subsection{Main results and outline of the paper}
Section~\ref{Gene Mk} is devoted to developing the main tools (Theorem~\ref{NiceMn}) for establishing in the subsequent sections $(M_k)$-type criteria for the existence of $\mathcal{U}$-frequently hypercyclic subspaces or of common hypercyclic subspaces.
Section~\ref{SecU} is devoted to $\mathcal{U}$-frequently hypercyclic subspaces. We show for instance that any operator on a complex Banach space that satisfies the Frequent Hypercyclicity Criterion (Theorem~\ref{T:FHC}) {\em has a $\mathcal{U}$-frequently hypercyclic subspace if and only if $T$ has a hypercyclic subspace} (Theorem~\ref{T:FHC2ufhcs}). We then answer Question~\ref{Q:P(D)ufhcs} in the affirmative. Indeed, we show that {\em for any $N\ge 1$ and any $a_0(z)$,\dots, $a_{N-1}(z)\in H(\mathbb C )$ and $0\ne a_N\in\mathbb C$ the linear differential operator
\[
L=a_N D^N+ a_{N-1}(z) D^{N-1}+\cdots +a_0(z) I
\]
has a $\mathcal{U}$-frequently hypercyclic subspace on $H(\mathbb C )$} (Corollary~\ref{C:ufhsNonConstant}). Concerning Question~\ref{Q:B_wUfhcs}, we show that {\em each $\mathcal{U}$-frequently hypercyclic unilateral weighted backward shift with a hypercyclic subspace on $\ell^p$ has a $\mathcal{U}$-frequently hypercyclic subspace} (Corollary~\ref{caraclp}).
As a consequence, we derive the existence of {\em a frequently hypercyclic operator that has a $\mathcal{U}$-frequently hypercyclic subspace but no frequently hypercyclic subspace} (Corollary~\ref{C:ufhcsNofhcs}). We complement our answer to Question~\ref{Q:B_wUfhcs} by showing that each $\mathcal{U}$-frequently hypercyclic bilateral weighted backward shift with a hypercyclic subspace on $\ell^p(\mathbb Z)$ has even a frequently hypercyclic subspace (Theorem~\ref{caraclpbi}).
We obtain the above results after establishing the following criterion for supporting $\mathcal{U}$-frequently hypercyclic subspaces (cf.Theorem~\ref{UM0} and Theorem~\ref{UMk}): {\em If the Fr\'{e}chet space $X$ supports a continuous norm then any $T\in L(X)$ satisfying the Frequent Hypercyclicity Criterion has a $\mathcal{U}$-frequently hypercyclic subspace provided there exists a strictly increasing sequence of positive integers $(n_k)$ with positive upper density satisfying either of the two conditions:
\begin{enumerate}
\item[(a)]\ $(T^{n_k})$ converges pointwise to zero on some closed and infinite-dimensional subspace $M_0$ of $X$, or
\item[(b)]\ $(T^{n_k})$ is equicontinuous along some non-increasing sequence $(M_k)$ of closed infinite-dimensional subspaces of $X$.
\end{enumerate}}
Section~\ref{Sec common} is devoted to common hypercyclic subspaces. We provide with Theorem~\ref{CritM0} a constructive proof of a Criterion $M_0$ due to Bayart \cite{Bayart} for the existence of common hypercyclic subspaces but on Fr\'{e}chet spaces that support a continuous norm, and use it to establish a corresponding Criterion $(M_k)$ which is simpler to apply (Theorem~\ref{Mk com}). Indeed, we use the latter to answer Question~\ref{Q:Bwchcs}
in the affirmative (Corollary~\ref{Ex:Costa}) and to partially anwser Question~\ref{Q:P(D)common}: {\em for any $N\ge 1$ and any $a_0(z)$,\dots , $a_{N-1}(z)\in H(\mathbb C )$, the family
\[
\{ \lambda L \}_{0\ne \lambda\in\mathbb C}
\]
of non-zero scalar multiples of the linear differential operator \[ L=D^n+a_{N-1}(z)D^{N-1}+\cdots +a_0(z)I\] has a common hypercyclic subspace on $H(\mathbb C )$} (Corollary~\ref{C:NonConstantCommon}). Finally, we seek in the last subsection spectral characterizations for the existence of common hypercyclic subspaces for certain families of operators, complementing the spectral characterization by Gonz\'{a}lez, Le\'{o}n and Montes \cite{Gonzalez} for the case of a single operator. In particular, we show that
{\em for a complex Banach space $X$ and $0< a<b\le \infty$, a family of non-zero scalar mutiples of a given $T\in L(X)$
\[
\{ \lambda T \}_{a<\lambda <b}
\]
that satisfies the Common Hypercyclicity Criterion has a common hypercyclic subspace if and only if the essential spectrum of $T$ contains an element of modulus no larger than $\frac{1}{b}$, with the convention that $\frac{1}{b}=0$ when $b=\infty$} (Corollary~\ref{C:a<lambda<b} and Corollary~\ref{C:Spectral}).
We finish this introduction with a brief subsection on the Common Hypercyclicity Criterion, which we use in Section~\ref{Sec common}.
\subsection{The Common Hypercyclicity Criterion} \label{Ss:CHC}
Several criteria have been used to show the existence of common hypercyclic vectors thanks to the works of Costakis and Sambarino \cite{Costakis2}, Bayart and Matheron \cite{Bayart2}, and others \cite{Bayart2004, BayartGrivaux, Bernal2009, Costakis, Gallardo, Grosse}. In this paper we use the following version of the Common Hypercyclicity Criterion based on \cite[Remark 11.10]{Grosse}.
\begin{definition}[{\bf Common Hypercyclicity Criterion}]\label{D:CHC}
Let $\Lambda \subseteq \mathbb{R}$ be an open interval. We say that a family $\{ (T_{n,\lambda})_{n\ge 0} \}_{\lambda\in \Lambda}$ of sequences in $L(X,Y)$ satisfies the \emph{Common Hypercyclicity Criterion} (CHC) provided
for each $(n, x)\in \mathbb{Z}^+\times X$ the vector $T_{n,\lambda}x$ depends continuously on $\lambda\in \Lambda$ and provided for each compact subset $K$ of $\Lambda$ there exist dense subsets $X_0$ and $Y_0$ of $X$ and $Y$, respectively, and mappings
\[
S_{n, \lambda}: Y_0\to X \ \ \ (\lambda\in K, \ n=0,1,\dots )
\]
so that for each $y_0\in Y_0$, each $x_0\in X_0$, we have
\begin{enumerate}
\item[(1)]\ $\sum_{k=0}^m T_{m, \lambda} S_{m-k, \mu_k} y_0$ converges unconditionally and uniformly on $\lambda \ge \mu_0\ge \mu_1\ge\dots \ge \mu_m$ in $K$ and $m\ge 0$.
\item[(2)]\ $\sum_{k=0}^\infty T_{m, \lambda} S_{m+k, \mu_k} y_0$ converges unconditionally and uniformly on $\lambda \le \mu_0\le \mu_1\le\dots $ in $K$ and $m\ge 0$.
\item[(3)]\ For each $\varepsilon>0$ and each continuous seminorm $q$ on $Y$, there exists a sequence $(\delta_{k})$ of positive real numbers such that $\sum_{k=0}^{\infty}\delta_k=\infty$ and for each $k\ge 0$ and $\lambda, \mu\in K$ we have
\[
0\le \mu-\lambda < \delta_{k} \ \ \Rightarrow \ \ \ q( T_{k, \lambda} S_{k,\mu} y_0 - y_0) < \epsilon.
\]
\item[(4)]\ $T_{k, \lambda} x_0 \xrightarrow[k\to\infty]{} 0$\ uniformly on $\lambda \in K$.
\item[(5)]\ $\sum_{k=0}^\infty \ S_{k, \mu_k} y_0$ converges unconditionally and uniformly on $\mu_0\le \mu_1\le \dots$ in $K$.
\end{enumerate}
\end{definition}
\begin{theorem}
\label{T:CHC}
Let $\{ (T_{n,\lambda})_n \}_{\lambda\in \Lambda}$ be a family of sequences in $L(X, Y)$ satisfying (CHC). Then the set of vectors in X that are hypercyclic for every sequence $(T_{n,\lambda})_n$ is residual in $X$.
\end{theorem}
Theorem~\ref{T:CHC} (follows from the same arguments and) slightly generalizes the Common Hypercyclicity Criterion given in \cite[Theorem~11.9, Remark 11.10(d)]{Grosse}.
\begin{remark} \label{R:CHC}\
\begin{enumerate}
\item[(a)]\ If $\{ T_{\lambda} \}_{\lambda \in \Lambda}$ is a family of operators satisfying (the assumptions of) the Common Hypercyclicity Criterion of Costakis and Sambarino~\cite{Costakis2}, then the family $\{ (T_{\lambda}^{n+1})_{n\ge 0} \}_{\lambda\in\Lambda}$ satisfies $(CHC)$ as defined above in Definition~\ref{D:CHC}.
\item[(b)]\ Given a scalar $\lambda_0\ge 0$ and $T\in L(X)$, the family $\{ (\lambda^n T^n)_{n\ge 1} \}_{\lambda >\lambda_0}$ satisfies $(CHC)$ if there exists a dense subset $A$ of $X$ that is contained in $\cup_{n\ge 1} \mbox{Ker}(T^n)$ and maps $S_n:A\to X$ so that for each $x\in A$ we have that $(i)$ $T^nS_nx=x$, that $(ii)$ $T^mS_{m+n}x=S_nx$ for each $m,n\ge 0$, and that $(iii)$ $\{\frac{1}{\lambda^n} S_nx \}_{n\ge 1}$ is bounded in $X$ for each $\lambda >\lambda_0$, see \cite[Section 11.2]{Grosse}.
\end{enumerate}
\end{remark}
\section{Main tool for the generalization of Criterion $(M_k)$}\label{Gene Mk}
In this section we develop a link between Criterion $M_0$ and Criterion $(M_k)$. Notice that Criterion $M_0$ immediately gives Criterion $(M_k)$, by the Banach-Steinhaus theorem. Conversely, if a sequence of operators $(T_n)$ satisfies Criterion $(M_k)$ along a given sequence of integers $(n_k)$, then it also satisfies Criterion $M_0$ for some subsequence $(m_k)$ of $(n_k)$ \cite{Menet1}. The section's main result, Theorem~\ref{NiceMn}, allows us to get this subsequence $(m_k)$ to inherit special properties of the original sequence $(n_k)$ (see Lemma~\ref{L:52} and Proposition~\ref{prophered}) which is a key ingredient for establishing $(M_k)$-type criteria for the existence of $\mathcal{U}$-frequently hypercyclic subspaces or common hypercyclic subspaces from the corresponding $M_0$-type criteria.
We need the following two definitions.
\begin{definition}
We say that $(\Lambda_n)_{n\ge 1}$ is a \emph{chain} of a given set $\Lambda$ if
the sequence $(\Lambda_n)_{n\ge 1}$ is non-decreasing and $\bigcup_{n\ge 1}\Lambda_n=\Lambda$.
\end{definition}
\begin{definition} Let $(M_n)$ be a sequence of infinite-dimensional closed subspaces of $X$ with $M_n\supseteq M_{n+1}$ for all $n$.
A family $\{ (T_{n, \lambda} ) \}_{\lambda\in \Lambda}$ of sequences in $L(X, Y)$ is said to be {\em uniformly equicontinuous along $(M_n)$} provided for each continuous seminorm $q$ on $Y$ there exists a continuous seminorm $p$ of $X$ so that for each $n\in\mathbb{N}$
\[
\mbox{sup}_{\lambda\in\Lambda} q(T_{n, \lambda}x) \le p(x) \ \ \mbox{ for each $x\in M_n$.}
\]
\end{definition}
\begin{theorem}\label{NiceMn} Let $X$ be an infinite-dimensional Fr\'{e}chet space with continuous norm, let $Y$ be a separable Fr\'echet space and let $\Lambda$ be a set.
Let $\{ (T_{k, \lambda})_{k\ge 1} \}_{\lambda\in\Lambda}$ be a family of sequences of operators in $L(X,Y)$. Suppose that there exist chains $(\Lambda^j_n)_{n\ge 1}$ of $\Lambda$ $(j=0,1,2)$ satisfying:
\begin{enumerate}
\item[(i)]\ for each $n\in\mathbb{N}$ and each $k\in \mathbb N$, the family
$
\{ T_{k, \lambda} \}_{\lambda\in \Lambda_n^0} \ \mbox{ is equicontinuous;}
$
\item[(ii)]\ for each $n\in\mathbb{N}$, there exists a dense subset $X_{n,0}$ of $X$ such that for any $x\in X_{n,0}$,
\[
T_{k,\lambda}x\xrightarrow[k\to\infty]{} 0 \ \ \ \mbox{uniformly on $\lambda\in \Lambda_n^1$;}
\]
\item[(iii)]\ there exists a non-increasing sequence of infinite-dimensional closed subspaces $(M_k)$ of $X$ such that for each $n\ge 1$, the family of sequences
\[
\{ (T_{k,\lambda})_{k\ge 1}\}_{\lambda\in \Lambda_n^2}
\]
is uniformly equicontinuous along $(M_k)$.
\end{enumerate}
Then for any map $\phi:\mathbb{N}\to \mathbb{N}$ there exist an increasing sequence of integers $(k_s)_{s\ge 1}$ and an infinite-dimensional closed subspace $M_0$ of $X$ such that for any $(x,\lambda)\in M_0\times \Lambda$,
\[T_{k,\lambda}x\xrightarrow[k\to \infty]{k\in I} 0,\]
where $I=\bigcup_{s\ge 1}[k_s,k_s+\phi(k_s)]$.
\end{theorem}
The choice of the function $\phi:\mathbb N\to \mathbb N$ in Theorem~\ref{NiceMn} is the main tool to generalize Criterion~$(M_k)$ to $\mathcal{U}$-frequently hypercyclic subspaces (Section~\ref{SecU}) and to common hypercyclic subspaces (Section~\ref{Sec common}).
The proof of Theorem~\ref{NiceMn} relies on the notion of a basic sequence, which has been commonly used for the construction of closed, infinite-dimensional subspaces whose vectors are to satisfy special properties. We refer to \cite{Menet1, Petersson} for more details about the construction of basic sequences in Fr\'{e}chet spaces with a continuous norm.
\begin{definition}
A sequence $(u_k)_{k\ge 1}$ in a Fr\'{e}chet space is called \emph{basic} if for every $x\in \overline{\text{span}}\{u_k:k\ge 1\}$, there exists a unique sequence $(a_k)_{k\ge 1}$ in $\mathbb{K}$ ($\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$) such that $x=\sum_{k=1}^{\infty}a_ku_k$.
\end{definition}
\begin{proof}[Proof of Theorem \ref{NiceMn}]\ Let $(p_j)$ be an increasing sequence of norms inducing the topology of $X$ and let $(q_j)$ be an increasing sequence of seminorms inducing the topology of $Y$.
We consider $\Lambda_n=\Lambda^0_n\cap \Lambda^1_n\cap \Lambda^2_n$ and a basic sequence $(u_i)_{i\ge1}$ in $X$ such that $u_i\in M_i$ and $p_1(u_i)=1$ for each $i\ge 1$, and so that for every $j\ge 1$, the sequence $(u_i)_{i\ge j}$ is basic in $X_j:=(X,p_j)$ with basic constant less than $2$.
By $(i)$, for each $n, k, j\ge 1$ , there exists $K_{n,k,j}>0$ and $m^0(n,k,j)\in\mathbb{N}$ such that for any $x\in X$,
\begin{equation}
\sup_{\lambda\in \Lambda_n}q_j(T_{k,\lambda}x)\le K_{n,k,j} \ p_{m^0(n,k,j)}(x).
\label{eqi)}
\end{equation}
By $(iii)$, for each $n,j\ge 1$ there exist $C_{n,j}>0$ and $m(n,j)\in\mathbb{N}$
such that for any $k\ge 1$ and any $x\in M_k$,
\begin{equation}
\sup_{\lambda\in \Lambda_n}q_j(T_{k,\lambda}x)\le C_{n,j} p_{m(n,j)}(x).
\label{eqiii)}
\end{equation}
Since each set $X_{i,0}$ is dense, we can construct a family $(x_{i,l})_{i,l\ge 1}\subset X$ and a sequence of integers $(k_l)_{l\ge 0}$ with $k_0=1$ such that for any $l\ge 1$,
\begin{enumerate}
\item for any $i\ge 1$, any ${n,k,j\le i}$,
\begin{equation}
p_{m^0(n,k,j)}(x_{i,l})<\frac{1}{2^{i+l}K_{n,k,j}} \quad \text{and} \quad p_i(x_{i,l})<\frac{1}{2^{i+l+2}};
\label{3prop f}
\end{equation}
\item for any $i,k\le k_{l-1}+\phi(k_{l-1})$,
\begin{equation}
\sup_{\lambda\in \Lambda_{k_{l-1}+\phi(k_{l-1})}}q_{k_{l-1}+\phi(k_{l-1})}(T_{k,\lambda}x_{i,l})<\frac{1}{2^{i+l}};
\label{3prop f2}
\end{equation}
\item for any $i\ge 1$, $u_i+\sum_{l'=1}^{l}x_{i,l'}\in X_{k_{l-1}+\phi(k_{l-1}),0}$;
\item $k_{l}> k_{l-1}+\phi(k_{l-1})$;
\item for any $i\le k_{l-1}+\phi(k_{l-1})$, any $k\in [k_l,k_l+\phi(k_l)]$,
\begin{equation}
\sup_{\lambda\in \Lambda_{k_{l-1}+\phi(k_{l-1})}}q_{k_{l-1}+\phi(k_{l-1})}\Big(T_{k,\lambda} \Big(u_i+\sum_{l'=1}^{l}x_{i,l'}\Big)\Big)<\frac{1}{2^{l+i}}.
\label{3prop k}
\end{equation}
\end{enumerate}
Indeed, in order to satisfy (1) and (2), it suffices to choose $x_{i,l}$ sufficiently small thanks to \eqref{eqi)} and since we choose $x_{i,l}$ such that $u_i+\sum_{l'=1}^{l}x_{i,l'}\in X_{k_{l-1}+\phi(k_{l-1}),0}$, we know that for any $i\ge 1$,
$T_{k,\lambda}\Big(u_i+\sum_{l'=1}^{l}x_{i,l'}\Big)$ tends uniformly to $0$ on $\lambda\in \Lambda_{k_{l-1}+\phi(k_{l-1})}$. We can thus find $k_l$ sufficiently big such that (4) and (5) are satisfied.
For any $n\ge 1$, we let $x_n=u_n+\sum_{l=1}^{\infty}x_{n,l}$. We deduce from \eqref{3prop f} that $(x_{k_l+\phi(k_l)})_{l\ge 1}$ is a basic sequence equivalent to the sequence $(u_{k_l+\phi(k_l)})_{l\ge 1}$. Let $M_0$ be the closed linear span of $(x_{k_l+\phi(k_l)})_{l\ge 1}$ in $X$ and $x$ a vector in $M_0$. We know that we have $x=\sum_{s=1}^{\infty} a_s x_{k_s+\phi(k_s)}$ where the sequence $(a_s)_{s\ge 1}$ is bounded by some constant $K$.
Let $n,j\ge 1$ and $l\ge 2$ with $n,j\le k_{l-1}+\phi(k_{l-1})$. Since
\begin{equation*}
{\sum_{s=l}^{\infty}a_s u_{k_s+\phi(k_s)}\in M_{k_l+\phi(k_l)}},
\end{equation*}
we deduce from \eqref{eqi)}, \eqref{eqiii)}, \eqref{3prop f}, \eqref{3prop f2} and \eqref{3prop k} that for any $\lambda\in \Lambda_n$, any $k\in [k_l,k_l+\phi(k_l)]$,
\begin{align*}
q_j(T_{{k},\lambda}x)&\le \sum_{s=1}^{l-1}|a_s| q_j(T_{{k},\lambda}x_{k_s+\phi(k_s)})
+ \sum_{s=l}^{\infty} |a_s| q_j(T_{{k},\lambda}(x_{k_s+\phi(k_s)}-u_{k_s+\phi(k_s)}))\\
&\quad+ q_j\Big(T_{{k},\lambda}\Big(\sum_{s=l}^{\infty}a_su_{k_s+\phi(k_s)}\Big)\Big)\\
&\le \sum_{s=1}^{l-1}|a_s| q_{k_{l-1}+\phi(k_{l-1})}\Big(T_{{k},\lambda}\Big(u_{k_s+\phi(k_s)}+\sum_{l'=1}^{l}x_{k_s+\phi(k_s),l'}\Big)\Big)\\
&\quad+ \sum_{s=1}^{l-1}\sum_{l'=l+1}^{\infty}|a_s| q_{k_{l'-1}+\phi(k_{l'-1})}(T_{{k},\lambda}x_{k_s+\phi(k_s),l'})\\
&\quad+ \sum_{s=l}^{\infty} \sum_{l'=1}^{\infty}|a_s| q_j(T_{{k},\lambda}x_{k_s+\phi(k_s),l'})+ q_j\Big(T_{{k},\lambda}\Big(\sum_{s=l}^{\infty}a_su_{k_s+\phi(k_s)}\Big)\Big)\\
&\le \sum_{s=1}^{l-1} \frac{K}{2^{l+k_s+\phi(k_s)}}+\sum_{s=1}^{l-1} \sum_{l'=l+1}^{\infty}\frac{K}{2^{k_s+\phi(k_s)+l'}}\\
&\quad +K \sum_{s=l}^{\infty} \sum_{l'=1}^{\infty} K_{n,k,j} p_{m^0(n,k,j)}(x_{k_s+\phi(k_s),l'}) + C_{n,j}p_{m(n,j)}\Big(\sum_{s=l}^{\infty}a_s u_{k_s+\phi(k_s)}\Big)\\
&\le \frac{lK}{2^{l-1}}
+ K\sum_{s=l}^{\infty} \frac{K_{n,k,j}}{2^{k_s+\phi(k_s)}K_{n,k,j}}+ C_{n,j}p_{m(n,j)}\Big(\sum_{s=l}^{\infty}a_s u_{k_s+\phi(k_s)}\Big)\\
&\le \frac{lK}{2^{l-1}}
+ K\sum_{s=l}^{\infty} \frac{1}{2^{k_s+\phi(k_s)}} + C_{n,j}p_{m(n,j)}\Big(\sum_{s=l}^{\infty}a_s u_{k_s+\phi(k_s)}\Big)
\underset{l\rightarrow \infty}{\longrightarrow} 0.
\end{align*}
\end{proof}
We finish this section by stating two particular cases of Theorem~\ref{NiceMn}. In Corollary~\ref{NiceMnBan} below we consider the case when $\{ T_{\lambda}\}_{\lambda\in \Lambda}$ is a family of operators on a Banach space $X$. We use Corollary~\ref{NiceMnBan} in Section~\ref{Sec:Charac} to study the existence of common hypercyclic subspaces for collections of scalar multiples of a fixed operator on a complex Banach space.
\begin{cor}\label{NiceMnBan}
Let $(X,\|\cdot\|)$ be a separable infinite-dimensional Banach space, let $\{ T_{\lambda}\}_{\lambda\in \Lambda}$ be a family of operators in $L(X)$, and let $(n_k)_{k\ge 1}$ be a strictly increasing sequence of positive integers.
Suppose there exist chains $(\Lambda^1_n)$ and $(\Lambda^2_n)$ of $\Lambda$ satisfying:
\begin{enumerate}
\item[(i)]\ for any $n\ge 1$, there exists a dense subset $X_{n,0}$ of $X$ such that for any $x\in X_{n,0}$, \[\sup_{\lambda\in \Lambda^1_n} \|T^{n_k}_{\lambda}x\|\xrightarrow[k\to \infty]{} 0;\]
\item[(ii)]\ there exists a non-increasing sequence $(M_k)$ of infinite-dimensional closed subspaces of $X$ such that for any $n\in\mathbb{N}$,
\[\sup_{(\lambda, k)\in \Lambda^2_n\times \mathbb{N}}\|T^{n_k}_{\lambda|M_k}\|<\infty.\]
\end{enumerate}
Then for any $\phi:\mathbb{N}\to \mathbb{N}$, there exist an increasing sequence of integers $(k_s)_{s\ge 1}$ and an infinite-dimensional closed subspace $M_0$ of $X$ such that for any $x\in M_0$ and any $\lambda\in \Lambda$,
\[T^{n_k}_{\lambda}x\xrightarrow[k\to \infty]{k\in I} 0\]
where $I=\bigcup_{s\ge 1}[k_s,k_s+\phi(k_s)]$.
\end{cor}
\begin{proof}
In view of Theorem~\ref{NiceMn}, it suffices to prove the existence of a chain $(\Lambda^0_n)$ of $\Lambda$ such that for any $n,k\ge 1$, there exists $K_{n,k}$ such that for any $x\in X$,
\[\sup_{\lambda\in \Lambda^0_n}\|T^{n_k}_{\lambda}x\|\le K_{n,k}\|x\|.\]
For each $n\in\mathbb{N}$, let $\Lambda^0_n=\{\lambda\in \Lambda:\|T_\lambda\|\le n\}$. So $(\Lambda^0_n)$ is a chain of $\Lambda$ and for any $n,k\in\mathbb{N}$ and $x\in X$, we have
\[\sup_{\lambda\in \Lambda^0_n}\|T^{n_k}_{\lambda}x\|
\le \sup_{\lambda\in \Lambda^0_n}\|T^{n_k}_{\lambda}\| \|x\|
\le \sup_{\lambda\in \Lambda^0_n}\|T_{\lambda}\|^{n_k} \|x\|
\le n^{n_k}\|x\|.\]
We conclude that the chain $(\Lambda^0_n)$ satisfied the desired inequalities for $K_{n,k}=n^{n_k}$.
\end{proof}
Finally, for the case of a single sequence of operators $(T_n)$ in $L(X,Y)$, Theorem~\ref{NiceMn} gives the following.
\begin{cor}\label{NiceMnone}
Let $X$ be an infinite-dimensional Fr\'{e}chet space with a continuous norm, let $Y$ be a separable Fr\'{e}chet space and let $(T_n)$ be a sequence in $L(X,Y)$. Suppose there exists a strictly increasing sequence of integers $(n_k)$ satisfying
\begin{enumerate}
\item[(a)]\ $T_{n_k}x\underset{k\to \infty}{\to} 0$ for each $x$ in some dense subset $X_0$ of $X$, and
\item[(b)]\ $(T_{n_k})_{k\ge 1}$ is equicontinuous along some non-increasing sequence $(M_k)$ of infinite-dimensional closed subspaces of $X$.
\end{enumerate}
Then for any $\phi:\mathbb{N}\to \mathbb{N}$ there exist an increasing sequence of integers $(k_s)_{s\ge 1}$ and an infinite-dimensional closed subspace $M_0$ such that if $I=\bigcup_{s\ge 1}[k_s,k_s+\phi(k_s)]$, then for any $x\in M_0$,
\[T_{n_k}x\xrightarrow[k\to \infty]{k\in I} 0.\]
\end{cor}
We use Corollary~\ref{NiceMnone} in Section~\ref{SecU} to obtain a version of Criterion~$(M_k)$ for the existence of $\mathcal{U}$-frequently hypercyclic subspaces.
\section{Existence of $\mathcal{U}$-frequently hypercyclic subspaces}\label{SecU}
The existence of frequently hypercyclic subspaces was first investigated by Bonilla and Grosse-Erdmann~\cite{Bonilla2}, who generalized Criterion~$M_0$ to one for obtaining frequently hypercyclic subspaces, which has in turn been applied to convolution operators and weighted composition operators, see \cite{Bonilla2}, \cite{Bes}, \cite{Bes2}.
We first recall the Frequent Hypercyclicity Criterion due to Bayart and Grivaux~\cite{Bayart04}; the version we use here is due to Bonilla and Grosse-Erdmann~\cite{Bonilla3}.
\begin{theorem}[{\bf Frequent Universality Criterion}] \label{T:FHC}
Let $X$ be a Fr\'{e}chet space, let $Y$ be a separable Fr\'{e}chet space and let $(T_n)$ be a sequence in $L(X,Y)$. If there exist a dense subset $Y_0\subset Y$ and $S_k:Y_0\rightarrow X$, $k\ge 0$ such that for each $y\in Y_0$,
\begin{enumerate}[\upshape 1.]
\item $\sum_{n=0}^{\infty}S_ny$ converges unconditionally in $X$,
\item $\sum_{n=1}^{k}T_{k}S_{k-n}y$ converges unconditionally in $Y$, uniformly in $k\ge 0$,
\item $\sum_{n=1}^{\infty}T_{k}S_{k+n}y$ converges unconditionally in $Y$, uniformly in $k\ge 0$,
\item $T_nS_ny\rightarrow y$,
\end{enumerate}
then $(T_n)$ is frequently hypercyclic.
\end{theorem}
We can now state (a slightly more general formulation of) the Bonilla and Grosse-Erdmann's criterion for the existence of frequently hypercyclic subspaces.
\begin{theorem} {\bf (Bonilla and Grosse-Erdmann)} \label{T:FHCS}
Let $X$ be a Fr\'{e}chet space with a continuous norm and let $Y$ be a separable Fr\'{e}chet space.
Let $(T_n)$ be a sequence in $L(X, Y)$ satisfying the Frequent Universality Criterion. Suppose that
\[
T_nx\xrightarrow[n\to\infty]{} 0
\]
for each vector $x$ in some closed, infinite-dimensional subspace $M_0$ of $X$. Then $(T_n)$ has a frequently hypercyclic subspace.
\end{theorem}
Under the assumption that $(T_n)$ satisfies the Frequent Universality Criterion, their proof of the above theorem (see \cite[Theorem 3]{Bonilla2}) can then be adapted to $\mathcal{U}$-frequently hypercyclic subspaces as follows:
\begin{theorem}[{\bf Criterion~$M_0$ for $\mathcal{U}$-Frequently Hypercyclic Subspaces}]\label{UM0}
Let $X$ be a Fr\'{e}chet space with a continuous norm and let $Y$ be a separable Fr\'{e}chet space. Let $(T_n)$ be a sequence in $L(X,Y)$ satisfying the Frequent Universality Criterion and for which there exist an infinite-dimensional closed subspace $M_0$ and a strictly increasing sequence $(n_k)_{k\ge 1}$ of positive upper density such that for any $x\in M_0$,
\[T_{n_k}x\xrightarrow[k\to \infty]{} 0.\]
Then $(T_n)$ has a $\mathcal{U}$-frequently hypercyclic subspace.
\end{theorem}
\begin{proof}
By applying Theorem~\ref{T:FHCS} to the sequence of operators $(T_{n_k})_{k\ge 1}$, we obtain an infinite-dimensional closed subspace $M$ such that for any non-zero vector $x\in M$ and any non-empty open set $U$ in $Y$ the return set $\{k\ge 1:T_{n_k} x\in U\}$ is a set of positive lower density. Since $(n_k)_{k\ge 1}$ is a set of positive upper density, the set $(n_k)_{k\in A}$ is a set of positive upper density for any set $A$ of positive lower density. Indeed, we have
\[\frac{\#(\{n_k:k\in A\}\cap[0,N])}{N+1}=\frac{\#(A\cap[1,j])}{j} \frac{\#(\{n_k:k\in \mathbb N\}\cap[0,N])}{N+1},\]
where $j=\# (\{n_k:k\in \mathbb N\}\cap[0,N])$. We conclude that $M$ is a $\mathcal{U}$-frequently hypercyclic subspace.
\end{proof}
We next derive the following $(M_k)$-criterion for the existence of $\mathcal{U}$-frequently hypercyclic subspaces, which is simpler to apply than Theorem~\ref{UM0}.
\begin{theorem}[{\bf Criterion~${\bf (M_k)}$ for $\mathcal{U}$-Frequently Hypercyclic Subspaces}]\label{UMk}
Let $X$ be an infinite-dimensional Fr\'{e}chet space with a continuous norm, let $Y$ be a separable Fr\'{e}chet space and let $(T_n)$ be a sequence in $L(X,Y)$ satisfying the Frequent Universality Criterion.
Suppose that there exists a strictly increasing sequence $(n_k)$ of positive upper density so that
\begin{enumerate}
\item \ $T_{n_k}x\underset{k\to\infty}\to 0$ for each $x$ in some dense subset $X_0$ of $X$, and
\item \ $(T_{n_k})_{k\ge 1}$ is equicontinuous along some non-increasing sequence $(M_k)$ of infinite-dimensional closed subspaces of $X$.
\end{enumerate}
Then $(T_n)$ has a $\mathcal{U}$-frequently hypercyclic subspace.
\end{theorem}
We will prove Theorem~\ref{UMk} with Theorem~\ref{UM0} and by applying Corollary~\ref{NiceMnone} with a suitable map $\phi$ whose existence is given by the following lemma.
\begin{lemma} \label{L:52} Let $(n_k)$ be a strictly increasing sequence in $\mathbb{N}$
with positive upper density.
Then there exists a map $\phi:\mathbb{N}\to \mathbb{N}$ so that for every strictly increasing sequence $(k_s)$ in $\mathbb{N}$ we have
\[
\overline{\text{\emph{dens}}}(\{ n_k:\ k\in \cup_{s\ge 1} [k_s, k_s+\phi (k_s)] \}) = \overline{\text{\emph{dens}}}(n_k).
\]
\end{lemma}
\begin{proof} Let $\delta = \overline{\text{dens}}(n_k)$, and choose $\phi:\mathbb{N}\to \mathbb{N}$ such that
for any $k\ge 1$,
\[\frac{\phi(k)+1}{n_{k+\phi(k)}}\ge\delta-\frac{\delta}{k}.\]
Such a map $\phi$ exists because for any $k\ge 1$,
\[\delta=\limsup_{N}\frac{\#\{j: n_j\in [n_k,N]\}}{N}=\limsup_{l}\frac{\#\{j: n_j\in [n_k,n_l]\}}{n_l}=\limsup_{l}\frac{l+1-k}{n_l}\]
and thus for any $k\ge 1$ there exists $\phi(k)$ such that
\[\frac{\phi(k)+1}{n_{k+\phi(k)}}\ge\delta-\frac{\delta}{k}.\]
The upper density of the subsequence $(m_l)=\{n_k: k\in\bigcup_{s\ge 1}[k_s,k_s+\phi(k_s)]\}$
of $(n_k)$ equals~{$\delta$.} Indeed, $\overline{\text{dens}}(m_l)\le \overline{\text{dens}}(n_k)=\delta$
since $(m_l)$ is a subsequence of $(n_k)$, and on the other hand
\begin{align*}
\overline{\text{dens}}(m_l)&=\limsup_{N}\frac{\#\{j: m_j\in [0,N]\}}{N}\\
&\ge \limsup_{s}\frac{\#\{j: m_j\in [n_{k_s},n_{k_s+\phi(k_s)}]\}}{n_{k_s+\phi(k_s)}}\\
&= \limsup_{s}\frac{\phi(k_s)+1}{n_{k_s+\phi(k_s)}}\ge \limsup_{s}\Big(\delta-\frac{\delta}{k_s}\Big)=\delta.
\end{align*}
So Lemma~\ref{L:52} follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{UMk}]\ Let $\phi$ as in Lemma~\ref{L:52}. By
Corollary~\ref{NiceMnone}, there exists a closed infinite-dimensional subspace $M_0$ of $X$ and a strictly increasing sequence $(k_s)$ in $\mathbb{N}$ such that for any $x\in M_0$,
\[
T_{n_k}x\xrightarrow[k\to\infty]{k\in I} 0
\]
where $I=\cup_{s\ge 0} [k_s, k_s+\phi (k_s)]$. Since $(n_k)$ is a sequence of positive upper density, the subsequence $(m_l)=\{n_k:k\in I\}$ has positive upper density (Lemma~\ref{L:52}) and the conclusion now follows from Theorem~\ref{UM0}.
\end{proof}
If we consider an operator $T\in L(X)$ and its iterates, we can consider the following version of the Frequent Universality Criterion.
\begin{theorem}[{\bf Frequent Hypercyclicity Criterion}]
Let $X$ be a separable Fr\'{e}chet space and $T\in L(X)$. If there are a dense subset $X_0$\ of $X$ and $S_k:X_0\rightarrow X$, $k\ge 1$, such that for each $x\in X_0$,
\begin{enumerate}[\upshape 1.]
\item $\sum_{n=1}^{\infty}T^nx$ converges unconditionally,
\item $\sum_{n=1}^{\infty}S_nx$ converges unconditionally,
\item $T^mS_nx=S_{n-m}x$ for any $m< n$, and
\item $T^nS_nx=x$ for any $n\ge 1$.
\end{enumerate}
Then $T$ is frequently hypercyclic.
\end{theorem}
We remark that if $T$ satisfies the Frequent Hypercyclicity Criterion, then its sequence of iterates $(T^n)$ satisfies the Frequent Universality Criterion and there exists a dense subset $X_0$ of $X$ such that $T^{n}x \to 0$ for each $x\in X_0$. Therefore, we obtain the following version of the Criterion~$(M_k)$ for $\mathcal{U}$-frequently hypercyclic subspaces.
\begin{theorem}\label{UMkT}
Let $X$ be a separable infinite-dimensional Fr\'{e}chet space with a continuous norm and $T\in L(X)$ an operator satisfying the Frequent Hypercyclicity Criterion.
Suppose that there exists a strictly increasing sequence of positive integers $(n_k)$ of positive upper density such that
$(T^{n_k})_{k\ge 1}$ is equicontinuous along some non-increasing sequence $(M_k)$ of infinite-dimensional closed subspaces of $X$. Then $T$ possesses a $\mathcal{U}$-frequently hypercyclic subspace.
\end{theorem}
In 2000, Gonz\'{a}lez, Le\'{o}n and Montes~\cite{Gonzalez} obtained a characterization of operators with hypercyclic subspaces on complex Banach spaces. In particular, we can deduce from their proof that if $T$ is an operator on a complex Banach space satisfying the Hypercyclicity Criterion, then $T$ possesses a hypercyclic subspace if and only if $(T^k)_{k\ge 0}$ is equicontinuous along some non-increasing sequence $(M_k)$ of infinite-dimensional closed subspaces of $X$, see \cite[Theorem 3.2]{Gonzalez}. In view of Theorem~\ref{UMkT} and of the characterization of Gonz\'{a}lez, Le\'{o}n and Montes, we conclude the following.
\begin{theorem} \label{T:FHC2ufhcs}
Let $X$ be a separable infinite-dimensional complex Banach space and $T\in L(X)$ be an operator satisfying the Frequent Hypercyclicity Criterion. Then $T$ has a hypercyclic subspace if and only if it has a $\mathcal{U}$-frequently hypercyclic subspace.
\end{theorem}
Theorem~\ref{UMkT} also gives the following characterization of unilateral weighted shifts with $\mathcal{U}$-frequently hypercyclic subspaces on real or complex Banach spaces $\ell^p$ ($1\le p <\infty$).
\begin{cor}\label{caraclp}
Let $B_w:\ell^p\to \ell^p$ be a $\mathcal{U}$-frequently hypercyclic unilateral weighted shift.
Then $B_w$ has a $\mathcal{U}$-frequently hypercyclic subspace if and only if
$B_w$ has a hypercyclic subspace. Thus a unilateral backward shift $B_w$ with weight sequence $w=(w_n)$ has a $\mathcal{U}$-frequently hypercyclic subspace if and only if
$(\frac{1}{w_1\cdots w_n})\in\ell^p$ and $\sup_{n\ge 1} \inf_{k\ge 0} \prod_{\nu=1}^{n} |w_{k+\nu}|\le 1$.
\end{cor}
\begin{proof
It is immediate that $B_w$ has a hypercyclic subspace if it has a $\mathcal{U}$-frequently hypercyclic subspace. Conversely, if $B_w$ has a hypercyclic subspace we can deduce from \cite{LeonMontes, Menet1} that there exists a non-increasing sequence of infinite-dimensional closed subspaces $(M_k)$ in $X$ such that
\[\sup_{k\ge 1}\|B^k_{w|M_k}\|<\infty.\]
We also know thanks to Bayart and Ruzsa \cite{Bayart3} that if $B_w:\ell^p\to \ell^p$ is a $\mathcal{U}$-frequently hypercyclic weighted shift, then $B_w$ satisfies the Frequent Hypercyclicity Criterion. We therefore conclude by Theorem~\ref{UMkT} that $B_w$ has a $\mathcal{U}$-frequently hypercyclic subspace. The second assertion follows now from the facts that $B_w$ is $\mathcal{U}$-frequently hypercyclic on $\ell^p$ if and only if $(\frac{1}{w_1\cdots w_n})\in\ell^p$ \cite{Bayart3} and that $B_w$ has a hypercyclic subspace if and only if $\sup_{n\ge 1} \inf_{k\ge 0} \prod_{\nu=1}^{n} |w_{k+\nu}|\le 1$ \cite{LeonMontes, Menet1}.
\end{proof}
This result may seem surprising given that a weighted shift on $\ell^p$ is $\mathcal{U}$-frequently hypercyclic if and only if it is frequently hypercyclic \cite{Bayart3}, and there exist frequently hypercyclic weighted shifts with hypercyclic subspaces and no frequently hypercyclic subspace~\cite{Menet2}. In particular, we deduce the following.
\begin{cor} \label{C:ufhcsNofhcs}
There exists a frequently hypercyclic operator that has $\mathcal{U}$-frequently hypercyclic subspaces but has no frequently hypercyclic subspace.
\end{cor}
The above results motivate the following.
\begin{problem}: Does there exist a $\mathcal{U}$-frequently hypercyclic operator possessing hypercyclic subspaces but no $\mathcal{U}$-frequently hypercyclic subspace? What if the operator is frequently hypercyclic?
\end{problem}
We complement our study of the existence of $\mathcal{U}$-frequently hypercyclic subspaces for weighted shifts by showing that every $\mathcal{U}$-frequently hypercyclic bilateral weighted shift on $\ell^p(\mathbb Z)$ possesses a frequently hypercyclic subspace and thus a $\mathcal{U}$-frequently hypercyclic subspace.
\begin{theorem}\label{caraclpbi}
Let $B_w$ be a bilateral weighted shift on $\ell^p(\mathbb Z)$. If $B_w$ is $\mathcal{U}$-frequently hypercyclic, then $B_w$ possesses a frequently hypercyclic subspace.
\end{theorem}
\begin{proof}
Let $B_w$ be a $\mathcal{U}$-frequently hypercyclic bilateral weighted shift on $\ell^p(\mathbb Z)$. We already know that $B_w$ satisfies the Frequent Hypercyclicity Criterion~\cite{Bayart3}. In view of Theorem~\ref{UM0}, it thus suffices to prove that there exists an infinite-dimensional closed subspace $M_0$ such that for any $x\in M_0$
\[B_w^{n}x\xrightarrow[n\to \infty]{} 0.\]
Since $B_w$ is $\mathcal{U}$-frequently hypercyclic, we also know that
$\sum_{n\le 0}|w_{0}\cdots w_n|^p<\infty$ ~\cite{Bayart3}. In particular, we have for any $k\ge 1$
\begin{equation}\label{wsUb}
\prod_{\nu=0}^{n}|w_{-k-\nu}|=\frac{\prod_{\nu=0}^{k+n}|w_{-\nu}|}{\prod_{\nu=0}^{k-1}|w_{-\nu}|}\xrightarrow[n\to \infty]{} 0.
\end{equation}
Let $k_0\ge 1$. We show that there exists $k_1\ge k_0$ such that for any $n\ge 0$,
\begin{equation*}
\prod_{\nu=0}^{n}|w_{-k_1-\nu}|\le 1.
\end{equation*}
Indeed, either for any $n\ge 0$, we have $\prod_{\nu=0}^{n}|w_{-k_0-\nu}|\le 1$ and we can consider $k_1=k_0$, or the set $F:=\{n\ge 0:\prod_{\nu=0}^{n}|w_{-k_0-\nu}|>1\}$ is non-empty.
We then remark that if $n\in F$, we have
\[\prod_{\nu=0}^{k_0+n}|w_{-\nu}|=\Big(\prod_{\nu=0}^{n}|w_{-k_0-\nu}|\Big)\Big(\prod_{\nu=0}^{k_0-1}|w_{-\nu}|\Big)>\prod_{\nu=0}^{k_0-1}|w_{-\nu}|.\]
We deduce from \eqref{wsUb} that $F$ has to be finite. Let $n_0:=\max F$ and $k_1=k_0+n_0+1$. We then have $k_1\ge k_0$ and for any $n\ge 0$,
\[\prod_{\nu=0}^{n}|w_{-k_1-\nu}|=\frac{\prod_{\nu=0}^{n_0+n+1}|w_{-k_0-\nu}|}{\prod_{\nu=0}^{n_0}|w_{-k_0-\nu}|}\le 1.\]
We conclude that there exists an increasing sequence of positive integers $(k_j)_{j\ge 0}$ such that for any $j\ge 0$, any $n\ge 0$,
\begin{equation}\label{wsUb2}
\prod_{\nu=0}^{n}|w_{-k_j-\nu}|\le 1.
\end{equation}
Let $M_0:=\overline{\text{span}}\{e_{-k_j}:j\ge 0\}$ and $x\in M_0$. We have $x=\sum_{j=0}^{\infty}a_je_{-k_j}$ and for any $n\ge 1$, any $J\ge 0$, we deduce from \eqref{wsUb} and \eqref{wsUb2} that
\begin{align*}
\|B_w^nx\|^p&=\sum_{j=0}^{\infty}\Big(\prod_{\nu=0}^{n-1}|w_{-k_j-\nu}|\Big)^p |a_j|^p
\\
&\le \sum_{j=0}^J\Big(\prod_{\nu=0}^{n-1}|w_{-k_j-\nu}|\Big)^p |a_j|^p+ \sum_{j=J+1}^{\infty}|a_j|^p
\xrightarrow[n\to \infty]{}\sum_{j=J+1}^{\infty}|a_j|^p.
\end{align*}
We thus have $\|B_w^n x\|\to 0$ for any $x\in M_0$ and we conclude by using Theorem~\ref{UM0}.
\end{proof}
We finish this section by investigating the operators $P(D)$ on the space $H(\mathbb C)$ of entire functions where $P$ is a non-constant polynomial and $D$ is the derivative operator.
\begin{cor}\label{C:P(D)ufhs}
Let $D:H(\mathbb C)\to H(\mathbb C)$ be the derivative operator and $P$ a non-constant polynomial.
Then $P(D)$ has a $\mathcal{U}$-frequently hypercyclic subspace.
\end{cor}
\begin{proof
We know that $P(D)$ satisfies the Frequently Hypercyclicity Criterion~\cite{Bonilla}. It is also shown in~\cite{Menet1} that if $P(D)$ satisfies the Hypercyclicity Criterion along a given strictly increasing sequence $(n_k)$, then there exists
a non-increasing sequence of infinite-dimensional closed subspaces $(M_k)$ in $H(\mathbb C)$ such that for any $j\ge 1$, there exist a positive number $C_{j}$ and two integers $m(j),k(j)$
such that for any $k\ge k(j)$, any $x\in M_k$,
\[p_j(P(D)^{n_k}x)\le C_{j} p_{m(j)}(x).\]
In other words, it is shown that $(P(D)^{n_k})$ is equicontinuous along $(M_k)$.
Since $P(D)$ satisfies the Frequent Hypercyclicity Criterion, $P(D)$ satisfies the Hypercyclicity Criterion along the whole sequence $(k)$ and we can conclude by applying Theorem~\ref{UMkT} along the whole sequence $(n_k)=(k)$.
\end{proof}
Thanks to the following result by Delsarte and Lions, Corollary~\ref{C:P(D)ufhs} extends
to
linear differential operators of finite order whose coefficients -except the leading one- may be non-constant.
\begin{lemma} {\rm (Delsarte and Lions \cite{Delsarte})} \label{L:Delsarte}
Let $T:H(\mathbb{C})\to H(\mathbb{C})$ be a differential operator of the form $T=D^N+a_{N-1}(z) D^{N-1}+\cdots + a_0(z) I$, where $N\ge 1$ and $a_j\in H(\mathbb{C})$ for $1\le j\le N$. Then there exists an onto isomorphism $U:H(\mathbb{C})\to H(\mathbb{C})$ so that $UT=D^NU$.
\end{lemma}
\begin{cor} \label{C:ufhsNonConstant}
For each $N\ge 1$ and $a_0,\dots, a_{N-1}\in H(\mathbb{C})$, the differential operator
\[
T=D^N+a_{N-1}(z) D^{N-1}+\cdots + a_0(z) I: H(\mathbb{C})\to H(\mathbb{C})
\]
has an $\mathcal{U}$-frequently hypercyclic subspace.
\end{cor}
\section{Existence of common hypercyclic subspaces}\label{Sec common}
\subsection{Criterion~$(M_k)$ for common hypercyclic subspaces}\label{Sec Com Mk}
In 2005, Bayart~\cite{Bayart} generalized Criterion~$M_0$ to the existence of common hypercyclic subspaces by using the Common Hypercyclicity Criterion of Costakis and Sambarino~\cite{Costakis2} and by using the approach introduced by Chan~\cite{Chan} to constructing hypercyclic subspaces via left-multiplication operators. With the same approach, Grosse-Erdmann and Peris showed that such Criterion~$M_0$ for common hypercyclic subspaces remains true with their own version of the Common Hypercyclicity Criterion~\cite{Grosse}. We note that a family of operators $\{ T_{\lambda}\}_{\lambda\in \Lambda}$ satisfying the Common Hypercyclicity Criterion of Costakis and Sambarino~\cite{Costakis2} also satisfies the Common Hypercyclicity Criterion given in~\cite{Grosse} and in particular it satifies $(CHC)$ given in Definition~\ref{D:CHC}. We give next a constructive proof of Criterion~$M_0$ for common hypercyclic subspaces for families of operators satisfying $(CHC)$.
\begin{theorem}[{\bf Criterion~${\bf M_0}$ for Common Hypercyclic Subspaces}]\label{CritM0}
Let $X$ be a Fr\'{e}chet space with a continuous norm, let $Y$ be a separable Fr\'{e}chet space and let $\{ (T_{n,\lambda})_{n\ge 0} \}_{\lambda\in \Lambda}$ be a family of sequences of operators in $L(X,Y)$ satisfying (CHC) and for which there exists an infinite-dimensional closed subspace $M_0$ such that for any $(\lambda, x) \in \Lambda\times M_0$,
\[T_{n,\lambda}x\xrightarrow[n\to \infty]{} 0.\]
Then $\{ (T_{n,\lambda})_{n\ge 0} \}_{\lambda\in \Lambda}$ has a common hypercyclic subspace.
\end{theorem}
\begin{proof}
Let $(p_j)_{j\ge 1}$ be an increasing sequence of norms inducing the topology of~$X$, let $(q_j)_{j\ge 1}$ be an increasing sequence of seminorms inducing the topology of $Y$ and let $K=[a,b]\subset \Lambda$.
We denote by $X_{K,0}$, $Y_{K,0}$ and $S_{K,n,\lambda}$ the dense subsets and maps given by (CHC) for $K$.
We first show that for any $\varepsilon>0$, any $j\ge 1$, any $N_0\ge 0$, and any $y\in Y_{K,0}$, there exist
$N_1\ge N_0$ and $x\in X$ with $p_j(x)<\varepsilon$ so that for any $\lambda \in [a,b]$ there exists $k\in [N_0,N_1]$ satisfying
\begin{equation}
q_j(T_{k,\lambda}x-y)<\varepsilon.
\label{goodx}
\end{equation}
Let $\varepsilon>0$, $j\ge 1$, $N_0\ge 0$, and $y\in Y_{K,0}$ be given.
We consider $C\ge N_0$ such that
\begin{enumerate}
\item for any finite set $F\subset [C,\infty[$, any $l\ge 0$, any $\lambda\ge \mu_0\ge \dots \ge \mu_l$ in $K$,
\[q_j\Big(\sum_{k\in F\cap [0,l]}T_{l,\lambda}S_{K,l-k,\mu_k}y\Big)<\varepsilon;\]
\item for any finite set $F\subset [C,\infty[$, any $l\ge 0$, any $\lambda\le \mu_0\le \mu_1\le \cdots$ in $K$,
\[q_j\Big(\sum_{k\in F}T_{l,\lambda}S_{K,l+k,\mu_k}y\Big)<\varepsilon;\]
\item for any finite set $F\subset [C,\infty[$, any $\mu_0\le \mu_1\le \dots$ in $K$.
\[p_j\Big(\sum_{k\in F}S_{K,l,\mu_k}y\Big)<\varepsilon.\]
\end{enumerate}
Let $(\delta_{l})_{l\ge 0}$ be a sequence of positive real numbers such that ${\sum_{l=0}^{\infty}\delta_l=\infty}$ and for any $\alpha,\lambda \in K$, any $l\ge 0$,
\[0\le \alpha-\lambda\le \delta_l \quad\Rightarrow\quad q_j(T_{l,\lambda} S_{l,\alpha}y-y)<\varepsilon;\]
Since ${\sum_{l=0}^{\infty}\delta_l=\infty}$, there exists an increasing sequence $(k_l)_{l\ge 1}$ such that $\max\{C,N_0\}\le k_1$, for any $l\ge 1$, $k_{l+1}-k_l\ge C$ and $\sum_{l=1}^{\infty}\delta_{k_l}=\infty$. Such a sequence exists because
$(\{NC+k:N\ge 1\})_{k=0,\dots,N-1}$ forms a partition of $[C,\infty[$ and ${\sum_{l=C}^{\infty}\delta_l=\infty}$.
We select $L\ge 1$ the smallest integer such that
\[\sum_{l=1}^{L}\delta_{k_l}\ge b-a.\]
Let $\lambda_0:=a$ and $\lambda_l:=\lambda_{l-1}+\delta_{k_l}$ for any $1\le l\le L$.
We deduce that $a=\lambda_0<\lambda_1<\cdots<\lambda_{L-1}\le b\le\lambda_L$.
We then consider
\[x=\sum_{l=0}^{L-1} S_{K,k_{l+1},\lambda_l}y \quad\text{and}\quad N_1=k_L\]
and we show that $x$ and $N_1$ satisfy the desired properties. We first remark that $x$ satisfies
\[p_j(x)=p_j\Big(\sum_{l=0}^{L-1} S_{K,k_{l+1},\lambda_l}y\Big)<\varepsilon.\]
On the other hand, we see that for any $\lambda\in [a,b]$, there exists $0\le s\le L-1$ such that $\lambda\in [\lambda_s,\lambda_{s+1}]$ and thus
\begin{align*}
q_j(T_{k_{s+1},\lambda}x-y)&\le q_j(T_{k_{s+1},\lambda}S_{K,k_{s+1},\lambda_s}y-y)\\
&\quad +q_j\Big(\sum_{0\le l<s}T_{k_{s+1},\lambda}S_{K,k_{l+1},\lambda_l}y\Big)
+q_j\Big(\sum_{s<l\le L-1}T_{k_{s+1},\lambda}S_{K,k_{l+1},\lambda_l}y\Big)\\
&< q_j(T_{k_{s+1},\lambda}S_{K,k_{s+1},\lambda_s}y-y)+2\varepsilon\\
&\le 3\varepsilon \quad\quad\text{because }0\le \lambda-\lambda_s\le \lambda_{s+1}-\lambda_s\le \delta_{k_{s+1}}.
\end{align*}
Since for any $0\le s\le L-1$, $k_{s+1}\in[N_0,N_1]$, we conclude that \eqref{goodx} is satisfied.
Let $M_0$ be an infinite-dimensional closed subspace such that for any $\lambda\in \Lambda$, any $x\in M_0$,
\[T_{k,\lambda}x\xrightarrow[k\to \infty]{} 0.\]
We consider a chain $(K_n)_{n\ge 1}$ of $\Lambda$ such that each $K_n$ is a compact subinterval, and
a basic sequence $(u_n)_{n\ge 1}$ in $M_0$ such that for every $n\ge 1$, we have $p_1(u_n)=1$ and the sequence $(u_k)_{k\ge n}$ is basic in $(X,p_n)$ with basic constant less than $2$. We remark that, since each $K_n$ is compact and $T_{l,\lambda}(x)$ depends continuously on $\lambda$, for any $n,k,j\ge 1$, there exist a positive number $C_{n,k,j}$ and a positive integer $m^0(n,k,j)$ such that for any $x\in X$,
\begin{equation}
\sup_{\lambda\in K_n}p_j(T_{k,\lambda}x)\le C_{n,k,j}p_{m^0(n,k,j)}(x). \quad\text{(Banach-Steinhaus)}
\label{cont Tlambda}
\end{equation}
In particular, if $\tilde{X}$ is a dense subset in $X$, we deduce from the previous reasoning that for any $\varepsilon>0$, any $j,n\ge 1$, any $N_0\ge 1$, any $y\in Y_{K_n,0}$, there exists $x\in \tilde{X}$ and $N_1\ge N_0$ such that $p_j(x)<\varepsilon$ and such that for any $\lambda \in K_n$, there exists $k\in [N_0,N_1]$ such that
\begin{equation}
q_j(T_{k,\lambda}x-y)<\varepsilon.
\label{superx}
\end{equation}
Let $(y_k)_{k\ge 1}$ be a dense sequence in $X_0$ and $\prec$ the order on $\mathbb{N}\times\mathbb{N}$ defined by $(i,j)\prec(i',j')$ if $i+j<i'+j'$ or if $i+j=i'+j'$ and $i<i'$. We construct a family $(z_{i,j})_{i,j\ge 1}\subset X$ and two families $(n^{0}_{i,j})_{i,j\ge 1}, (n^{1}_{i,j})_{i,j\ge 1} \subset \mathbb{N}$ such that for any $i,j\ge 1$, $n^{0}_{i,j}\le n^{1}_{i,j}$ and for any $i\ge 1$, $(n^{0}_{i,j})_{j\ge 1}, (n^{1}_{i,j})_{j\ge 1}$ are increasing. If $z_{i',j'}$, $n^{0}_{i',j'}$ and $n^{1}_{i',j'}$ are already constructed for every $(i',j')\prec(i,j)$, then we choose $z_{i,j}\in X$ and $n^0_{i,j}, n^1_{i,j}$ with $n^{0}_{i,j}>\max\{n^{1}_{i',j'}:(i',j')\prec(i,j)\}$
such that
\begin{itemize}
\item we have,
\begin{gather}
\sum_{j'\le j} z_{i,j'}\in X_{K_{i+j+1},0},
\label{X0}
\end{gather}
\item for any $n\ge n^{0}_{i,j}$, any $i'\ge 1$ we have
\begin{gather}
\sup_{\lambda\in K_{i+j}}q_{i+j}\Big(\sum_{j':(i',j')\prec (i,j)}T_{n,\lambda}z_{i',j'}\Big)<\frac{1}{2^{i'+j}}, \label{3n grand}
\end{gather}
\item we have
\begin{gather}
p_{i+j}(z_{i,j})<\frac{1}{2^{i+j+2}},\label{3z petit}
\end{gather}
\item for any $(i',j')\prec(i,j)$, any $\lambda\in K_{i+j}$, any $n\in[n^{0}_{i',j'},n^{1}_{i',j'}]$, we have
\begin{equation}
q_{i+j}(T_{n,\lambda}z_{i,j})<\frac{1}{2^{i+j+j'}}, \label{3T z petit}\\
\end{equation}
\item for any $\lambda\in K_{i+j}$, there exists $n\in[n^{0}_{i,j},n^{1}_{i,j}]$, such that we have
\begin{equation}
q_{i+j}(T_{n,\lambda}z_{i,j}-y_j)<\frac{1}{2^{j}}.
\label{close y}
\end{equation}
\end{itemize}
Satisfying \eqref{3n grand} is possible by choosing $n^{0}_{i,j}$ sufficiently big because for any $i'\ge 1$, if $i'<i$,
\[\sum_{j':(i',j')\prec (i,j)}z_{i',j'}=\sum_{j'\le i+j-i'}z_{i',j'}\in X_{K_{i+j+1},0}\]
and if $i'\ge i$,
\[\sum_{j':(i',j')\prec (i,j)}z_{i',j'}=\sum_{j'\le i+j-i'-1}z_{i',j'}\in X_{K_{i+j},0}.\]
Satisfying \eqref{3z petit} and \eqref{3T z petit} is possible by choosing $z_{i,j}$ sufficiently close to $0$ thanks to \eqref{cont Tlambda}, and we can choose $z_{i,j}$ and $n^{1}_{i,j}$ such that \eqref{X0} and \eqref{close y} are satisfied thanks to \eqref{superx}.
We define, for any $i\ge 1$,
\[z_i:=u_i+\sum_{j=1}^{\infty} z_{i,j}.\]
By \eqref{3z petit}, these series are convergent and we deduce that the sequence $(z_i)_{i\ge 1}$ is a basic sequence equivalent to $(u_i)_{i\ge 1}$ in $X$. Let $M$ be the closed linear span of $(z_i)$ and $z\in M\backslash\{0\}$. We need to show that $z$ is hypercyclic for each sequence $(T_{n,\lambda})$. Since $(z_i)$ is a basic sequence, we can write $z=\sum_{i=1}^{\infty}\alpha_i z_i$ for some scalar sequence $(\alpha_i)$, and re-scaling $z$ if necessary we can further assume that $\alpha_{k}=1$ for some $k$. By the equivalence between the basic sequences $(z_i)$ and $(u_i)$, we deduce that $\sum_{i=1}^{\infty}\alpha_i u_i$ also converges and that there exists $K>0$ such that for any $i\ge 1$ we have $|\alpha_i|\le K$.
Let $l\ge 1$, $\lambda\in \Lambda$ and $r\ge l$ be given. If $\lambda\in K_r$ we deduce by \eqref{3n grand}, \eqref{3T z petit} and \eqref{close y} that there exists $n\in[n^{0}_{k,r},n^{1}_{k,r}]$ such that
\begin{align*}
q_l(T_{n,\lambda}z-y_r)&\le q_l\Big(\sum_{(i,j)\prec(k,r)} \alpha_iT_{n,\lambda}z_{i,j}\Big) + q_l\Big(\sum_{(i,j)\succ(k,r)} \alpha_iT_{n,\lambda}z_{i,j}\Big)\\
&\quad + q_l(T_{n,\lambda}z_{k,r}-y_r) + q_l\Big(T_{n,\lambda}\Big(\sum_{i\ge1} \alpha_iu_i\Big)\Big)\\
&\le\sum_{i=1}^{\infty}K\Big(q_{k+r}\Big(\sum_{j:(i,j)\prec (k,r)}T_{n,\lambda}z_{i,j}\Big)+\sum_{j:(i,j)\succ(k,r)}q_{i+j}(T_{n,\lambda} z_{i,j})\Big)\\
&\quad+ q_{k+r}(T_{n,\lambda}z_{k,r}-y_r) + q_l\Big(T_{n,\lambda}\Big(\sum_{i\ge 1} \alpha_iu_i\Big)\Big)\\
&\le \sum_{i=1}^{\infty} K \Big(\frac{1}{2^{i+r}}+\sum_{j=1}^{\infty}\frac{1}{2^{i+j+r}}\Big) + \frac{1}{2^r} +q_l\Big(T_{n,\lambda}\Big(\sum_{i\ge 1} \alpha_iu_i\Big)\Big)\\
&\le \frac{2K+1}{2^{r}}+ q_l\Big(T_{n,\lambda}\Big(\sum_{i\ge 1} \alpha_iu_i\Big)\Big)
\end{align*}
The last quantity can be made arbitrarily small as long as $r$ is sufficiently large, since $\sum_{i\ge 1} \alpha_iu_i\in M_0$. So $z$ is hypercyclic for $(T_{n,\lambda})_{n\ge 1}$.
\end{proof}
\begin{theorem}[{\bf Criterion~${\bf (M_k)}$ for Common Hypercyclic Subspaces}]\label{Mk com}
Let $X$ be a Fr\'{e}chet space with a continuous norm, let $Y$ be a separable Fr\'{e}chet space and let $\{ (T_{n,\lambda})_{n\ge 0} \}_{\lambda\in \Lambda}$ a family of sequences in $L(X,Y)$ satisfying $(CHC)$. Suppose there exist a chain $(\Lambda_n)$ of $\Lambda$ and a non-increasing sequence $(M_k)$ of infinite-dimensional closed subspaces of $X$
so that for each $n\in\mathbb{N}$,
\[
\{ (T_{k,\lambda})_{k\ge 1}\}_{\lambda\in\Lambda_n} \ \ \mbox{is uniformly equicontinuous along $(M_k)$.}
\]
Then $\{ (T_{n,\lambda})_{n\ge 0} \}_{\lambda\in \Lambda}$ has a common hypercyclic subspace.
\end{theorem}
For the proof of Theorem~\ref{UMk} we used a map $\phi$ to guarantee that the subsequence
$\{ n_k:\ k\in I=\bigcup_{s\ge 1} [k_s,k_{s}+\phi(k_s)]$
given by Corollary~\ref{NiceMnone} kept the positiveness of the upper density of $(n_k)$. In the proof of Proposition~\ref{prophered} below we use the fact that given a divergent series $\sum_{k\ge 1} \delta_k $ of positive real numbers there exists a map $\phi:\mathbb N\to\mathbb N$ so that for each increasing sequence of integers $(k_s)_{s\ge 1}$ in $\mathbb N$ the set $I=\bigcup_{s\ge 1} [k_s,k_{s}+\phi(k_s)]$ satisfies
\[\sum_{j\in I}\delta_{j}=\infty.\]
We prove Theorem~\ref{Mk com} after the proof of Proposition~\ref{prophered}.
\begin{prop}\label{prophered}
Let $X$ be a Fr\'{e}chet space supporting a continuous norm, let $Y$ be a separable Fr\'{e}chet space and let $\{ (T_{n,\lambda})_{n\ge 0 } \}_{\lambda\in \Lambda}$ be a family of sequences in $L(X,Y)$ satisfying the $(CHC)$. Then there exists a map $\phi:\mathbb N\to \mathbb N$ such that for any increasing sequence $(k_s)$ in $\mathbb{N}$, if $(m_k)=\bigcup_{s\ge 1}[k_s,k_s+\phi(k_s)]$, then $\{ (T_{m_k,\lambda})_{k\ge 1} \}_{\lambda\in \Lambda}$ satisfies $(CHC)$.
\end{prop}
\begin{proof}
Let $(q_j)_{j\ge 1}$ be an increasing sequence of seminorms inducing the topology of $Y$. Let $(K_n)_{n\ge 1}$ be a chain of $\Lambda$ such that each $K_n$ is a compact subinterval. We denote by $X^{(n)}_{0}$, $Y^{(n)}_{0}$ and $S^{(n)}_{k,\lambda}$ the sets and maps given by $(CHC)$ for the compact set $K_n$. We remark that without loss of generality we can assume that for each $n\ge1$, $Y^{(n)}_{0}$ is countable. Moreover, for any $N\ge 1$, $j\ge 1$, $n\ge 1$, and $y_0\in Y^{(n)}_0$ we denote by $(\delta^{(n)}_{y_0,N,j,k})_k$ a sequence of positive real numbers such that $\sum_{k=0}^\infty \delta^{(n)}_{y_0,N,j,k} =\infty$ and for each $k\ge 0$ and $\lambda, \mu\in K_N$ we have
\[
0\le \mu-\lambda < \delta_{y_0,N,j,k} \ \ \Rightarrow \ \ \ q_j( T_{k, \lambda} S^{(n)}_{k,\mu} y_0 - y_0 ) < \frac{1}{N}.
\]
We then choose $\phi$ such that for any $N\ge 1$, $j\ge 1$, $n\ge 1$, any $y_0\in Y^{(n)}_0$, if $I=\bigcup_{s\ge 1}[k_s,k_s+\phi(k_s)]$, then
\[\sum_{k\in I} \delta^{(n)}_{y_0,N,j,k} =\infty.\]
Such a map exists because, if $Y^{(n)}_0=(y_{n,l})_{l\ge 1}$, then it suffices to let $\phi(k)$ such that for any $n,j,l,N\le k$,
\[\sum_{i=k}^{k+\phi(k)} \delta^{(n)}_{y_{n,l},N,j,i}\ge 1.\]
Let $(k_s)$ be an increasing sequence and $(m_k)=\bigcup_{s\ge 1}[k_s,k_s+\phi(k_s)]$. We show that $\{(T_{m_k,\lambda})_{k\ge 1}\}_{\lambda\in \Lambda}$ satisfies $(CHC)$. We first remark that for any compact subset $K$ in $\Lambda$, there exists $n\ge 1$ such that $K_n\supset K$.
It thus suffices to prove that each condition of $(CHC)$ is satisfied for the compact subintervals $K_n$.
Let $x_0\in X^{(n)}_0$, $y_0\in Y_0^{(n)}$ and $q$ a continuous seminorm on $Y$.
\begin{enumerate}
\item[(1)]\ Assume that for any finite set $F\subset [C,\infty[$, any $l\ge 0$, any $\lambda\ge \mu_0\ge \dots \ge \mu_l$ in $K_n$,
\[q\Big(\sum_{k\in F\cap [0,l]}T_{l,\lambda}S^{(n)}_{l-k,\mu_k}y_0\Big)<\varepsilon.\]
We remark that for any finite set $F\subset [C,\infty[$, any $l\ge C$, any $\lambda\ge \mu_0\ge \dots \ge \mu_l$ in $K_n$, we have
\begin{align*}
q\Big(\sum_{k\in F\cap [0,l-1]}T_{m_l,\lambda}S^{(n)}_{m_{l-k},\mu_k}y_0\Big)
& = q\Big(\sum_{k'\in F'\cap [0,m_l]} T_{m_l,\lambda}S^{(n)}_{m_l-k',\nu_{k'}}y_0\Big),
\end{align*}
where $F'=\{k'\ge 0: k'= m_l-m_{l-k}, k\in F\cap [0,l-1]\}$ and for any $k'\in F'$, if $k'=m_l-m_{l-k}$, then $\nu_{k'}=\mu_{k}$. In particular, we have $F'\subset[m_l-m_{l-C},\infty[\subset [C,\infty[$ and thus we deduce that
\[
q\Big(\sum_{k\in F\cap [0,l-1]}T_{m_l,\lambda}S^{(n)}_{m_{l-k},\mu_k}y_0\Big)=
q\Big(\sum_{k'\in F'\cap [0,m_l]} T_{m_l,\lambda}S^{(n)}_{m_l-k',\nu_{k'}}y_0\Big)<\varepsilon.\]
\item[(2)]\ Assume that for any finite set $F\subset [C,\infty[$, any $l\ge 0$, any $\lambda\le \mu_0\le \mu_1\le \cdots$ in $K_n$,
\[q\Big(\sum_{k\in F}T_{l,\lambda}S^{(n)}_{l+k,\mu_k}y_0\Big)<\varepsilon.\]
We remark that for any finite set $F\subset [C,\infty[$, any $l\ge 1$, and any $\lambda\le \mu_0\le \mu_1\le \cdots$ in $K_n$, we have
\begin{align*}
q\Big(\sum_{k\in F}T_{m_l,\lambda}S^{(n)}_{m_{l+k},\mu_k}y_0\Big)
& = q\Big(\sum_{k'\in F'} T_{m_l,\lambda}S^{(n)}_{m_l+k',\nu_{k'}}y_0\Big),
\end{align*}
where $F'=\{k'\ge 0: k'= m_{l+k}-m_{l}, k\in F\}$ and for any $k'\in F'$, if $k'=m_{l+k}-m_{l}$, then $\nu_{k'}=\mu_{k}$. In particular, we have $F'\subset[m_{l+C}-m_{l},\infty[\subset [C,\infty[$ and thus we deduce that
\[
q\Big(\sum_{k\in F}T_{m_l,\lambda}S^{(n)}_{m_{l+k},\mu_k}y_0\Big)=
q\Big(\sum_{k'\in F'} T_{m_l,\lambda}S^{(n)}_{m_l+k',\nu_{k'}}y_0\Big)<\varepsilon.\]
\item[(3)]\ Let $\varepsilon>0$, $j\ge1$ and $n\ge 1$. We consider $N\ge n$ such that $\frac{1}{N}<\varepsilon$. We know by choice of $\phi$ that $\sum_{k=1}^\infty \delta_{y_0,N,j,m_k} =\infty$, and we have, by definition of $\delta_{y_0,N,j,m_k}$, that for each $k\ge 1$ and $\lambda, \mu\in K_N$,
\[0\le \mu-\lambda < \delta_{y_0,N,j,m_k} \ \ \Rightarrow \ \ \ q_j(T_{m_k, \lambda} S^{(n)}_{m_k,\mu} y_0 - y_0) < \epsilon.
\]
Since $K_N\supset K_n$, we have the desired result.
\end{enumerate}
Conditions $(4)$ and $(5)$ are clear.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{Mk com}]\
Let $(K_n)$ be a chain of $\Lambda$ such that each $K_n$ is a compact subinterval.
We remark that the assumptions of Theorem~\ref{NiceMn} are satisfied if we consider $(\Lambda^0_n)=(\Lambda^1_n)=(K_n)$ and $(\Lambda^2_n)=(\Lambda_n)$. Indeed, we deduce that $(\Lambda^0_n)$ satisfies the required properties by using the Banach-Steinhaus theorem and the fact that $T_{n,\lambda}$ depends continuously on $\lambda$.
On the other hand, as $\{ (T_{n,\lambda})_{n\ge 0} \}_{\lambda\in \Lambda}$ satisfies $(CHC)$, we deduce from Condition $(4)$ in Definition~\ref{D:CHC} that for any subinterval $[a,b]\subset \Lambda$, there exists a dense subset $X_{0}$ such that for any $x\in X_0$, $T_{k,\lambda}x\xrightarrow[k\to \infty]{}0$ uniformly on $\lambda\in [a,b]$. The chain $(\Lambda^1_n)$ thus satisfies the required properties.
We conclude by Proposition~\ref{prophered} and Theorem~\ref{NiceMn} that there exist an increasing sequence of positive integers $(m_k)$ and an infinite-dimensional closed subspace $M_0$ of $X$
such that $(T_{m_k,\lambda})_{k\ge 1}$ satisfies $(CHC)$
and such that for each $x\in M_0$ and $\lambda\in \Lambda$,
\[T_{m_k, \lambda}x\xrightarrow[k\to \infty]{} 0.\]
We obtain the desired result by applying Theorem~\ref{CritM0}.
\end{proof}
\subsection{Applications of Criterion $(M_k)$ for common hypercyclic subspaces}
In general, Criterion $(M_k)$ is much easier to use than Criterion~$(M_0)$ because in a lot of cases, when Criterion~$(M_k)$ is satisfied, it is satisfied along the whole sequence $(k)$. We illustrate the use of Criterion $(M_k)$ for common hypercyclic subspaces on families of weighted shifts on K\"{o}the sequence spaces.
\begin{definition}
Let $A=(a_{j,k})_{j\ge 1,k\ge 0}$ be a matrix such that for any $j\ge 1$ and $k\ge 0$, we have $a_{j,k}>0$ and $a_{j,k}\le a_{j+1,k}$.
The (real or complex) \emph{K\"{o}the sequence space} $\lambda^p(A)$ is defined as
\begin{align*}
\lambda^p(A)&:=\Big\{(x_k)_{k\ge 0}\in \omega : p_j((x_k)_k)=\Big(\sum_{k=0}^{\infty}|x_ka_{j,k}|^p\Big)^{\frac 1 p}<\infty, \ j\ge 1\Big\},
\end{align*}
endowed with the sequence of norms $(p_j)$.
\end{definition}
Let $\Lambda$ be an open interval of $\mathbb R$ and $w_{\lambda}$ be a sequence of non-zero scalars.
The weighted shift $B_{w_{\lambda}}$ is defined as $B_{w_{\lambda}}e_n=w_{\lambda,n}e_{n-1}$, where $e_{-1}=0$ and $(e_n)_{n\ge 0}$ is the canonical basis. We provide in Proposition~\ref{Bwlambda} a sufficient condition for a family of shifts
$\{ B_{w,\lambda} \}_{\lambda\in \Lambda}$ satisfying the $(CHC)$ on $\lambda^p(A)$ to support a common hypercyclic subspace. We first note the following.
\begin{remark}\label{remequi}
Let $(p_j)_{j\ge 1}$ and $(q_j)_{j\ge 1}$ be increasing sequences of seminorms inducing the topologies of $X$, and $Y$, respectively.
If $T_{n,\lambda}$ depends continuously on $\lambda$ and $K$ is a compact subset of $\Lambda$, then $\{(T_{n,\lambda})_{n\ge 1}\}_{\lambda\in K}$ is uniformly equicontinuous along $(M_k)$ if and only if for any $j\ge 1$, there exists $C_j>0$ and $k(j),m(j)\ge 1$ such that for any $k\ge k(j)$, any $\lambda\in K$ and any $x\in M_k$,
\[q_j(T_{k,\lambda}x)\le C_j p_{m(j)}(x).\]
\end{remark}
\begin{prop} \label{Bwlambda}
Let $\lambda^p(A)$ be a K\"{o}the sequence space, $\Lambda$ an open interval of $\mathbb R$ and let $\{ B_{w_\lambda}\}_{\lambda\in \Lambda}$ be a family of weighted shifts on $\lambda^p(A)$ satisfying $(CHC)$. If there exist a chain of compact sets $(K_n)_{n\ge 1}$ of $\Lambda$, and for each $n\ge 1$ a sequence $(C_{n,j})_{j}$ of positive scalars and a sequence $(m(n,j))_j$ of integers such that for any $l\ge 1$ and any $N\ge 0$,
\[\inf_{k\ge N}\max_{1\le n,j,m\le l}\sup_{\lambda\in K_n}\frac{p_j(B^m_{w_{\lambda}}e_k)}{C_{n,j}p_{m(n,j)}(e_k)}\le 1,\]
then $\{ B_{w_\lambda}\}_{\lambda\in \Lambda}$ has a common hypercyclic subspace.
\end{prop}
\begin{proof}
Let $(K_n)$, $(C_{n,j})$ and $(m(n,j))$ be sequences satisfying the above assumptions. For any $l\ge 1$, we consider $e_{n_l}$ such that $n_l>n_{l-1}$ and for any $n,j,m\le l$,
\[\sup_{\lambda\in K_n}\frac{p_j(B^m_{w_{\lambda}}e_{n_l})}{p_{m(n,j)}(e_ {n_l})}\le 2C_{n,j},\]
Let $M_k=\overline{\text{span}}\{e_{n_l}:l\ge k\}$.
We deduce that for any $n,j\ge 1$, any $\lambda\in K_n$, any $k\ge \max\{n,j\}$ and any $x\in M_k$ we have
\begin{align*}
p_j(B^{k}_{w_{\lambda}}x)^p&=
p_j\Big(B^{k}_{w_{\lambda}}\Big(\sum_{l=k}^{\infty}x_{n_l}e_{n_l}\Big)\Big)^p\\
&=\sum_{l=k}^{\infty}|x_{n_l}|^p p_j(B^{k}_{w_{\lambda}}e_{n_l})^p\\
&\le (2C_{n,j})^p \sum_{l=k}^{\infty}|x_{n_l}|^p p_{m(n,j)}(e_{n_l})^p\\
&\le (2C_{n,j})^p p_{m(n,j)}(x)^p.
\end{align*}
Since each $K_n$ is compact and $B_{w_{\lambda}}$ depends continuously on $\lambda$, we conclude by Remark~\ref{remequi} that each $\{ B_{w_{\lambda}}\}_{\lambda\in K_r}$ is uniformly equicontinuous along $(M_k)$. The conclusion now follows by Theorem~\ref{Mk com}.
\end{proof}
\begin{cor}\label{easycrit}
Let $\lambda^p(A)$ be a K\"{o}the sequence space. Let $\Lambda$ be an open interval of $\mathbb R$ and let $\{ B_{w_\lambda} \}_{\lambda\in \Lambda}$ be a family of weighted shifts on $\lambda^p(A)$ satisfying $(CHC)$. Suppose that for each compact subset $K$ of $\Lambda$ there exist a sequence $(C_{K,j})$ of positive scalars and a sequence $(m(K,j))$ of positive integers such that for any $j\ge 1$ and $n\ge 1$ we have
\[\limsup_{k\to \infty}\sup_{\lambda\in K}\frac{p_j(B^n_{w_{\lambda}}e_k)}{C_{K,j}\ p_{m(K,j)}(e_k)}\le 1.\]
Then $\{ B_{w_\lambda} \}_{\lambda\in \Lambda}$ has a common hypercyclic subspace.
\end{cor}
Corollary~\ref{easycrit} gives an answer to a question of Costakis and Sambarino~{\cite[Section 8]{Costakis2}}:
\begin{cor}\label{Ex:Costa}
For each $\lambda >1$, let $B_{w_\lambda}$ be the backward shift on $\ell^p$ $(1\le p<\infty)$ with weight sequence $w_{\lambda}=(1+\frac{\lambda}{k})_{k\ge 1}$.
Then $\{ B_{w_\lambda} \}_{\lambda >1}$ has a common hypercyclic subspace.
\end{cor}
\begin{proof}
Costakis and Sambarino \cite{Costakis2} have shown that the family
$\{ B_{w_{\lambda}}\}_{\lambda>1}$ satisfies (CHC).
But for each $K=[a,b]\subset (0,\infty )$ and $n\ge 1$ we have
\begin{align*}
\limsup_{k\to\infty}\sup_{\lambda\in K}\|B^n_{w_{\lambda}}e_k\|& = \limsup_{k\to\infty}\sup_{\lambda\in K}
\Big(\prod_{\nu=0}^{n-1} |w_{\lambda,k-\nu}|\Big)\\
&= \limsup_{k\to\infty}\sup_{\lambda\in K}\Big(\prod_{\nu=0}^{n-1} \big(1+\frac{\lambda}{k-\nu}\big)\Big)\\
&\le \limsup_{k\to\infty}\ \big(1+\frac{b}{k-n+1}\big)^n=1.
\end{align*}
That is, the assumptions of Corollary~\ref{easycrit} are satisfied.
\end{proof}
Bayart and Ruzsa \cite{Bayart3} showed that a shift $B_w$ with weight sequence $w=(w_n)$ is frequently hypercyclic on $\ell^p$ if and only it is $\mathcal{U}$-frequently hypercyclic and if and only if $(\frac{1}{\prod_{1\le j\le n} w_j} )_{n\ge 1} \in \ell^p$.
So each shift $B_{w_{\lambda}}$ from Corollary~\ref{Ex:Costa} is frequently hypercyclic on $\ell^p$ for any $1\le p<\infty$. Moreover, by Corollary~\ref{caraclp}, each $B_{w_\lambda}$ has a $\mathcal{U}$-frequently hypercyclic subspace on $\ell^p$. This motivates the following.
\begin{problem}
For each $\lambda \in \mathbb C$, let $\{ B_{w_{\lambda}}\}_{\lambda>1}$ be the shift operator on $\ell^p$ with weight sequence $w_\lambda=(1+\frac{\lambda}{n})$. Does
$
\{ B_{w_\lambda} \}_{\lambda >1}
$
have a common $\mathcal{U}$-frequently hypercyclic subspace?
\end{problem}
We note that two operators having $\mathcal{U}$-frequently hypercyclic subspaces may fail to have a common hypercyclic subspace.
\begin{example} \label{E:T1T2}
Let $B_w$ be the backward shift operator on $\ell^2$ of weight sequence $w=(\frac{n+1}{n})$. Then each of the operators $T_1=B_w\oplus 2B$ and $T_2=2B\oplus B_w$ on $X=\ell^2\oplus \ell^2$ has a $\mathcal{U}$-fhc subspace, but $\{ T_1, T_2 \}$ has no common hypercyclic subspace. The latter follows from an argument used in \cite[Example 2.1]{Aron}: For $j=1,2$, let $P_j:\ell^2\oplus\ell^2\to\ell^2$, $P_j(x_1, x_2)=x_j$ be the $j$-th orthogonal projection. If $M\subset \ell^2\oplus\ell^2$ is a hypercyclic subspace for both $T_1$ and $T_2$, then $P_1(M)\cup P_2(M)$ consists (but zero) of common hypercyclic vectors for $B_w$ and $2B$. But since $M$ is closed and infinite-dimensional and $X$ is a Hilbert space, one of $P_1(M)$, $P_2(M)$ must {\em contain} a closed and infinite-dimensional subspace, contradicting the fact that $2B$ has no hypercyclic subspaces. It remains to see that each of $T_1$ and $T_2$ has a $\mathcal{U}$-fhc subspace. Each of $T_1$, $T_2$ satisfies the FHC on $\ell^2\oplus\ell^2$, since each of their summands $B_w$ and $2B$ satisfies the FHC on $\ell^2$. Also, $B_w$ has a hypercyclic subspace on $\ell^2$, and
there exists a non-increasing sequence $(M_k)$ of closed and infinite-dimensional subspaces of $\ell^2$
for which $\mbox{sup}_{k\ge 1} \| {B_w^k}_{|_{M_k}} \| <\infty$. Thus $(T_1^k)$ and $(T_2^k)$ are equicontinuous along $(M_k\oplus\{ 0\})$ and $(\{ 0\}\oplus M_k)$, respectively, and the conclusion follows from Theorem~\ref{UMkT}
\end{example}
In 2010, Shkarin~\cite{ShkarinD} showed that the derivative operator $D$ on $H(\mathbb C)$ has a hypercyclic subspace.
By considering $H(\mathbb C)$ as the K\"{o}the sequence space $\Lambda^1(A)$ for $a_{j,k}=j^k$, we can now prove the existence of a common hypercyclic subspace for the family of operators $\{ \lambda D\}_{\lambda\in \mathbb C\backslash\{0\}}$ on $H(\mathbb C)$.
\begin{cor}\label{mult D}
Let $D$ be the derivative operator on $H(\mathbb C)$.
Then the family $\{ \lambda D\}_{\lambda\in \mathbb C\backslash\{0\}}$ has a common hypercyclic subspace.
\end{cor}
\begin{proof}
We know that the family $\{ \lambda D\}_{\lambda> 0}$ satisfies $(CHC)$ \cite{Costakis2}. Moreover, for any $K=[a,b]\subset ]0,\infty[$ and any $n\ge 1$ we have
\begin{align*}
\limsup_{k\to\infty}\sup_{\lambda\in K}\frac{p_j(\lambda^nD^ne_k)}{p_{2j}(e_k)}
&=\limsup_{k\to\infty}\sup_{\lambda\in K}\frac{p_j(\lambda^nD^ne_{k+n})}{p_{2j}(e_{k+n})}\\
&= \limsup_{k\to\infty}\sup_{\lambda\in K}\frac{\lambda^n \prod_{\nu=1}^{n}(k+\nu)p_j(e_k)}{p_{2j}(e_{k+n})}\\
&\le \limsup_{k\to\infty}\frac{b^n (k+n)^n j^k}{(2j)^{k+n}}=0.
\end{align*}
Thus by Corollary~\ref{easycrit} the family $\{ \lambda D\}_{\lambda>0}$ has a common hypercyclic subspace. The conclusion now follows from the fact that for any complex scalar $\lambda$ and operator $T$ the operators
$\lambda T$ and $|\lambda| T$ have the same set of hypercyclic vectors \cite{Leon2}.
\end{proof}
Corollary~\ref{mult D} extends to operators $P(D)$ where $P$ is a non-constant polynomial.
\begin{prop} \label{P:P(D)common}
Let $D$ be the derivative operator on $H(\mathbb C)$ and let $P$ be a non-constant polynomial.
Then $\{ \lambda P(D)\}_{\lambda\in \mathbb C\backslash\{0\}}$ has a common hypercyclic subspace.
\end{prop}
\begin{proof}
The family $\{ (\lambda^k P(D)^k)_{k\ge 1} \}_{\lambda>0}$ satisfies $(CHC)$ (see the proof of \cite[Proposition 2.2]{Costakis} and Remark~\ref{R:CHC}).
On the other hand, we know~\cite{Menet1} that there exists a non-increasing sequence of infinite-dimensional closed subspaces $(M_k)$ in $H(\mathbb C)$ such that for any $j\ge 1$, there exist a positive number $C_{j}$ and two integers $m(j),k(j)$ such that for any $k\ge k(j)$, any $x\in M_k$,
\[p_j(P(D)^{k}x)\le C_{j} p_{m(j)}(x).\]
Without loss of generality, we can assume that for any $k\ge 1$, the subspace $M_k$ is included in $\overline{\text{span}}\{e_n:n\ge k\}$. We recall that the sequence $(p_j)$ of norms considered here is given by $p_j((x_k)_k)=\sum_{k=0}^{\infty}|x_kj^k|$. In particular, we deduce from the previous inclusion that for any $x\in M_k$, any $j\ge 1$, any $n\ge 1$, \[p_j(x)\le n^{-k}p_{nj}(x).\]
Let $n\ge 1$. If we consider $\Lambda^{2}_n=[\frac{1}{n},n]$, $C_{n,j}=C_j$, $m(n,j)=nm(j)$, $k(n,j)=k(j)$, then for
any $\lambda\in \Lambda^2_n$, any $k\ge k(n,j)$, any $x\in M_k$, we have
\[p_j((\lambda P(D))^{k}x)\le \lambda^k C_{j} p_{m(j)}(x)\le \lambda^k C_{j} n^{-k}p_{nm(j)}(x)\le C_{j}p_{m(n,j)}(x).\]
By Remark~\ref{remequi} and Theorem~\ref{Mk com}, the family $\{ \lambda P(D)\}_{\lambda>0}$ has a common hypercyclic subspace. So $\{ \lambda P(D)\}_{\lambda\in \mathbb C\backslash\{0\}}$ has a common hypercyclic subspace, as for any operator $T$ and any complex scalar $\lambda$ the operators $\lambda T$ and $|\lambda |T$ have the same set of hypercyclic vectors \cite{Leon2}.
\end{proof}
As with Corollary~\ref{C:ufhsNonConstant}, Lemma~\ref{L:Delsarte} by Delsarte and Lions allows to extend Proposition~\ref{P:P(D)common} to any linear differential operators of finite order whose coefficients -other than the leading one- may be non-constant.
\begin{cor} \label{C:NonConstantCommon}
For each $N\ge 1$ and $a_0,\dots, a_{N-1}\in H(\mathbb{C})$, consider the differential operator
$
T=D^N+a_{N-1}(z) D^{N-1}+\cdots + a_0(z) I: H(\mathbb{C})\to H(\mathbb{C})
$.
Then
\[
\{ \lambda T \}_{\lambda >0}
\] has a common hypercyclic subspace.
\end{cor}
Recall that for any non-constant entire function $\phi$ of exponential type the family $\{ \lambda \phi (D) \}_{\lambda >0}$ has a common hypercyclic vector. This is due to Costakis and Mavroudis~\cite{Costakis} for the case $\phi$ is a polynomial, to Bernal \cite{Bernal2009} when $\phi$ has exponential growth order no larger than $\frac{1}{2}$, and due to Shkarin \cite{Shkarin} in the general case.
Proposition~\ref{P:P(D)common} motivates the following.
\begin{problem}
Let $\phi$ be entire, transcendental, and of exponential type. Does the family $\{ \lambda \phi (D) \}_{\lambda \in \mathbb C\setminus\{ 0 \} }$ have a common hypercyclic subspace?
\end{problem}
\subsection{Multiples of an operator on a complex Banach space}\label{Sec:Charac}
A characterization by Gonz\'{a}lez, Le\'{o}n and Montes~\cite{Gonzalez} asserts that an operator $T$ on a complex Banach space satisfying the Hypercyclicity Criterion has a hypercyclic subspace if and only if its essential spectrum intersects the closed unit disc. We recall that if the essential spectrum of $T$ does not intersect the closed unit disk then for any infinite-dimensional closed subspace $M$, there exists $x\in M$ such that $\|T^n x\|\to \infty$. We denote by $\sigma_e(T)$ the essential spectrum of $T$, i.e. the set of complex scalars $\lambda$ such that $\lambda Id -T$ is not a Fredholm operator, where an operator is said to be Fredholm if its range is closed and its kernel and cokernel are finite-dimensional.
While it is likely not possible to extend such spectral characterization to arbitrary families of operators (even if they consist of two operators) as evidenced in \cite{Bes4}, we show that it indeed extends to certain
families of the form $\{ \lambda T\}_{\lambda\in\Lambda}$ thanks to Criterion~$(M_k)$ for common hypercyclic subspaces.
Throughout this section, we let $X$ be a complex Banach space, $T\in L(X)$ and let $\mathcal{P}=(P_\lambda)_{\lambda\in \Lambda}$ be a family of polynomials.
We denote by
\[r_{\mathcal{P}}:=\inf\{r> 0: \exists \lambda\in \Lambda,\ P_\lambda(B(0,r)^c)\cap \overline{B(0,1)}= \emptyset\}.\]
We start by stating a sufficient condition for the non-existence of common hypercyclic subspaces for the family $\{ P_{\lambda}(T)\}_{\lambda\in\Lambda}$ in terms of the essential spectrum of~$T$.
\begin{prop}\label{prop E}
If $\sigma_e(T)\cap \overline{B(0,r_{\mathcal{P}})}=\emptyset$ (with $\overline{B(0,0)}=\{0\}$), then
there exists $\lambda\in \Lambda$ such that each infinite-dimensional closed subspace $M$ of $X$ contains some $x$ for which
\[\|P^n_{\lambda}(T)x\|\xrightarrow[n\to \infty]{} \infty.\]
In particular, there exists $\lambda\in \Lambda$ such that $P_{\lambda}(T)$ has no hypercyclic subspace.
\end{prop}
\begin{proof}
Since $\sigma_e(T)$ is compact and $\sigma_e(T)\cap \overline{B(0,r_{\mathcal{P}})}=\emptyset$, there exists ${r>r_{\mathcal{P}}}$ such that
\[\sigma_e(T)\cap B(0,r)=\emptyset.\]
By definition of $r_{\mathcal{P}}$, we know that there exists $\lambda\in \Lambda$ such that
\[P_\lambda(B(0,r)^c)\cap \overline{B(0,1)}= \emptyset.\]
Therefore, since $\sigma_e(P_\lambda(T))=P_\lambda(\sigma_e(T))$, we deduce that $\sigma_e(P_\lambda(T))\subset P_\lambda(B(0,r)^c)$ and thus
\[\sigma_e(P_\lambda(T))\cap \overline{B(0,1)}= \emptyset,\]
i.e. the essential spectrum of $P_{\lambda}(T)$ does not intersect the closed unit disk.
We conclude that for any infinite-dimensional closed subspace $M\subset X$, there exists $x\in M$ such that
\[\|P^n_{\lambda}(T)x\|\xrightarrow[n\to \infty]{} \infty.\]
\end{proof}
Under certain conditions on the essential spectrum of~$T$, we can construct a non-increasing sequence $(M_n)$ of closed infinite-dimensional subspaces of $X$ so that the assumptions of Criterion~$(M_k)$ for common hypercyclic subspaces are met.
\begin{prop}\label{Mk}
Suppose $\sup_{\lambda\in \Lambda}|P_{\lambda}(\mu)|\le 1$ and that
$\text{\emph{Ran}}(T-\mu Id)$ is dense in $X$, for some $\mu\in \sigma_{e}(T)$.
Then there exists a non-increasing sequence of infinite-dimensional closed subspaces $(M_n)$ in $X$ and a chain $(\Lambda_n)$ of $\Lambda$ such that for any $n\ge 1$, any $\lambda\in \Lambda_n$, any $m\le n$, we have
\[\|P^m_{\lambda}(T)x\|\le 2\|x\| \quad \text{for any $x\in M_n$.}\]
\end{prop}
\begin{proof}
Since $\text{Ran}(T-\mu Id)$ is dense and $\mu\in \sigma_{e}(T)$, either $\dim \ker(T-\mu Id)=\infty$ or $\text{Ran}(T-\mu Id)$ is not closed. If $\dim \ker(T-\mu Id)=\infty$, then we consider $M_n=\ker(T-\mu Id)$. Therefore, for any $n\ge 1$, any $x\in M_n$, and any $\lambda\in \Lambda$ we have
\[\|P^n_{\lambda}(T)x\|=\|(P_{\lambda}(\mu))^n x\|\le \|x\|.\]
We now suppose that $\dim \ker(T-\mu Id)<\infty$ and $\text{Ran}(T-\mu Id)$ is not closed. We can then show that there exists an infinite-dimensional closed subspace $M$ in $X$ and a compact operator $K$ such that $T_{|M}=\mu Id + K$ (see \cite{Gonzalez}). Therefore, for any $n\ge 0$ we have $T^n_{|M}=\mu^n Id +K_n$ where $K_n$ is a compact operator. Since $K_n$ is compact, we know that for any $\varepsilon>0$, there exists a closed subspace of finite-codimension $E_{n,\varepsilon}$ in $X$ such that
$\|K_{n|E_{n,\varepsilon}}\|\le \varepsilon$.
We consider \[\Lambda_n=\{\lambda\in \Lambda: \text{deg}P_{\lambda}\le n \ \text{and}\ P_{\lambda}(x)=\sum_{k=0}^na_kx^k \ \text{with}\ \sum_{k=0}^{n}|a_k|\le n\}\] and we let $M_0:=M$ and
\[M_n:=M_{n-1}\cap \bigcap_{k \le n^2}E_{k,\varepsilon_n} \quad\text{with}\quad \varepsilon_n=\frac{1}{n^{n}}.\]
We remark that for any $n\ge 1$, any $\lambda\in \Lambda_n$, any $m\ge 0$,
\[P^m_\lambda(T)=\sum_{k=0}^{mn}c_kT^k \quad \text{with }\sum_{k=0}^{mn}|c_k|\le n^m.\]
We deduce that for any $n\ge 1$, any $\lambda\in \Lambda_n$, any $m\le n$, and any $x\in M_n$,
\begin{align*}
\|P^m_\lambda(T)x\|&= \Big\|\sum_{k=0}^{mn}c_kT^kx\Big\|=
\Big\|\sum_{k=0}^{mn}c_k(\mu^kx+K_kx)\Big\|\\
&\le |P_\lambda(\mu)|^m\cdot\|x\|+ \sum_{k=0}^{mn}(|c_k|\cdot \|K_{k|M_n}\|)\|x\|
\le \|x\| + n^{n} \varepsilon_n \|x\|\le 2\|x\|.
\end{align*}
\end{proof}
Thanks to Proposition~\ref{prop E} and Proposition~\ref{Mk}, we can now generalize the characterization of Gonz\'{a}lez, Le\'{o}n and Montes as follows:
\begin{theorem}\label{Charac com}
Let $X$ be a separable infinite-dimensional complex Banach space, $T\in L(X)$ and $\{P_{\lambda}\}_{\lambda\in \Lambda}$ a family of polynomials.
If $\{ P_{\lambda}(T)\}_{\lambda\in \Lambda}$ satisfies (CHC) and
if for any $\mu\in \overline{B(0,r_{\mathcal{P}})}$, $\text{\emph{Ran}}(T-\mu Id)$ is dense and ${\sup_{\lambda\in \Lambda}|P_{\lambda}(\mu)|\le 1}$,
then the following assertions are equivalent:
\begin{enumerate}
\item for any $\lambda\in \Lambda$, $P_{\lambda}(T)$ has a hypercyclic subspace;
\item $\{ P_{\lambda}(T)\}_{\lambda\in \Lambda}$ has a common hypercyclic subspace;
\item $\{ P_{\lambda}(T))\}_{\lambda\in \Lambda}$ satisfies Criterion~$M_0$ for common hypercyclic subspaces;
\item $\{ P_{\lambda}(T)\}_{\lambda\in \Lambda}$ satisfies Criterion~$(M_k)$ for common hypercyclic subspaces;
\item the essential spectrum of $T$ intersects $\overline{B(0,r_{\mathcal{P}})}$.
\end{enumerate}
\end{theorem}
\begin{proof}
$\neg(5)\Rightarrow \neg (1)$ follows from Proposition~\ref{prop E}.
$(5)\Rightarrow (4)$. If the essential spectrum of $T$ intersects $\overline{B(0,r_{\mathcal{P}})}$, then by assumption, there exists $\nu\in \sigma_e(T)$ such that $\text{Ran}(T-\nu Id)$ is dense and ${\sup_{\lambda\in \Lambda}|P_{\lambda}(\nu)|\le 1}$. Since $(P_{\lambda}(T))_{\lambda\in \Lambda}$ satisfies (CHC), we deduce from Proposition~\ref{Mk} that
$(P_{\lambda}(T))_{\lambda\in \Lambda}$ satisfies the Criterion~$(M_k)$ for common hypercyclic subspaces.
$(4)\Rightarrow (3)$ and $(3)\Rightarrow (2)$ follow from Theorem~\ref{Mk com} and Theorem~\ref{CritM0}.
$(2)\Rightarrow (1)$ is immediate.
\end{proof}
In the case of scalar multiples of an operator, Theorem~\ref{Charac com} gives us the following two interesting characterizations.
\begin{cor} \label{C:a<lambda<b}
Let $T\in L(X)$, where $X$ is a separable infinite-dimensional complex Banach space,
and
let $0< a<b$.
If $\{ \lambda T\}_{\lambda\in ]a,b[}$ satisfies (CHC),
then the following assertions are equivalent:
\begin{enumerate}
\item the family $\{ \lambda T\}_{\lambda\in ]a,b[}$ has a common hypercyclic subspace;
\item for any $\lambda\in ]a,b[$, the operator $\lambda T$ has a hypercyclic subspace;
\item the essential spectrum of $T$ intersects $\overline{B(0,\frac{1}{b})}$.
\end{enumerate}
\end{cor}
\begin{proof}
Let $\mathcal{P}=\{\lambda Id:\lambda\in ]a,b[\}$. We remark that $r_{\mathcal{P}}=\frac{1}{b}$ and
for any $\mu\in \overline{B(0,r_{\mathcal{P}})}=\overline{B(0,\frac{1}{b})}$,
we have
\[\sup_{\lambda\in ]a,b[}|\lambda \mu|=b|\mu|\le 1.\]
Moreover, since $\{ \lambda T\}_{\lambda\in ]a,b[}$ satisfies (CHC), we know in particular that
$\lambda T$ is hypercyclic for any $\lambda\in ]a,b[$ and therefore that $\text{Ran}(T-\mu Id)$ is dense for any $\mu\in \mathbb C$~\cite{Bourdon}. The conclusion follows by Theorem~\ref{Charac com}.
\end{proof}
\begin{cor} \label{C:Spectral}
Let $T\in L(X)$, where
$X$ is a separable infinite-dimensional complex Banach space, and let $a> 0$.
If $\{ \lambda T\}_{\lambda\in ]a,\infty[}$ satisfies (CHC),
then the following assertions are equivalent:
\begin{enumerate}
\item the family $\{ \lambda T\}_{\lambda\in ]a,\infty[}$ has a common hypercyclic subspace;
\item for any $\lambda> a$, the operator $\lambda T$ has a hypercyclic subspace;
\item $0\in \sigma_{e}(T)$.
\end{enumerate}
\end{cor}
\begin{proof}
If $\mathcal{P}=\{\lambda Id:\lambda\in ]a,\infty[\}$, we have $r_{\mathcal{P}}=0$ and thus $\overline{B(0,r_{\mathcal{P}})}=\{0\}$. We conclude as previously by using Theorem~\ref{Charac com}.
\end{proof}
We note that Corollary~\ref{C:Spectral} applies to several interesting examples by Gallardo and Partington \cite[Section 3]{Gallardo} of families of scalar multiples of either adjoint multiplication operators on the Hardy space, adjoint multiplier operators on weighted $\ell^2$-spaces, adjoint convolution operators on weighted $L^2(0,\infty)$ spaces, or adjoint composition operators on the reduced Hardy space.
This is because any family $\{ \lambda T \}_{|\lambda | >a}$ satisfying the assumptions of \cite[Theorem 2.1]{Gallardo} must satisfy (CHC). We illustrate with the following.
\begin{example}
Suppose $\varphi\in H^\infty(\mathbb{D})$ is univalent and bounded below on $\mathbb{T}$ but is not an outer function. Suppose that zero belongs to the boundary of $\varphi (\mathbb{D})$.
Then
\[
\{ \lambda M_\varphi^* \}_{ | \lambda | > a }
\]
has a common hypercyclic subspace on $H^2(\mathbb{D})$, where $a=\| \frac{1}{\varphi } {\|}_{ L_{ \mathbb{T} }^\infty }$ .
\end{example}
\begin{proof} We know that the spectrum of $M_\varphi^*$ is the closure of $\{ \overline{\varphi (z) }: \ z\in \mathbb{D} \}$ and that its essential spectrum is
$
\sigma_e(M_\varphi^*)=\partial \{ \overline{\varphi (z) }: \ z\in \mathbb{D} \}
$
thanks to $\varphi$ being univalent, see \cite[Proposition 4.4(b)]{Godefroy}. Hence the assumption
$0\in \partial \varphi (\mathbb{D})$ gives that $0\in \sigma_{e}(M_\varphi^*)$, and the conclusion follows by Corollary~\ref{C:Spectral}.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,309 |
\section{Introduction}
\label{sec:introduction}
\IEEEPARstart{F}{ollowing} its remarkable success in artificial intelligence, learning from data is becoming a popular topic in various engineering domains \cite{Hou_2013}. This concept is by no means a new idea for control engineering. The system identification paradigm has been widely used in control applications, where data are used to fit an a priori parametrized model structure \cite{LjungBook2}. The control strategy is then designed with the identified nominal model based on the certainty equivalence principle \cite{chow1975analysis}, or with the uncertainty model based on robust/stochastic control frameworks, possibly with adaptive elements also learned from online data \cite{goodwin2014adaptive}.
However, this conventional scheme of learning dynamical systems is challenged by increasing complexity of systems and the large amount of data available. In particular, a low-dimensional model structure that is suitable to design compact, closed-form control strategies can be very hard and costly to obtain for complex systems \cite{Hjalmarsson_2005}. In fact, low dimensionality is not really required in modern optimization-based control frameworks, and may limit the predictive power of big data \cite{Sutton_2019}. Therefore, alternative paths are investigated to facilitate control design directly from raw measurement data from dynamical systems. Early attempts in this direction include unfalsified control \cite{Safonov_1997}, iterative feedback tuning \cite{Hjalmarsson_1998}, and virtual reference feedback tuning \cite{Campi_2002}. Reinforcement learning techniques are also widely pursued in this area \cite{Recht_2019}, including policy search \cite{lagoudakis2003least} and approximate dynamic programming \cite{powell2007approximate}.
The above approaches typically avoid predicting the behavior of systems explicitly but aim at the control strategy directly. On the other hand, it is possible to directly replace the conventional model with a data-driven predictor \cite{Van_Waarde_2020}. In the seminal work from Willems \textit{et al. }\cite{Willems_2005}, a single input-output trajectory of the linear system is shown to be able to characterize all possible trajectories of length up to the order of persistency of excitation by constructing Hankel matrices from data. This result is known as the Willems' fundamental lemma. With this result, the behavior of the system can be simulated and thus controlled by selecting a suitable combination of sections from the known trajectory that satisfies the initial condition constraints \cite{Markovsky_2008, DePersis_2020, vanWaarde_2020}.
This observation is especially suitable for optimal trajectory tracking. In this regard, model predictive control (MPC) is known to be very effective when an accurate model of the system is available \cite{camacho2016model}. From the Willems' fundamental lemma, the future output prediction step in MPC can be achieved by using known trajectories of the system directly, instead of an explicit model. This data-driven alternative to MPC algorithms, known as data-enabled predictive control (DeePC) \cite{Coulson_2019}, has lead to multiple successful applications \cite{Huang_2019, Coulson_2019_reg, Alpago_2020} with a recent stability and robustness proof \cite{Berberich_2020}.
Although this data-driven approach is often promoted as model-free, it effectively provides a model of system trajectory based on known data. With a low-rank approximation, this approach directly leads to the intersection algorithm in subspace identification where state-space models can be derived \cite{Moonen_1989}. The main differences of the data-driven approach compared to the conventional model-based methods are: 1) the model is implicit and as such there is no closed-form solution of the model in general; 2) the model is over-parametrized in that it does not impose any assumption on the system structure other than linearity. In this paper, this implicit and over-parametrized model is called the data-driven model.
However, it is well-known that when data are noisy, over-parametrized models may lead to high variances and overfitting \cite{Geman_1992}. In data-driven modeling, finding a combination of known trajectory sections that give reliable prediction is an ill-conditioned problem for datasets with stochastic noise. In current data-driven control schemes empirical regularizers \cite{Berberich_2020,Coulson_2019} or least-norm problems \cite{Favoreel_1999,Huang_2019,Sedghizadeh_2018} are introduced to select a reasonable combination for prediction. Yet, it is not clear what is the optimal way to combine a large set of known trajectory sections to achieve the most reliable prediction. In addition, the hyperparameters in the empirical regularizers are difficult to tune.
Another application of data-driven modeling is to simulate the system response \cite{Markovsky_2005b,Quintana_Carapia_2020}. The main advantage of applying this approach in system identification is that it gives the correct estimation of nonparametric models in the noise-free case.
Again in this scenario, the best practice for solving the underdetermined linear system in the Willems' fundamental lemma in the noisy case is not understood. For computational simplicity, the Moore-Penrose pseudoinverse solution that solves the least-norm problem is often the default choice \cite{Huang_2019}, leading to the data-driven subspace predictor \cite{Sedghizadeh_2018}.
As can be seen from the above discussion, one of the central questions in data-driven approaches based on the Willems' fundamental lemma is how to obtain the optimal data-driven model from a large noise-corrupted dataset \cite{van2020noisy}. Therefore, in the first part of the paper, we propose a maximum likelihood estimation (MLE) framework to estimate such an optimal model with noise in both offline data and online measurements. This optimal model is named the signal matrix model\ (SMM). This framework optimizes the combination of offline trajectories by maximizing the conditional probability of observing the predicted output trajectory and the measured past outputs.
In addition, a preconditioning strategy is proposed based on singular value decomposition (SVD) to compress the data matrix such that the complexity of the algorithm is only dependent on the trajectory length to be simulated, to the benefit of large datasets.
In the second part of the paper, we present two scenarios where the SMM\ leads to effective algorithms: 1) estimating finite impulse response (FIR) models with the kernel-based method in system identification; and 2) obtaining a tuning-free data-driven predictive control scheme. In the first scenario, the kernel-based method is used to derive a maximum a posteriori (MAP) estimator based on the impulse response estimate learned from data. Carefully designed kernels encode appropriate characteristics of the dynamical system. Conventionally, the data-based estimate is obtained by least-squares regression. In this work, it is replaced by the signal matrix model\ simulated with an impulse, which guarantees that an unbiased estimate is obtained. Results show that the model fitting is enhanced when the transient response is unknown or the truncation error of the impulse response is large.
In the second scenario, we replace the prediction part in the DeePC algorithm with the SMM. This predictor is shown to be superior to the pseudoinverse subspace predictor in predictive control. The main advantage of the proposed algorithm is that it avoids the difficult hyperparameter tuning problem in regularized DeePC. The control performance of the proposed algorithm is shown to be better than the DeePC algorithm with optimal hyperparameters when the noise is significant, and similar in the low noise scenario.
The remainder of the paper is organized as follows. Section~\ref{sec:2} defines the notions and preliminaries used in the paper. Section~\ref{sec:3} reviews the Willems' fundamental lemma for finite-dimensional systems and its application to deterministic systems. Section~\ref{sec:4} derives the signal matrix model\ with MLE and presents an optimal data-driven simulation algorithm with a data compression scheme. This model is then applied to two problems: Section~\ref{sec:5} identifies an FIR model using SMM\ simulation and the kernel-based regularization; Section~\ref{sec:6} applies the SMM\ predictor in predictive control. Section~\ref{sec:7} concludes the paper.
\section{Notation \& Preliminaries}
\label{sec:2}
For a vector $x$, the weighted $l_2$-norm $(x^\mathsf{T}Px)^{\frac{1}{2}}$ is denoted by $\norm{x}_P$. The symbol $\pazocal{N}(\mu,\Sigma)$ indicates a Gaussian distribution with mean $\mu$ and covariance $\Sigma$. The expectation and the covariance of a random vector $x$ are denoted by $\mathbb{E}(x)$ and $\text{cov}(x)$ respectively. For a matrix $X$, the vectorization operator stacks its columns in a single vector and is denoted by $\text{vec}(X)$; $X^\dagger$ indicates the Moore-Penrose pseudoinverse; $(X)_{i,j}$ denotes the $(i,j)$-th entry of $X$. The symbol $\mathbb{S}_{++}^{n}$ indicates the set of $n$-by-$n$ positive definite matrix. For a sequence of matrices $X_1,\dots,X_n$, we denote $[X_1^\mathsf{T}\ \dots\ X_n^\mathsf{T}]^\mathsf{T}$ by $\text{col}\left(X_1,\dots,X_n\right)$. Given a signal $x:\mathbb{Z}\to \mathbb{R}^n$, its trajectory from $k$ to $k+N-1$ is denoted as $(x_i)_{i=k}^{k+N-1}$, and in the vector form as $\mathbf{x}=\text{col}(x_k,\dots,x_{k+N-1})$.
Consider a discrete-time linear time-invariant (LTI) system with output noise, given by
\begin{equation}
\begin{cases}
x_{t+1}&=\ A x_t+B u_t,\\
\hfil y_t&=\ C x_t + D u_t + w_t,
\label{eq:sys}
\end{cases}
\end{equation}
where $x_t \in \mathbb{R}^{n_x}$, $u_t \in \mathbb{R}^{n_u}$, $y_t \in \mathbb{R}^{n_y}$, $w_t \in \mathbb{R}^{n_y}$ are the states, inputs, outputs, and output noise respectively. The system is denoted compactly by $(A,B,C,D)$. The pair $(A,B)$ is controllable if $[B\ AB\ \dots\ A^{n_x-1}B]$ has full row rank; the pair $(A,C)$ is observable if $\text{col}\left(C,CA,\dots,CA^{n_x-1}\right)$ has full column rank. The system is minimal if $(A,B)$ is controllable and $(A,C)$ is observable.
The notion of persistency of excitation is defined as follows.
\begin{defi}
A signal trajectory $(x_i)_{i=0}^{N-1}\in \mathbb{R}^n\times\{0,\dots,N-1\}$ is said to be persistently exciting of order $L$ if the block Hankel matrix
\begin{equation}
X=\begin{bmatrix}
x_0&x_1&\cdots&x_{M-1}\\
x_1&x_2&\cdots&x_{M}\\
\vdots&\vdots&\ddots&\vdots\\
x_{L-1}&x_L&\cdots&x_{N-1}
\end{bmatrix}\in \mathbb{R}^{Ln\times M}
\end{equation}
\label{def:1}
\end{defi}
has full row rank, where $M=N-L+1$ \cite{Willems_2005}.
Intuitively, this definition means that sections of length $L$ of the trajectory span $\mathbb{R}^{Ln}$. When used as the input to a linear dynamical system, it can thus excite all the possible behaviors of the system in a window of length $L$. A necessary condition of Definition~\ref{def:1} is $N\geq L(n+1)-1$, which gives a lower bound on the trajectory length.
\section{Data-Driven Modeling for Finite-Dimensional Systems}
\label{sec:3}
In this section, we first review the Willems' fundamental lemma and a few related results in a state-space formulation, followed by an overview of deterministic data-driven simulation and control.
\subsection{Willems' Fundamental Lemma}
Built on the notion of the persistency of excitation, the Willems' fundamental lemma shows that all the behavior of a linear system can be captured by a single persistently exciting trajectory of the system when no noise is present. This lemma was originally proposed in the context of behavioral system theory \cite{Willems_2005,willems1997introduction}, where systems are characterized by the subspace that contains all possible trajectories. It was later reformulated in the state-space context \cite{DePersis_2020,vanWaarde_2020}. In the state-space formulation, the output trajectory is unique to a particular input trajectory when a past sufficiently exciting input-output trajectory is specified as the initial condition. The length of the past trajectory should not be shorter than the state dimension. This idea has strong ties with the intersection algorithm in subspace identification \cite{Moonen_1989}, where a low-order subspace of the data matrices that corresponds to a low state dimension is sought.
We summarize the available results on data-driven modeling based on the Willems' fundamental lemma for finite-dimensional LTI systems, which are the foundation for the data-driven methods discussed in this paper. These results hold exactly only when the system is noise-free, i.e., $\forall i, w_i=0$.
\begin{theorem}
Consider a finite-dimensional LTI system $(A,B,C,D)$. Let $(u_i^d, x_i^d, y_i^d)_{i=0}^{N-1}$ be a triple of input-state-output trajectory of the system. If the pair $(A,B)$ is controllable and the input is persistently exciting of order $(L+n_x)$, then
\begin{itemize}
\item[(a)]the matrix
\begin{equation}
\begin{bmatrix}X\\U\end{bmatrix}:=\begin{bmatrix}
x_0^d&x_1^d&\cdots&x_{M-1}^d\\\hline
u_0^d&u_1^d&\cdots&u_{M-1}^d\\
\vdots&\vdots&\ddots&\vdots\\
u_{L-1}^d&u_L^d&\cdots&u_{N-1}^d\\
\end{bmatrix}
\end{equation} has full row rank (Corollary 2 in \cite{Willems_2005}, Theorem 1(i) in \cite{vanWaarde_2020}, Lemma 1 in \cite{DePersis_2020});
\item[(b)] the pair $(u_i,y_i)_{i=0}^{L-1}$ is an input-output trajectory of the system iff there exists $g$, such that
\begin{equation}
\begin{bmatrix}u_0\\\vdots\\u_{L-1}\\\hline y_0\\\vdots\\y_{L-1}\end{bmatrix}=\begin{bmatrix}U\\Y\end{bmatrix}g:=\begin{bmatrix}
u_0^d&u_1^d&\cdots&u_{M-1}^d\\
\vdots&\vdots&\ddots&\vdots\\
u_{L-1}^d&u_L^d&\cdots&u_{N-1}^d\\\hline
y_0^d&y_1^d&\cdots&y_{M-1}^d\\
\vdots&\vdots&\ddots&\vdots\\
y_{L-1}^d&y_L^d&\cdots&y_{N-1}^d\\
\end{bmatrix}g
\end{equation}
(Theorem 1 in \cite{Willems_2005}, Theorem 1(ii) in \cite{vanWaarde_2020}, Lemma 2 in \cite{DePersis_2020});
\item[(c)] $\text{rank}\left(\begin{bmatrix}U\\Y\end{bmatrix}\right)=n_x+n_u L$ (Theorem 2 in \cite{Moonen_1989}).
\end{itemize}
Furthermore, if the system is minimal,
\begin{itemize}
\item[(d)] the immediate past input-output trajectory $(u_i,y_i)_{i=-L_0}^{-1}$ uniquely determines the initial condition $x_0$, if $L_0\geq n_x$ (Lemma 1 in \cite{Markovsky_2008});
\item[(e)] let $L'=L-L_0$, then $(y_i)_{i=0}^{L'-1}$ is the unique output trajectory of the system with past trajectory $(u_i,y_i)_{i=-L_0}^{-1}$ and input trajectory $(u_i)_{i=0}^{L'-1}$, iff there exists $g$, such that
\begin{equation}
\begin{bmatrix}u_{-L_0}\\\vdots\\u_{L'-1}\\\hline y_{-L_0}\\\vdots\\y_{L'-1}\end{bmatrix}=\begin{bmatrix}U\\Y\end{bmatrix}g:=\begin{bmatrix}
u_0^d&u_1^d&\cdots&u_{M-1}^d\\
\vdots&\vdots&\ddots&\vdots\\
u_{L-1}^d&u_L^d&\cdots&u_{N-1}^d\\\hline
y_0^d&y_1^d&\cdots&y_{M-1}^d\\
\vdots&\vdots&\ddots&\vdots\\
y_{L-1}^d&y_L^d&\cdots&y_{N-1}^d\\
\end{bmatrix}g
\label{eqn:fund}
\end{equation}
(Proposition 1 in \cite{Markovsky_2008}).
\end{itemize}
\label{thm:1}
\end{theorem}
In Theorem \ref{thm:1}, parts (a) and (b) state the original Willems' fundamental lemma; part (c) draws the connection between data-driven modeling and subspace identification methods; and parts (d) and (e) further give the uniqueness of the trajectory by fixing a sufficiently long past trajectory. Parts (d) and (e) allow the formulation to be applied in simulation and predictive control.
\subsection{Deterministic Data-Driven Simulation and Control}
In the noise-free case, the system can be simulated solely based on a known trajectory by applying Theorem \ref{thm:1}(e) \cite{Markovsky_2005b}. Define
\begin{equation}
U_p = \begin{bmatrix}
u_0^d&u_1^d&\cdots&u_{M-1}^d\\
\vdots&\vdots&\ddots&\vdots\\
u_{L_0-1}^d&u_{L_0}^d&\cdots&u_{M+L_0-2}^d\\
\end{bmatrix}\in\mathbb{R}^{L_0 n_u\times M},
\label{eqn:Up}
\end{equation}
\begin{equation}
U_f = \begin{bmatrix}
u_{L_0}^d&u_{L_0+1}^d&\cdots&u_{M+L_0-1}^d\\
\vdots&\vdots&\ddots&\vdots\\
u_{L-1}^d&u_L^d&\cdots&u_{N-1}^d\\
\end{bmatrix}\in\mathbb{R}^{L' n_u\times M},
\label{eqn:Uf}
\end{equation}
\begin{equation}
\mathbf{u}_{\text{ini}} = \begin{bmatrix}u_{-L_0}\\\vdots\\u_{-1}\end{bmatrix}\in\mathbb{R}^{L_0 n_u},
\mathbf{u} = \begin{bmatrix}u_{0}\\\vdots\\u_{L'-1}\end{bmatrix}\in\mathbb{R}^{L' n_u},
\end{equation}
and similarly for $Y_p$, $Y_f$, $\mathbf{y}_{\text{ini}}$, and $\mathbf{y}$. Then we interpret (\ref{eqn:fund}) as an implicit model of the system trajectory parametrized by $g$, namely
\begin{subequations}
\begin{empheq}[left=\empheqlbrace]{align}
\mathbf{u}=U_f g, \label{eqn:uu}\\
\mathbf{y}=Y_f g, \label{eqn:yy}
\end{empheq}
\label{eqn:uy}%
\end{subequations}
subject to the initial condition requirement
\begin{subequations}
\begin{empheq}[left=\empheqlbrace]{align}
\mathbf{u}_{\text{ini}} = U_p g, \label{eqn:up}\\
\mathbf{y}_{\text{ini}} = Y_p g. \label{eqn:yp}
\end{empheq}
\label{eqn:uyp}%
\end{subequations}
Thus, the system can be simulated by means of a two-step approach with $g$ as the intermediate parameter as shown in Algorithm~\ref{al:1}. The system identification process is performed online for a particular input by estimating $g$. This algorithm effectively gives an implicit model of the system in the form of
\begin{equation}
\mathbf{y}=f(\mathbf{u};\mathbf{u}_{\text{ini}},\mathbf{y}_{\text{ini}}, U_p,U_f,Y_p,Y_f),
\label{eqn:ddmodel}
\end{equation}
where $U_p,U_f,Y_p,Y_f$ are offline data that describe the behaviors of the system, and $\mathbf{u}_{\text{ini}}, \mathbf{y}_{\text{ini}}$ are online data that estimate the initial condition.
\begin{algorithm}[htb]
\caption{Noise-free data-driven simulation \cite{Markovsky_2005b}}
\begin{algorithmic}[1]
\State \textbf{Given: }$U_p,U_f,Y_p,Y_f$.
\State \textbf{Input: }$\mathbf{u}_{\text{ini}},\mathbf{y}_{\text{ini}},\mathbf{u}$.
\State Solve the linear system
\begin{equation}
\begin{bmatrix}\mathbf{u}_{\text{ini}}\\\mathbf{y}_{\text{ini}}\\\mathbf{u}\end{bmatrix}=\begin{bmatrix}U_p\\Y_p\\U_f\end{bmatrix}g
\label{eqn:con}
\end{equation}
for $g$.
\State \textbf{Output: }$\mathbf{y}=Y_f g$.
\end{algorithmic}
\label{al:1}
\end{algorithm}
Multiple control algorithms have been developed based on this model structure \cite{DePersis_2020,Markovsky_2008}. In this work, we focus on the optimal trajectory tracking problem, which optimizes the following control cost over a horizon of length $L'$ at each time instant $t$ \cite{camacho2016model}:
\begin{equation}
J_\text{ctr}(\mathbf{u},\mathbf{y})=\sum_{k=0}^{L'-1}\left(\norm{y_k-r_{t+k}}_Q^2+\norm{u_k}_R^2\right),
\end{equation}
where $\mathbf{r}$ is the reference trajectory, and $Q$ and $R$ are the output and the input cost matrices respectively \cite{camacho2016model}. At each time instant, the first entry in the newly optimized input trajectory is applied to the system in a receding horizon fashion.
Algorithm~\ref{al:1} can be applied as the predictor in place of the model-based predictor in conventional MPC algorithms. This leads to the following optimization problem
\begin{equation}
\begin{matrix}
\underset{\mathbf{u},\mathbf{y},g}{\text{minimize}} & J_\text{ctr}(\mathbf{u},\mathbf{y})\\
\text{subject to} & (\ref{eqn:uy}),(\ref{eqn:uyp}),\mathbf{u} \in \pazocal{U}, \mathbf{y} \in \pazocal{Y},
\end{matrix}
\label{eqn:deepc0}
\end{equation}
where $\pazocal{U}$ and $\pazocal{Y}$ are the constraint sets of the inputs and the outputs respectively. Vectors $\mathbf{u}_{\text{ini}}$ and $\mathbf{y}_{\text{ini}}$ are the immediate past input-output measurements online. This method is known as the unregularized DeePC algorithm \cite{Coulson_2019}.
\section{Maximum Likelihood Data-Driven Model: Signal Matrix Model}
\label{sec:4}
The linear system (\ref{eqn:con}) is highly underdetermined when a large dataset is available. When the data are noise-free, this parameter estimation problem is trivial, where any solution to (\ref{eqn:con}) gives an exact output model of the system, according to Theorem~\ref{thm:1}(e).
However, the problem of finding the model (\ref{eqn:ddmodel}) becomes ill-conditioned when the data are noisy. In this case, Theorem~\ref{thm:1}(c) is no longer satisfied. Instead, $\text{col}(U,Y)$ has full row rank almost surely. If we still follow Algorithm~\ref{al:1}, any output trajectory $\mathbf{y}$ can be obtained by choosing different solutions to (\ref{eqn:con}). In fact, Theorem \ref{thm:1}(e) does not hold exactly for the noisy case, so satisfying condition (\ref{eqn:con}) is not guaranteed to be statistically optimal. An empirical remedy to this problem is to use the Moore-Penrose pseudoinverse solution of $g$, namely
\begin{equation}
g_{\text{pinv}}=\begin{bmatrix}U_p\\Y_p\\U_f\end{bmatrix}^\dagger\begin{bmatrix}\mathbf{u}_{\text{ini}}\\\mathbf{y}_{\text{ini}}\\\mathbf{u}\end{bmatrix},
\label{eqn:pinv}
\end{equation}
which solves the least-norm problem
\begin{equation}
\begin{matrix}
\underset{g}{\text{minimize}} & \norm{g}_2^2 \\
\text{subject to} & (\ref{eqn:con}).
\end{matrix}
\label{eqn:ln}
\end{equation}
This solution is known as the subspace predictor related to the prediction error method \cite{Huang_2019,Sedghizadeh_2018}. However, this predictor fails to appropriately encode the effects of noise in the data matrices. To the best of our knowledge, there is no existing statistical framework for estimating $g$. In what follows, we will derive a maximum likelihood estimator of $g$. As opposed to existing algorithms, this estimator obtains a statistically optimal data-driven model for systems with noise. Since this model is expressed purely in terms of matrices of signal trajectories, we name this model the signal matrix model. For simplicity of exposition, the results in the section are stated for the single-input single-output case, but they seamlessly hold for the multiple-input multiple-output case.
\subsection{Derivation of the Maximum Likelihood Estimator}
In this work, we consider a scenario where the output errors are i.i.d. Gaussian noise for both offline data and online data, i.e.,
\begin{equation}
y_i^d = y_i^{d,0}+w_i^d,\,(w_i^d)_{i=0}^{N-1}\sim \pazocal{N}(0,\sigma^2\mathbb{I}),
\end{equation}
\begin{equation}
\mathbf{y}_{\text{ini}} = \mathbf{y}_{\text{ini}}^{0}+\mathbf{w}_{p},\,\mathbf{w}_{p}\sim \pazocal{N}(0,\sigma_{p}^2\mathbb{I}).
\end{equation}
\begin{remark}
The formulation can be extended to other noise models in a straightforward way, including correlated noise, input noise, and alternative noise distributions. For example, when the noise is Laplacian, it would lead to an $l_1$-norm penalization in the estimator similar to the regularizer proposed in \cite{Coulson_2019}.
\end{remark}
\vspace{1em}
Under this noise model, the equations (\ref{eqn:uu}) and (\ref{eqn:up}) still hold exactly, but the past output equation (\ref{eqn:yp}) includes noise on both sides, which leads to a total least squares problem. In this work, the maximum likelihood interpretation of the total least squares problem is used \cite{Markovsky_2007}.
Define
\begin{equation}
\hat{\mathbf{y}}=\begin{bmatrix}\epsilon_y\\\mathbf{y}\end{bmatrix}=Yg-\begin{bmatrix}\mathbf{y}_{\text{ini}}\\\mathbf{0}\end{bmatrix},
\label{eqn:yhat}
\end{equation}
where $\epsilon_y:=Y_pg-\mathbf{y}_{\text{ini}}$ is the residual of the past output relation (\ref{eqn:yp}). Then we want to construct an estimator that maximizes the conditional probability of observing the realization $\hat{\mathbf{y}}$ corresponding to the available data given $g$. Applying vectorization on $Yg$ in (\ref{eqn:yhat}), we have
\begin{equation}
\hat{\mathbf{y}}=\left(g^\mathsf{T}\otimes \mathbb{I}\right)\text{vec}(Y)-\begin{bmatrix}\mathbf{y}_{\text{ini}}\\\mathbf{0}\end{bmatrix},
\end{equation}
where we make use of the property of the Kronecker product
\begin{equation}
\text{vec}(ABC) = (C^\mathsf{T}\otimes A)\text{vec}(B).
\end{equation}
Denote the noise-free version of $Y_p$ and $Y_f$ by $Y_p^0$ and $Y_f^0$ respectively. Then for a given $g$, we have
\begin{equation}
\begin{aligned}
\mathbb{E}(\hat{\mathbf{y}}|g)&=\mathbb{E}(Y)g-\begin{bmatrix}\mathbb{E}(\mathbf{y}_{\text{ini}})\\\mathbf{0}\end{bmatrix}=\begin{bmatrix}Y_p^0g-\mathbf{y}_{\text{ini}}^0\\Y_f^0g\end{bmatrix}=\begin{bmatrix}\mathbf{0}\\Y_f^0g\end{bmatrix},\\
\text{cov}(\hat{\mathbf{y}}|g)&=\left(g^\mathsf{T}\otimes \mathbb{I}\right)\Sigma_{yd}\left(g\otimes \mathbb{I}\right)+\begin{bmatrix}\sigma_p^2\mathbb{I}&\mathbf{0}\\\mathbf{0}&\mathbf{0}\end{bmatrix}=: \Sigma_y,
\label{eqn:stats}
\end{aligned}
\end{equation}
where $\Sigma_{yd}=\text{cov}\left(\text{vec}(Y)\right)$. According to the noise model of $\left(y_i^d\right)$ and accounting for the Hankel structure of $Y$, we have
\begin{equation}
\left(\Sigma_{yd}\right)_{i,j}=\begin{cases}
\sigma^2,&\left(\text{vec}(Y)\right)_i=\left(\text{vec}(Y)\right)_j\\
0,&\text{otherwise}
\end{cases}.
\label{eqn:ry}
\end{equation}
Then, $\Sigma_y$ can be calculated as
\begin{equation}
\left(\Sigma_y\right)_{i,j}=\sigma^2\sum_{k=1}^{M-|i-j|}g_k g_{k+|i-j|}+\begin{cases}\sigma_p^2,&i=j\leq L_0\\0,&\text{otherwise}\end{cases}.
\label{eqn:py}
\end{equation}
where $g_k$ denotes the $k$-th entry of $g$. The derivation is given in Appendix~\ref{sec:app1}. Thus, due to the linearity of the normal distribution, we have
\begin{equation}
\hat{\mathbf{y}}|g\sim \pazocal{N}\left(\begin{bmatrix}\mathbf{0}\\Y_f^0g\end{bmatrix},\Sigma_y\right),
\label{eqn:condprob}
\end{equation}
which has the distribution
\begin{equation}
\begin{split}
p(\hat{\mathbf{y}}|g)=&(2\pi)^{-\frac{L}{2}}\det{(\Sigma_y)}^{-\frac{1}{2}}\\
&\exp{\left(-\frac{1}{2}\begin{bmatrix}Y_pg-\mathbf{y}_{\text{ini}}\\Y_fg-Y_f^0g\end{bmatrix}^\mathsf{T}\Sigma_y^{-1}\begin{bmatrix}Y_pg-\mathbf{y}_{\text{ini}}\\Y_fg-Y_f^0g\end{bmatrix}\right)}.
\end{split}
\label{eqn:prob1}
\end{equation}
Note that here the true output data matrix $Y_f^0$ is also unknown, and can be estimated with the maximum likelihood approach.
In this way, we are ready to derive the signal matrix model\ by solving the following optimization problem.
\begin{equation}
\underset{g\in\pazocal{G},Y_f^0}{\text{minimize}}\ -\log p(\hat{\mathbf{y}}|g,Y_f^0),
\label{eqn:opt1}
\end{equation}
where $\pazocal{G}$ is the parameter space defined by the known noise-free input trajectory, namely
\begin{equation}
\pazocal{G} = \left\{g\in \mathbb{R}^M\left|\begin{bmatrix}
U_p\\U_f
\end{bmatrix}g=\begin{bmatrix}
\mathbf{u}_{\text{ini}}\\\mathbf{u}
\end{bmatrix}\right.\right\}.
\end{equation}
Substituting (\ref{eqn:prob1}) into (\ref{eqn:opt1}), we have the equivalent optimization problem,
\begin{equation}
\begin{split}
\underset{g\in\pazocal{G},Y_f^0}{\text{minimize}}\ &\text{logdet}(\Sigma_y(g))\\[-1em]
&+\begin{bmatrix}Y_pg-\mathbf{y}_{\text{ini}}\\Y_fg-Y_f^0g\end{bmatrix}^\mathsf{T}\Sigma_y^{-1}(g)\begin{bmatrix}Y_pg-\mathbf{y}_{\text{ini}}\\Y_fg-Y_f^0g\end{bmatrix}.
\end{split}
\label{eqn:opt2}
\end{equation}
It is easy to see that the optimal value of $Y_f^0$ is $Y_f$ regardless of the choice of $g$. So (\ref{eqn:opt2}) is equivalent to
\begin{equation}
\underset{g\in\pazocal{G}}{\text{minimize}}\ \text{logdet}(\Sigma_y(g))+\begin{bmatrix}Y_pg-\mathbf{y}_{\text{ini}}\\\mathbf{0}\end{bmatrix}^\mathsf{T}\Sigma_y^{-1}(g)\begin{bmatrix}Y_pg-\mathbf{y}_{\text{ini}}\\\mathbf{0}\end{bmatrix}.
\label{eqn:opt0}
\end{equation}
In this objective function, the first term indicates how accurate the output estimates are. The second term represents how much the estimate deviates from the past output observations.
\subsection{Iterative Computation of the Estimator}
To find a computationally efficient algorithm to solve (\ref{eqn:opt0}), we relax the problem and solve it with sequential quadratic programming (SQP) \cite{boggs1995sequential}. First, the covariance matrix $\Sigma_y$ is approximated with its diagonal part, denoted by $\bar{\Sigma}_y$, i.e.,
\begin{equation}
\left(\bar{\Sigma}_y\right)_{i,j} = \begin{cases}
\left(\Sigma_y\right)_{i,j},&i=j\\
0,&i\neq j
\end{cases}.
\end{equation}
\begin{remark}
This approximation holds exactly when the data matrices are constructed by truncating $(u_i^d, y_i^d)_{i=0}^{N-1}$ into sections of length $L$ with no overlap, or using multiple independent trajectories of length $L$, instead of forming Hankel structures. This construction is known as the Page matrix \cite{damen1982approximate} and it was shown in \cite{vanWaarde_2020} that similar results to Theorem~\ref{thm:1} still hold for Page matrices. The Hankel construction is able to use the data more efficiently, but leads to complex noise correlation, which is reflected in the non-diagonal structure of $\Sigma_y$. The comparison between the Hankel construction and the Page construction is, however, beyond the scope of this paper. See \cite{IM-FD,damen1982approximate} for more on this topic.
\end{remark}
\begin{remark}
This approximation gives an upper bound on the log-det terms. According to Hadamard's inequality, since $\Sigma_y\in \mathbb{S}_{++}^{L}$,
\begin{equation}
\det{(\Sigma_y)}\leq \prod_{i=1}^{L}\left(\Sigma_y\right)_{i,i}.
\end{equation}
So we have,
\begin{equation}
\mathrm{logdet}(\Sigma_y(g))\leq\mathrm{logdet}(\bar{\Sigma}_y(g)).
\end{equation}
\end{remark}
\vspace{1em}
In this way, problem (\ref{eqn:opt0}) is approximated as
\begin{equation}
\begin{split}
\underset{g\in\pazocal{G}}{\text{minimize}}\ L'\log\left(\norm{g}_2^2\right)&+L_0\log\left(\sigma^2\norm{g}_2^2+\sigma_p^2\right)\\
&+\dfrac{1}{\sigma^2\norm{g}_2^2+\sigma_p^2}\norm{Y_pg-\mathbf{y}_{\text{ini}}}_2^2.
\end{split}
\label{eqn:optapp}
\end{equation}
This problem can be readily solved by SQP. For each iteration, the following quadratic programming problem is solved.
\begin{equation}
\begin{matrix}
\quad\quad g^{k+1}=&
\text{arg}\underset{g}{\text{min}} & \lambda(g^k)\norm{g}_2^2+\norm{Y_pg-\mathbf{y}_{\text{ini}}}_2^2 \\
&\text{subject to} & \begin{bmatrix}
U_p\\U_f
\end{bmatrix}g=\begin{bmatrix}
\mathbf{u}_{\text{ini}}\\\mathbf{u}
\end{bmatrix},
\end{matrix}
\label{eqn:optsqp}
\end{equation}
where
\begin{equation}
\lambda(g^k) = \dfrac{L'\sigma_p^2}{\norm{g^k}_2^2}+L \sigma^2.
\end{equation}
The objective function in (\ref{eqn:optapp}) is approximated by a quadratic function around $g^k$, making use of the local expansion $\log x\approx \log x_0 + \frac{1}{x_0}(x-x_0)$. The optimality conditions of (\ref{eqn:optsqp}) are:
\begin{equation}
\begin{bmatrix}
F(g^k)&U^\mathsf{T}\\
U&\mathbf{0}
\end{bmatrix}\begin{bmatrix}g^{k+1}\\\nu^{k+1}\end{bmatrix}=
\begin{bmatrix}Y_p^\mathsf{T}\mathbf{y}_{\text{ini}}\\\tilde{\mathbf{u}}\end{bmatrix},
\end{equation}
where $\tilde{\mathbf{u}}=\text{col}(\mathbf{u}_{\text{ini}},\mathbf{u})$, $F(g^k)=\lambda(g^k)\mathbb{I}+Y_p^\mathsf{T}Y_p$, and $\nu_{k+1}\in\mathbb{R}^L$ is the Lagrange multiplier. The closed-form solution is thus given by
\begin{equation}
\begin{aligned}
g^{k+1} &= \left(F^{-1}-F^{-1}U^\mathsf{T}(U F^{-1}U^\mathsf{T})^{-1}UF^{-1}\right)Y_p^\mathsf{T}\mathbf{y}_{\text{ini}}\\&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad+F^{-1}U^\mathsf{T}(U F^{-1}U^\mathsf{T})^{-1}\tilde{\mathbf{u}}\\
&=:\pazocal{P}(g^k)\mathbf{y}_{\text{ini}}+\pazocal{Q}(g^k)\tilde{\mathbf{u}}.
\end{aligned}
\label{eqn:clsol}
\end{equation}
\subsection{Maximum Likelihood Data-Driven Simulation}
Based on the derived maximum likelihood estimator of $g$, the step of solving the linear system (\ref{eqn:con}) in Algorithm~\ref{al:1} can be replaced by solving the SQP problem (\ref{eqn:optsqp}). For simulation, the SQP problem can be initialized at the pseudoinverse solution $g_\text{pinv}$ (\ref{eqn:pinv}). This leads to the following algorithm for maximum likelihood data-driven simulation.
\begin{algorithm}[htb]
\caption{Maximum likelihood data-driven simulation: the signal matrix model}
\begin{algorithmic}[1]
\State \textbf{Given: }$U_p,U_f,Y_p,Y_f,\sigma,\sigma_p,\epsilon$.
\State \textbf{Input: }$\mathbf{u}_{\text{ini}},\mathbf{y}_{\text{ini}},\mathbf{u}$.
\State $k \leftarrow 0$, $g^0 \leftarrow g_{\text{pinv}}$ from (\ref{eqn:pinv})
\Repeat
\State Calculate $g^{k+1}$ with (\ref{eqn:clsol}).
\State $k \leftarrow k+1$
\Until{\norm{g^k-g^{k-1}}<\epsilon\norm{g^{k-1}}}
\State \textbf{Output: }$g_{\text{SMM}}=g^k$, $\mathbf{y}=Y_f g^k$.
\end{algorithmic}
\label{al:2}
\end{algorithm}
This algorithm gives the signal matrix model\ in the form of (\ref{eqn:ddmodel}). The approximate maximum likelihood estimator (\ref{eqn:optsqp}) has the same $\norm{g}_2^2$ penalization term as the least-norm problem (\ref{eqn:ln}). However, the estimate $g_{\text{SMM}}$ does not lie in the solution space of the underdetermined system (\ref{eqn:con}). The total least squares structure in (\ref{eqn:yp}) leads to the penalization term $\norm{Y_pg-\mathbf{y}_{\text{ini}}}_2^2$ in place of the hard constraint in (\ref{eqn:con}).
\subsection{Preconditioning of Data Matrices}
\label{sec:precon}
In data-driven applications, it is usually assumed that abundant data are available, i.e., $N\gg L$. Under this scenario, the dimension of the parameter vector $g\in \mathbb{R}^M$, which needs to be optimized online, would be much larger than the length of the predicted output trajectory. This leads to high online computational complexity even to estimate a very short trajectory. On the other hand, only $2L$ independent basis vectors are needed to describe all the possible input-output trajectories of length $L$. It is possible to precondition the data matrices such that only $2L$ basis trajectories are used.
To do this, we propose the following strategy based on the SVD to compress the data such that the dimension of the parameter vector $g$ is $2L$ regardless of the raw data length. Let
\begin{equation}
\begin{bmatrix}U\\Y\end{bmatrix}=WSV^\mathsf{T}\in\mathbb{R}^{2L\times M}
\end{equation}
be the SVD of the data matrix. Define the compressed data matrices $\tilde{U}_p,\tilde{Y}_p\in\mathbb{R}^{L_0\times 2L}$ and $\tilde{U}_f,\tilde{Y}_f\in\mathbb{R}^{L'\times 2L}$ such that
\begin{equation}
\text{col}\left(\tilde{U}_p,\tilde{U}_f,\tilde{Y}_p,\tilde{Y}_f\right)=WS_{2L}\in\mathbb{R}^{2L\times 2L},
\end{equation}
where $S_{2L}$ is the first $2L$ columns of $S$.
It is shown in the following proposition that Algorithm~\ref{al:2} with compressed data matrices obtains exactly the same output trajectory $\mathbf{y}$ as with raw data matrices.
\begin{prop}
Let the simulated trajectories with data matrices $(U_p, Y_p, U_f, Y_f)$ and $(\tilde{U}_p, \tilde{Y}_p, \tilde{U}_f, \tilde{Y}_f)$ from Algorithm~\ref{al:2} be $\mathbf{y}$ and $\tilde{\mathbf{y}}$ respectively. Then we have
\begin{equation}
\tilde{\mathbf{y}}=\mathbf{y}.
\end{equation}
\label{prop:1}
\end{prop}
\begin{proof} Define transformed data matrices $\bar{U}_p$, $\bar{Y}_p$, $\bar{U}_f$, and $\bar{Y}_f$ such that
\begin{equation}
\text{col}\left(\bar{U}_p, \bar{Y}_p, \bar{U}_f, \bar{Y}_f\right)=WS.
\end{equation}
Then the relations between the data matrices are given by
\begin{equation}
\begin{aligned}
\text{col}\left(U_p, Y_p, U_f, Y_f\right)&=\text{col}\left(\tilde{U}_p,\tilde{U}_f,\tilde{Y}_p,\tilde{Y}_f\right)V_{2L}^\mathsf{T},\\
\text{col}\left(U_p, Y_p, U_f, Y_f\right)&=\text{col}\left(\bar{U}_p, \bar{Y}_p, \bar{U}_f, \bar{Y}_f\right)V^\mathsf{T},\\
\text{col}\left(\bar{U}_p, \bar{Y}_p, \bar{U}_f, \bar{Y}_f\right)&=\left[\text{col}\left(\tilde{U}_p,\tilde{U}_f,\tilde{Y}_p,\tilde{Y}_f\right)\ \mathbf{0}\right].
\end{aligned}
\end{equation}
where $V_{2L}$ denotes the first $2L$ columns of $V$.
Denote the variables with the compressed data matrices by a tilde, and the variables with the transformed data matrices by a bar. Since $V_{2L}^\mathsf{T}V_{2L}=\mathbb{I}$, we have
\begin{equation}
g_{\text{pinv}}=V_{2L}\,\tilde{g}_{\text{pinv}}.
\end{equation}
This leads to $\norm{g_{\text{pinv}}}_2^2=\norm{\tilde{g}_{\text{pinv}}}_2^2$, and thus $\lambda(g^0)=\lambda(\tilde{g}^{0})$.
Suppose at the $k$-th iteration, $\lambda(g^k)=\lambda(\tilde{g}^{k})$. Due to the orthogonality of $V$ and the sparsity structure of $\bar{U}$ and $\bar{Y}_p$, we have
\begin{equation}
g^{k+1} = V\bar{g}^{k+1},\,\bar{g}^{k+1}=\left[\tilde{g}^{k+1}\ \mathbf{0}\right].
\end{equation}
This leads to
\begin{equation}
g^{k+1} = V_{2L}\,\tilde{g}^{k+1},\,\norm{g^{k+1}}_2^2=\norm{\tilde{g}^{k+1}}_2^2.
\label{eqn:proofnorm}
\end{equation}
Thus for all $k$, we have $g^{k} = V_{2L}\,\tilde{g}^{k}$. Therefore, the simulated trajectory satisfies
\begin{equation}
\mathbf{y}=Y_f g^k=\tilde{Y}_f \tilde{g}^{k}=\tilde{\mathbf{y}}.
\end{equation}
\end{proof}
\begin{remark}
It can be seen from the proof that $\bar{\Sigma}_y(g)=\bar{\Sigma}_y(\tilde{g})$. So with compressed data matrices, the output trajectory estimate has the same covariance as the raw data matrices when Page matrices are used, and the same diagonal components when Hankel matrices are used.
\end{remark}
The signal matrix model\ (Algorithm~\ref{al:2}), together with the above data compression scheme, can be applied to various problems in system identification and control. In the following two sections, two approaches are discussed to apply this model to problems in kernel-based identification and receding horizon predictive control.
\section{Kernel-Based Identification With the Signal Matrix Model}
\label{sec:5}
We propose here a kernel-based system identification method that identifies an FIR model of the system \cite{Pillonetto_2010}. By combining the kernel-based regularization with the signal matrix model, model fitting is improved compared to the conventional kernel method based on least squares, when the truncation error is large or the past inputs are unknown.
\subsection{Impulse Response Estimation}
Consider the problem of estimating the impulse response model $(h_i)_{i=0}^\infty$ of a system from data. The output $y_t$ is given by
\begin{equation}
y_t = \sum^\infty_{i=0} h_i u_{t-i}.
\end{equation}
The conventional approach is to formulate a linear regression to estimate a finite truncation of the impulse response
\begin{equation}
\underbrace{\begin{bmatrix}y^d_0\\y^d_1\\\vdots\\y^d_{N-1}\end{bmatrix}}_{y_N}=\underbrace{\begin{bmatrix}u^d_0&u^d_{-1}&\cdots&u^d_{1-n}\\u^d_1&u^d_{0}&\cdots&u^d_{2-n}\\\vdots&\vdots&\ddots&\vdots\\u^d_{N-1}&u^d_{N-2}&\cdots&u^d_{N-n}\end{bmatrix}}_{\Phi_N}\underbrace{\begin{bmatrix}h_0\\h_1\\\vdots\\h_{n-1}\end{bmatrix}}_{h},
\end{equation}
where $n$ is the length of the impulse response to be estimated. The regression problem can then be solved by least squares. There are two main assumptions underlying this formulation: 1) the truncation error of the finite impulse response is negligible, i.e., $h_i\approx 0$ for all $i\geq n$; and 2) the past input trajectory $(u_i^d)_{i=1-n}^{-1}$ is known. With these two assumptions, the least-squares solution is known to be the best unbiased estimator when i.i.d. Gaussian output noise is present \cite{LjungBook2}.
However, these assumptions may not be satisfied in practice. When the internal dynamics matrix $A$ has a large condition number, a very long impulse response sequence is needed to remove the truncation error even for a low-order system. In this case, the least-squares algorithm may become impractical due to limited data length and/or computation power. If the truncation error is not negligible, the estimator is not correct, i.e., in the noise-free case, the estimate does not coincide with the true system. When the past inputs are unknown, the first $(n-1)$ output measurements cannot be used. In non-parametric estimation, the size $n$ is usually comparable to $N$, in which case the data efficiency is substantially affected. Similar assumptions are required for non-parametric methods in the frequency domain as well, where periodic input signals are assumed and a sufficiently long dataset is needed to avoid aliasing.
In this work, we propose using the signal matrix model\ to estimate the impulse response with
\begin{equation}
\mathbf{u}_{\text{ini}}=\mathbf{0},\,\mathbf{y}_{\text{ini}}=\mathbf{0},\,\mathbf{u}=\text{col}(1,\mathbf{0}),\,\sigma_p=0,\,L'=n.
\label{eqn:ddimp}
\end{equation}
Then the output trajectory $\mathbf{y}$ is an estimate of the impulse response $h$ of length $n$ of the system \cite{Markovsky_2005b}. It is noted that this approach requires neither of the assumptions for the least-squares method. Instead of requiring a long past input sequence, this approach uses a small part of the data to estimate the initial condition of each trajectory section. In fact, the estimator is correct and unbiased for an arbitrary length $n$ and unknown past inputs as shown in Theorem~\ref{thm:1}(e), as long as the persistency of excitation condition is satisfied.
\subsection{Kernel-Based Regularization}
The FIR model is widely used in kernel-based identification methods, where the estimates learned from data are combined with a prior of the impulse response that encodes typical system characteristics including low-complexity and exponential stability. Consider an estimate from data with the following statistics
\begin{equation}
h\sim\pazocal{N}(\hat{h},\Sigma_d),
\end{equation}
and assume a prior distribution on $h$ defined by the kernel $\Sigma_k$ as
\begin{equation}
h\sim \pazocal{N}(\mathbf{0},\Sigma_k).
\end{equation}
The kernel $\Sigma_k$ is usually parametrized by a small number of design parameters $\eta$ \cite{Chen_2018}.
Then the optimal linear combination of the kernel-based prior and the data-based estimator in the minimum mean square error sense is given by
\begin{equation}
h\sim\pazocal{N}(K\hat{h}, \Sigma_\text{opt}),
\label{eqn:postdist}
\end{equation}
where $K=\Sigma_k\left(\Sigma_k+\Sigma_d\right)^{-1}$ is the Kalman gain and $\Sigma_\text{opt}=\Sigma_k-\Sigma_k\left(\Sigma_k+\Sigma_d\right)^{-1}\Sigma_k$. The least-squares solution is typically used as the data-based estimate with
\begin{equation}
\hat{h} = \left(\Phi_N^\mathsf{T}\Phi_N\right)^{-1}\Phi_N^\mathsf{T}y_N,\,\Sigma_d = \sigma^2\left(\Phi_N^\mathsf{T}\Phi_N\right)^{-1}.
\label{eqn:LS}
\end{equation}
The posterior mean of (\ref{eqn:postdist}) then gives the well-known closed-form solution of the kernel-based regularization \cite{Chen_2012}
\begin{equation}
h_\text{LS-ker}=K\hat{h}=\left(\Phi_N^\mathsf{T}\Phi_N+\sigma^2\Sigma_k^{-1}\right)^{-1}\Phi_N^\mathsf{T}y_N,
\label{eqn:lsker}
\end{equation}
which is also the solution to the following regularized least-squares problem
\begin{equation}
h_\text{LS-ker}=\text{arg}\underset{h}{\text{min}}\ \sigma^{-2}\norm{y_N-\Phi_N h}_2^2+h^\mathsf{T}\Sigma_k^{-1}h.
\end{equation}
In fact, any stochastic estimator of the impulse response can be combined with the kernel prior. In this work, to overcome the limitations of the least-squares regression discussed in the previous subsection, we apply the impulse response trajectory $\mathbf{y}$ simulated with the signal matrix model\ as the data-based estimate. Then according to (\ref{eqn:stats}), we have
\begin{equation}
\hat{h} = \mathbf{y},\,\Sigma_d = \Sigma_{y_f},
\end{equation}
where $\Sigma_{y_f}\in\mathbb{S}_{++}^{L'}$ denotes the last $L'$ rows and columns of $\Sigma_y$.
Then the kernel-based estimate of the FIR model with the signal matrix model\ is given by
\begin{equation}
\begin{split}
h_\text{SMM-ker}&=\text{arg}\underset{h}{\text{min}}\ (h-\mathbf{y})^\mathsf{T}\Sigma_{y_f}^{-1}(h-\mathbf{y})+h^\mathsf{T}\Sigma_k^{-1}h\\
&=\Sigma_k\left(\Sigma_k+\Sigma_{y_f}\right)^{-1}\mathbf{y}.
\end{split}
\end{equation}
To determine the hyperparameters in the kernel design $\Sigma_k=\Sigma_k(\eta)$, the empirical Bayes method is applied \cite{Pillonetto_2010}. This method maximizes the marginal probability of observing $\hat{h}$ given the kernel design parameters $\eta$:
\begin{equation}
\hat{h}|\eta \sim \pazocal{N}(\mathbf{0},\Sigma_k(\eta)+\Sigma_{y_f}).
\end{equation}
This leads to
\begin{equation}
\begin{split}
\eta^*&=\text{arg}\underset{\eta}{\text{min}}\ -\log p(\hat{h}|\eta)\\
&=\text{arg}\underset{\eta}{\text{min}}\ \text{logdet}(\Sigma_k(\eta)+\Sigma_{y_f})+\mathbf{y}^\mathsf{T}(\Sigma_k(\eta)+\Sigma_{y_f})^{-1}\mathbf{y}.
\end{split}
\label{eqn:eta}
\end{equation}
When the dimension of $\eta$ is small, this optimization problem can be solved by grid search, otherwise nonlinear optimization can be employed.
Building upon the previous steps, the following algorithm is proposed to identify the FIR model with the data-driven kernel-based method. Note that when $\sigma_p=0$, the adaptive weighting $\lambda(g^k)$ is fixed. Therefore, only one iteration is needed in Algorithm~\ref{al:2}.
\begin{algorithm}[htb]
\caption{Kernel-based impulse response estimation with the signal matrix model}
\begin{algorithmic}[1]
\State \textbf{Given: }$U_p,U_f,Y_p,Y_f,\sigma,\epsilon$.
\State $\mathbf{u}_{\text{ini}}\leftarrow\mathbf{0},\,\mathbf{y}_{\text{ini}}\leftarrow\mathbf{0},\,\mathbf{u}\leftarrow\text{col}(1,\mathbf{0}),\,\sigma_p\leftarrow0$
\State Compute $\mathbf{y}$ and $g_{\text{SMM}}$ by Algorithm~\ref{al:2}.
\State Calculate $\Sigma_{y_f}(g_{\text{SMM}})$ by (\ref{eqn:py}).
\State Solve (\ref{eqn:eta}) for $\eta^*$.
\State \textbf{Output: }$h_\text{SMM-ker}=\Sigma_k(\eta^*)\left(\Sigma_k(\eta^*)+\Sigma_{y_f}(g_{\text{SMM}})\right)^{-1}\mathbf{y}$.
\end{algorithmic}
\label{al:3}
\end{algorithm}
\subsection{Numerical Results}
In this subsection, the proposed algorithm is tested against least-squares-based estimates by applying it to numerical examples. The tuned/correlated (TC) kernel structure
\begin{equation}
\left(\Sigma_k\right)_{i,j} = \alpha\min\left(\beta^i,\beta^j\right),\,\eta=\text{col}(\alpha,\beta)
\end{equation}
is used \cite{Chen_2012}. We compare the following four algorithms: 1) \textit{LS}: least-squares estimate (\ref{eqn:LS}), 2) \textit{LS-TC}: kernel-based/least-squares estimate (\ref{eqn:lsker}) with the TC kernel, 3) \textit{SMM}: signal matrix model\ estimate (Algorithm~\ref{al:2} with (\ref{eqn:ddimp})), and 4) \textit{SMM-TC}: kernel-based/signal matrix model\ estimate (Algorithm~\ref{al:3}) with the TC kernel. The parameters used in the simulation are
$
N=50,\,L_0=4,\,n=L'=11,\,\sigma^2=0.01.
$
The identification data are generated with unit i.i.d Gaussian input signals. For each case, 100 Monte Carlo simulations are conducted.
In the first example, we consider the following fourth-order LTI system tested in \cite{Pillonetto_2010}
\begin{equation}
G_1(z) = \dfrac{0.1159(z^3+0.5z)}{z^4-2.2z^3+2.42z^2-1.87z+0.7225}.
\label{eqn:sys1}
\end{equation}
This system is relatively slow. The truncation error is significant when $n=11$ is selected. First, the data-based estimates \textit{LS} and \textit{SMM} are compared under the noise-free case, and the results are shown in Figure~\ref{fig:1}(a). It can be clearly seen that the least-squares estimator is not correct due to the presence of truncation errors, whereas the SMM\ estimator is correct. When the noise is present, all four algorithms are compared in Figure~\ref{fig:1}(b). The estimators with the signal matrix model\ have smaller variance compared to the least-squares ones.
\begin{figure}[htb]
\centerline{\includegraphics[width=\columnwidth]{1new.eps}}\vspace{-0.5em}
\centerline{\footnotesize (a) Noise-free case}
\centerline{\includegraphics[width=\columnwidth]{2new.eps}}\vspace{-0.5em}
\centerline{\footnotesize (b) Noisy case with $\sigma^2=0.01$}
\caption{Comparison of impulse response estimation with truncation errors. Colored area: estimates within two standard deviation.}
\label{fig:1}
\end{figure}
In the second example, we focus on the effect of unknown past inputs by investigating a faster LTI system used in \cite{Pillonetto_2010}
\begin{equation}
G_2(z) = \dfrac{0.9183z}{z^2+0.24z+0.36}.
\end{equation}
In this case, the truncation error is already negligible at $n=11$, but we assume the past inputs are unknown. The results of the estimation are illustrated in Figure~\ref{fig:2}. As can be observed from Figure~\ref{fig:2}(a), the result of the \textit{SMM} algorithm is more accurate than the \textit{LS} algorithm, especially for the first four coefficients. Regularizing with the TC kernel prior improves the estimation quality for the tail of the impulse response as shown in Figure~\ref{fig:2}(b).
\begin{figure}[htb]
\centerline{\includegraphics[width=\columnwidth]{3new.eps}}\vspace{-0.5em}
\centerline{\footnotesize (a) Comparison between SMM\ and least-squares estimates}\vspace{0.5em}
\centerline{\includegraphics[width=\columnwidth]{4new.eps}}\vspace{-0.5em}
\centerline{\footnotesize (b) Effect of kernel regularization on the SMM\ method}
\caption{Comparison of impulse response estimation with unknown past inputs. Colored area: estimates within two standard deviation.}
\label{fig:2}
\end{figure}
To quantitatively assess the performance of different algorithms, we quantify the model fitting by
\begin{equation}
W=100\cdot \left(1-\left[\frac{\sum_{i=1}^{n}(h_i-\hat{h}_i)^2}{\sum_{i=1}^{n}(h_i-\bar{h})^2}\right]^{1/2}\right),
\end{equation}
where $h_i$ are the true impulse response coefficients, $\hat{h}_i$ are the estimated coefficients, and $\bar{h}$ is the mean of the true coefficients. The boxplots of model fitting for both examples are plotted in Figure~\ref{fig:3}. For comparison, the case with known past inputs is also plotted for example 2. The \textit{SMM} and \textit{SMM-TC} algorithms perform better than the \textit{LS} and \textit{LS-TC} algorithms when the truncation error is large or the past inputs are unknown. However, when both assumptions of the least squares are satisfied, the least-squares-based methods perform slightly better than the methods with the signal matrix model. This is due to the fact that part of the data is used to estimate the initial condition in Algorithm~\ref{al:2}, whereas it is known for the least-squares method.
\begin{figure}[htb]
\centering
\begin{tabular}{ c @{\hspace{5pt}} c }
\includegraphics[width=1.65in]{5new} &
\includegraphics[width=1.65in]{6new} \\
\footnotesize (a) Example 1&
\footnotesize (b) Example 2
\end{tabular}
\caption{Comparison of model fitting for both examples with 100 simulations. In (b), yellow: unknown past input, cyan: known past inputs.}
\label{fig:3}
\end{figure}
\section{Data-Driven Predictive Control With the Signal Matrix Model}
\label{sec:6}
In this section, the signal matrix model\ is used as the predictor in receding horizon predictive control. As discussed in Section~\ref{sec:4}, the predictor in the unregularized DeePC problem (\ref{eqn:deepc0}) becomes ill-conditioned when noise is present. In the following subsections, we will present two existing methods to remedy this problem, and compare the control performance with the optimal predictor proposed in this paper.
\subsection{Pseudoinverse and Regularized Algorithms}
There are mainly two types of existing algorithms to extend (\ref{eqn:deepc0}) to the noisy case: the data-driven subspace predictive control and the regularized DeePC algorithm.
The subspace predictive control approach \cite{Sedghizadeh_2018} uses the pseudoinverse solution of $g$ in the predictor instead of the underdetermined linear equality constraints as follows
\begin{equation}
\begin{aligned}
\underset{\mathbf{u},\mathbf{y}}{\text{minimize}} & \quad\quad\quad\quad\quad\quad\quad J_\text{ctr}(\mathbf{u},\mathbf{y})\\
\text{subject to} &\quad \mathbf{y}=Y_f\, g_\text{pinv}(\mathbf{u;\mathbf{u}_{\text{ini}},\mathbf{y}_{\text{ini}}}),\mathbf{u} \in \pazocal{U}, \mathbf{y} \in \pazocal{Y},
\end{aligned}
\label{eqn:subpc}
\end{equation}
where $g_\text{pinv}(\cdot)$ is defined in (\ref{eqn:pinv}). Multiple applications have been studied with similar algorithms (e.g., \cite{Hallouzi_2008,Kadali_2003}). However, as discussed in Section~\ref{sec:4}, $g_\text{pinv}(\cdot)$ is not guaranteed to be an effective choice of $g$ for systems with noise.
The regularized DeePC algorithm \cite{Coulson_2019} adds additional regularization terms in the objective in order to regularize both the norm of $g$ and the slack variable needed to satisfy (\ref{eqn:yp}):
\begin{equation}
\begin{aligned}
\underset{\mathbf{u},\mathbf{y},g,\hat{\mathbf{y}}_{\text{ini}}}{\text{minimize}} & \quad J_\text{ctr}(\mathbf{u},\mathbf{y})+\lambda_g\norm{g}_p^p+\lambda_y\norm{\hat{\mathbf{y}}_{\text{ini}}-\mathbf{y}_{\text{ini}}}_p^p \\
\text{subject to} &\quad\ \ \begin{bmatrix}
U_p\\Y_p\\U_f\\Y_f
\end{bmatrix}g=\begin{bmatrix}
\mathbf{u}_{\text{ini}}\\\hat{\mathbf{y}}_{\text{ini}}\\\mathbf{u}\\\mathbf{y}
\end{bmatrix}, \mathbf{u} \in \pazocal{U}, \mathbf{y} \in \pazocal{Y},
\end{aligned}
\label{eqn:deepc}
\end{equation}
where $p$ is usually selected as 1 or 2. This algorithm can be interpreted as an MPC algorithm acting on the implicit parametric model structure (\ref{eqn:uy}) and (\ref{eqn:uyp}), where the objective is a trade-off between the control performance objective $J_{\text{ctl}}$ and the parameter estimation objective
\begin{equation}
J_{\text{id,DeePC}}:=\lambda\norm{g}_p^p+\norm{\hat{\mathbf{y}}_{\text{ini}}-\mathbf{y}_{\text{ini}}}_p^p,\,\lambda = \frac{\lambda_g}{\lambda_y}.
\label{eqn:iddeepc}
\end{equation}
The set of underdetermined model parameters $(g,\hat{\mathbf{y}}_{\text{ini}})$ are then estimated adaptively in the MPC algorithm. This algorithm is also shown to be effective in multiple applications (e.g., \cite{Coulson_2019_reg,Huang_2019}). However, there are two main problems associated with it. First, the model parameters are optimized in an optimistic manner in that as regularization terms, the estimator is biased towards those that predict better control performance. However, the system behaviors should be independent of the control task. Second, tuning of the regularization parameters is a very hard problem. To the best of our knowledge, there is no practical approach proposed to tune $\lambda_g$ and $\lambda_y$ beforehand, and unfortunately the control performance is known to be very sensitive to the regularization parameters \cite{Huang_2019}.
\begin{remark}
The same data compression scheme as discussed in \ref{sec:precon} is applicable to these two algorithms as well.
\end{remark}
\subsection{An Optimal Tuning-Free Approach}
To address the concerns regarding the two existing methods discussed in the previous subsection, we propose a receding horizon predictive control scheme with the signal matrix model\ as the predictor. This directly leads to
\begin{equation}
\begin{aligned}
\underset{\mathbf{u},\mathbf{y}}{\text{minimize}} & \quad\quad\quad\quad\quad\quad\quad J_\text{ctr}(\mathbf{u},\mathbf{y})\\
\text{subject to} &\quad \mathbf{y}=Y_f\, g_\text{SMM}(\mathbf{u;\mathbf{u}_{\text{ini}},\mathbf{y}_{\text{ini}}}),\mathbf{u} \in \pazocal{U}, \mathbf{y} \in \pazocal{Y},
\end{aligned}
\label{eqn:mlepc0}
\end{equation}
where $g_\text{SMM}(\cdot)$ is obtained by Algorithm~\ref{al:2}. However, unlike the pseudoinverse predictor where $g_{\text{pinv}}(\cdot)$ is linear with respect to $\mathbf{u}$, the maximum likelihood predictor $g_\text{SMM}(\cdot)$ involves an iterative algorithm which cannot be expressed as an equality constraint explicitly.
To solve this problem, we notice that the $l_2$-norm of $g$ does not change much throughout the receding horizon control, and the algorithm is only iterative with respect to $\norm{g}_2^2$. So in a receding horizon context, it makes sense to warm-start the optimization problem from the the $g$-value at the previous time instant. Then, $g_{\text{SMM}}(\cdot)$ can be closely approximated by the solution of (\ref{eqn:clsol}) after the first iteration, i.e.,
\begin{equation}
g^t(\mathbf{u;\mathbf{u}_{\text{ini}},\mathbf{y}_{\text{ini}}},g^{t-1})=\pazocal{P}(g^{t-1})\,\mathbf{y}_{\text{ini}}+\pazocal{Q}(g^{t-1})\,\tilde{\mathbf{u}},
\end{equation}
where, with an abuse of notation, $g^t(\cdot)$ denotes the approximation of $g_{\text{SMM}}(\cdot)$ with one iteration at time instant $t$. In this way, the SMM\ predictor is approximated by a linear equality constraint that can be efficiently solved within a quadratic program. Thus, the proposed approach solves the following optimization problem at each time step
\begin{equation}
\begin{aligned}
\underset{\mathbf{u},\mathbf{y}}{\text{minimize}} &\quad\ J_\text{ctr}(\mathbf{u},\mathbf{y})\\
\text{subject to} &\quad\begin{aligned}\mathbf{y}&=Y_f\, g^t,\\g^t&=\pazocal{P}(g^{t-1})\,\mathbf{y}_{\text{ini}}+\pazocal{Q}(g^{t-1})\,\tilde{\mathbf{u}},\\\mathbf{u} &\in \pazocal{U}, \mathbf{y} \in \pazocal{Y}.\end{aligned}
\end{aligned}
\label{eqn:mlepc}
\end{equation}
The parameter estimation part (\ref{eqn:iddeepc}) in regularized DeePC has the same form as the maximum likelihood estimator (\ref{eqn:optsqp}) with $p=2$, which leads to the predictor in (\ref{eqn:mlepc}). However, our proposed method avoids the aforementioned shortcomings with the regularized DeePC algorithm: 1) the parameter estimation part is isolated from the control performance objective and placed in the predictor itself; 2) the problem of hyperparameter tuning is avoided by deriving the coefficients statistically, which requires only information about the noise levels of the offline data $\sigma^2$ and the online measurements $\sigma_p^2$.
\subsection{Numerical Results}
In this subsection, we compare the control performance of three receding horizon predictive control algorithms: 1) \textit{Sub-PC}: subspace predictive control (\ref{eqn:subpc}), 2) \textit{DeePC}: regularized DeePC (\ref{eqn:deepc}), and 3) \textit{SMM-PC}: predictive control with the signal matrix model\ (\ref{eqn:mlepc}). In \textit{DeePC}, the algorithm is tested on a nine-point logarithmic grid of $\lambda_g$ between 10 and 1000. In this example, the control performance is found to not be sensitive to the value of $\lambda_y$, so a fixed value of $\lambda_y=1000$ is used. In \textit{SMM-PC}, it is assumed that the noise levels $\sigma^2$ and $\sigma_p^2$ are known. To benchmark the performance, we also consider the ideal MPC algorithm (denoted by \textit{MPC}), where both the true state-state model and the noise-free state measurements are available. The result of this benchmark algorithm is thus deterministic and gives the best possible control performance with receding horizon predictive control.
In this example, we consider the LTI system (\ref{eqn:sys1}). Unless otherwise specified, the following parameters are used in the simulation
\begin{equation*}
N=200,\,L_0=4,\,L'=11,\,\sigma^2=\sigma_p^2=1,\,Q=R=1.
\end{equation*}
No input and output constraints are enforced, i.e., $\pazocal{U}=\mathbb{R}^{L' n_u}$ and $\pazocal{Y}=\mathbb{R}^{L' n_y}$. A sinusoidal reference trajectory $r_t = 0.5\sin\left(\frac{\pi}{10} t\right)$ is to be tracked. The offline data are generated with unit i.i.d Gaussian input signals. For each case, 100 Monte Carlo simulations are conducted. In each run, 60 time steps are simulated. The control performance is assessed by the true stage cost over all time steps, i.e.,
\begin{equation}
J=\sum_{k=0}^{N_c-1}\left(\norm{y_k^0-r_k}_Q^2+\norm{u_k}_R^2\right),
\end{equation}
where $N_c=60$ and $y_k^0$ is the noise-free measurement at time $k$. When comparing the closed-loop performance, the best choices of $\lambda_g$ in \textit{DeePC} are selected with an oracle for each run as plotted in Figure~\ref{fig:7} for different noise levels. It can be seen that, even for the same control task, the optimal value of this hyperparameter is not only sensitive to the noise level but also to the specific realization of the noise. This makes the tuning process difficult in practice.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{11}
\caption{The best choice of $\lambda_g$ in \textit{DeePC} for different noise levels. Colored area: values within one standard deviation.}
\label{fig:7}
\end{figure}
The optimization problems are all formulated as quadratic programming problems and solved by MOSEK. The computation time for all three algorithms is similar. The effect of the proposed data compression scheme in Section~\ref{sec:precon} is illustrated in Figure~\ref{fig:8} with the example of \textit{SMM-PC}. By applying the preconditioning, the computational complexity no longer depends on the data size $N$.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{120}
\caption{Average computation time of \textit{SMM-PC} with and without the data compression scheme.}
\label{fig:8}
\end{figure}
The closed-loop input-output trajectories of different control algorithms are plotted in Figure~\ref{fig:4}. The closed-loop trajectories are characterized by the range within one standard deviation of the Monte-Carlo simulation. The \textit{SMM-PC} algorithm obtains the closest match with the benchmark trajectory \textit{MPC}. \textit{Sub-PC} applies more aggressive control inputs which results in much larger input costs, whereas the control strategy in \textit{DeePC} is more conservative which results in significantly larger tracking errors. The \textit{SMM-PC} trajectories also have the smallest variance. The boxplot of the control performance measure $J$ is shown in Figure~\ref{fig:5}, which confirms that \textit{SMM-PC} performs better than \textit{Sub-PC} and \textit{DeePC} in this control task.
\begin{figure*}[htb]
\centering
\includegraphics[width=7.16in]{7new}
\footnotesize (a) Output trajectory \hspace{23em} (b) Input trajectory
\caption{Comparison of closed-loop input-output trajectories with different control algorithms. Colored area: trajectories within one standard deviation.}
\label{fig:4}
\end{figure*}
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{8new2}
\caption{Comparison of the control performance in terms of total stage costs $J$ with different control algorithms with 100 simulations ($\sigma^2=\sigma_p^2=1, N=200$).}
\label{fig:5}
\end{figure}
The effects of different offline data sizes $N$ and noise levels $\sigma^2,\sigma_p^2$ are investigated in Figure~\ref{fig:6}. As shown in Figure~\ref{fig:6}(a), the control performance of \textit{SMM-PC} is not sensitive to the number of datapoints and performs uniformly better among the three algorithms. In fact, good performance is already obtained at only $N=50$. \textit{DeePC} does not perform very well with small data sizes but gets a similar performance to \textit{SMM-PC} when $N\geq 600$. \textit{Sub-PC} cannot achieve a satisfying result even with a large data size because a larger $N$ can only average out the effect of offline noise $\sigma^2$ but not online measurement noise $\sigma_p^2$, which is problematic to deal with in \textit{Sub-PC}. Figure~\ref{fig:6}(b) shows that all three algorithms perform similarly at low noise levels as they are all stochastic variants of the noise-free algorithm (\ref{eqn:deepc0}). \textit{SMM-PC} obtains slightly worse results under low noise levels ($\sigma^2=\sigma_p^2 < 0.3$) compared to the optimally tuned \textit{DeePC} with an oracle, but the performance improvement is significant for higher noise levels.
\begin{figure}[htb]
\centering
\includegraphics[width=\columnwidth]{9new2}
\footnotesize (a) Performance as a function of the number of datapoints
\includegraphics[width=\columnwidth]{10new2}
\footnotesize (b) Performance as a function of the noise level
\caption{The effect of different offline data sizes and noise levels on the control performance.}
\label{fig:6}
\end{figure}
\section{Conclusions}
\label{sec:7}
In this work, we propose a novel statistical framework to estimate data-driven models from large noise-corrupted datasets. This is formulated as a maximum likelihood estimation problem. The problem is solved efficiently by approximating it as a sequential quadratic program with data compression. This framework extends the current works on data-driven methods to noisy data by providing an optimal solution to the underdetermined implicit model structure and establishing the signal matrix model.
With the signal matrix model, two approaches in kernel-based system identification and receding horizon control are developed. They obtain an impulse response estimate with less restrictive assumptions and an effective tuning-free data-driven receding horizon control algorithm respectively. The results from these two approaches demonstrate that the proposed framework can improve the state-of-the-art methods in both data-driven simulation and control in the presence of noisy data.\\
\appendices
\section{Derivation of $\Sigma_y$}
\label{sec:app1}
Let $\zeta_i\in \mathbb{R}^L$, $i=1,\dots,LM$ be the $i$-th column of $\left(g^\mathsf{T}\otimes \mathbb{I}\right)$, $\pazocal{S}=\left\{(i,j)\left|\left(\text{vec}(Y)\right)_i=\left(\text{vec}(Y)\right)_j\right.\right\}$, and $\Sigma_{y1}=\left(g^\mathsf{T}\otimes \mathbb{I}\right)\Sigma_{yd}\left(g\otimes \mathbb{I}\right)$. According to (\ref{eqn:ry}), we have
\begin{equation}
\Sigma_{y1}=\sigma^2\sum_{(i,j)\in\pazocal{S}}\zeta_i\zeta_j^\mathsf{T}.
\end{equation}
Let the $i$-th and the $j$-th entries of $\text{vec}(Y)$ correspond to the $(q,r)$-th and the $(s,t)$-th entries of $Y$ respectively, i.e., $i=(r-1)L+q$, $j=(t-1)L+s$. From the Hankel structure, the pair $(i,j)\in \pazocal{S}$ iff $q+r=s+t$. According to the structure of $\left(g^\mathsf{T}\otimes \mathbb{I}\right)$, we have $\zeta_i=g_r\mathbf{e}_q$, $\zeta_j=g_t\mathbf{e}_s$, where $\mathbf{e}_q\in \mathbb{R}^L$ is the unit vector with $q$-th non-zero entry, and similarly for $\mathbf{e}_s$. Thus,
\begin{equation}
\Sigma_{y1}=\sigma^2\sum_{q+r=s+t}g_r g_t\mathbf{e}_q\mathbf{e}_s^\mathsf{T}.
\end{equation}
So the $(q,s)$-th entry of $\Sigma_{y1}$ is given by
\begin{equation}
\left(\Sigma_{y1}\right)_{q,s}=\sigma^2\sum_{q+r=s+t}g_r g_t,
\end{equation}
which directly leads to (\ref{eqn:py}).
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,443 |
A FINANCIAL advisor from East Renfrewshire has been inspired to go into business for himself after overcoming a brain tumour.
Gregory Jenkins, 40, has just launched his own firm – SM Financial Advice – to offer advice on mortgages and insurance.
And the subject of protection is especially close to his heart, as he knows from first-hand experience how important it can be when the worst happens.
Gregory told the Barrhead News: "I've been in the industry for 15 years, during which I've been heavily involved in protection, talking to people about what would happen to them, their business or their families if they were to become critically ill.
"Then, in March of 2010, I had a seizure and was diagnosed with a brain tumour. I had an operation to remove virtually all of it and, since then, I've been doing well.
"Periodically I have to have treatments like radiotherapy and chemotherapy but I've stayed fairly healthy since.
The married dad-of-two said, while being diagnosed with a brain tumour was frightening, he was able to maintain a positive attitude.
"It was a shock," he admitted. "It's scary to go through treatment.
"People have this perception of cancer and certainly brain tumours of being a death sentence, but it doesn't have to be.
Gregory, of Newton Mearns, believes his own experience helps him talk to people about how they'd cope if 'the worst was to happen'. | {
"redpajama_set_name": "RedPajamaC4"
} | 713 |
Q: Elements are moving outside of container in HTML/CSS I have a looping video header that I implemented using a snippet of code I found. I overlaid text and a button on top of it. It looks alright until I get to a certain break point, and then the text and button move outside of the video container.
I tried putting into the div that I thought the video was in, but it just removed the text and button from view altogether.
The site is kgshowroom.com/test and it's the coral-colored "Hello" button and text that says "KG Showroom You're on our site".
This is the code I think I'm supposed to be tweaking:
(Video Container?)
<div class="header-video">
<img src="img/masthead.jpg"
class="header-video--media"
data-video-src="0CxWLQxvY9g"
data-teaser-source="video/masthead-teaser"
data-provider="Youtube"
data-video-width="500"
data-video-height="281">
<a href="http://vimeo.com/87701971">
</div>
(Text and Button Section?)
<div id="header">
<!-- Inner -->
<div class="inner" style="position: relative; z-index: 999; top: 0px; width: 50%; margin: 0 auto; text-align: center;">
<header>
<h1>
<a href="index.html" id="logo"><img src="images/KG-logo.png"><br />KG Showroom</a></h1>
<p>You're on our site.</p>
</header>
<footer>
<a href="#banner" class="button circled scrolly">Hello!</a>
</footer>
</div>
I tried moving the "Text and Button Section" into the place above the closing div tag in the "Video Container" but it just made the Text and Button disappear.
Anyone know where I am going wrong on this? I want it to stay overlaid and centered on the video on all breakpoints, no matter the size of the screen.
Additionally, the looping video plays on my Samsung Note II, but it doesn't play on my brother's Samsung Galaxy (5?). It plays all the way through the different sizes on my desktop when testing; is there something in the code that is making it not work on my brother's phone?
Thanks!
A: It looks like the HeaderVideo function in script.js is dynamically resizing the header-videoclass which explains your sizing issues. You might want to rework the logic in that function or utilize media queries.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 765 |
El Escudo de armas de Alderney tiene la siguiente descripción heráldica: en un campo de azur (azul), un león rampante de oro, linguado, uñado, armado de gules (rojo) y coronado de oro con la Corona de San Eduardo que porta en su diestra una rama de plata, hojada del mismo metal (color) y afrutada de gules.
La Corona de San Eduardo es la empleada por los monarcas del Reino Unido, consiste un círculo de oro, engastado de piedras preciosas, compuesta de ocho florones, visibles cinco, la mitad con forma de cruz que se alternan con otros tantos con forma de flores de lis e interpolados de perlas. De los florones con forma de cruz salen cuatro diademas sumadas de perlas, que convergen en el mundo de azur o azul, con el semimeridiano y el ecuador en oro, sumado de cruz de oro. La corona forrada de gules o rojo y de armiño en su base.
En la bandera de Alderney figuran, con modificaciones, los elementos del escudo. Aparece representado el león rampante coronado pero se sustituye el color azul (azur en terminología heráldica) por el verde (sínople).
Enlaces externos
Bandera de Alderney, Flags of the World.
Escudo de Alderney, Civic Heraldry.com.
Alderney
Cultura de Alderney | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,205 |
0 B966BS-390M=P3 parts found for "B966BS-390M=P3"
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"redpajama_set_name": "RedPajamaC4"
} | 1,835 |
Now I am a proud owner of an iPad. So far, I have really liked it. My new HP laptop has to go to repair, and now I finally have a second device I can use more or less like a computer; at least considering communication and the Internet. And of course I plan to download as many versions of Angry Birds as possible; the touch-screen experience is something totally different than using a mouse. This post as well has been written with the help of the virtual keyboard, at least partly.
Additional update on Helsinki: we have no snow and today it was +7 degrees Celsius. Last year at this time, it was already a proper winter, but this year it is warm and rainy. In a way, I like it as the past years have been so very snowy and cold. Well, for Christmas and New Year it would be nice to get a bit of snow, but just a bit. Have a nice beginning of the last month of the year - Finland is so very Christmasy already, but well, Santa is supposed to be from here anyway.
Turku is a small nice town, sure. I've heard that Turun saaristo (the Turku archipelago) is nice, too. :) Oh, and we have rain as well. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,588 |
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established in 1933 manufactures vehicles in 20 countries and areas around the world. It also deals with sales and related business of automotive products and marine equipment. Nissan provides unique and innovative automotive products and services that deliver superior measurable values to all stakeholders in alliance with Renault. The company has a vision to enrich people's lives.
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\section{Introduction}\label{Sect1}
\begin{figure*}
\centering
\includegraphics[width=170mm, angle=0]{3color_a.jpg}
\caption{($left$)$^{12}$CO (1--0) and $^{13}$CO (1--0) emission observed toward the B211/B213 filament and ($right$) schematic picture of the velocity components. The $^{12}$CO and $^{13}$CO data are from \citet{Goldsmith08}. The panel ($left$) is adopted from \citet{Palmeirim13}.
In panel ($left$), the red color shows the distribution of the $^{12}$CO (1--0) emission with a velocity range of 6.6--7.4 km s$^{-1}$, the green color shows $^{13}$CO (1--0) emission with a velocity range of 5.6--6.4 km s$^{-1}$, and the blue color shows $^{12}$CO (1--0) emission with a velocity range of 4.2--5.5 km s$^{-1}$. The white box perpendicular to the filament axis shows the cut line for the position velocity diagrams shown in Fig. \ref{fig:pv}.}
\label{fig1}
\label{fig:3color}
\end{figure*}
\begin{figure}[t]
\centering
\includegraphics[width=90mm, angle=0]{tau.jpg}
\caption{Map of $^{12}$CO (1--0) optical depth derived from the \citet{Goldsmith08} $^{12}$CO (1--0) and $^{13}$CO (1--0) data.}
\label{fig:tau}
\end{figure}
The observations of the {\it Herschel} Gould Belt survey (HGBS) have revealed an omnipresence of parsec-scale filaments in molecular clouds and emphasized their importance for solar-type star formation \citep[e.g. ][]{Andre10, Menshchikov10,Arzoumanian11,Palmeirim13}.
In particular, most $Herschel$ prestellar cores are found to lie in dense (thermally supercritical) filaments, suggesting that cores generally form by filament fragmentation \citep[eg. ][]{Konyves15,Marsh16,Benedettini18}.
Molecular line observations of the velocity field around cores and filaments further support this view \citep{Tafalla15}.
Based on the HGBS results, \citet{Andre14} proposed a filament paradigm for star formation, whereby large-scale compression of interstellar material in supersonic flows generates a quasi-universal web of $\sim$0.1-pc wide filaments in the cold interstellar medium (ISM) and then the denser filaments fragment into prestellar cores by gravitational instability.
Recently, \citet{Shimajiri17} found that the star formation efficiency in dense molecular gas ($A_{\rm v}$ > 8), where filamentary structures dominate the mass budget, is remarkably uniform over a wide range of scales from 1-10 pc to >10 kpc \citep[see also,][]{Gao04,Lada10,Lada12,Chen15}.
Furthermore, \citet{Shimajiri17} proposed that this common star formation efficiency in dense gas results from the microphysics of star formation in filaments \citep[see also][]{Andre14}. This result suggests the existence of a universal "star formation law" converting dense molecular gas into stars along filaments. Therefore, unveiling how molecular filaments grow in mass and fragment is crucial to understanding star formation in filaments.
The B211/B213 filament system is located in the Taurus molecular cloud, which is one of the nearest star-forming regions \citep[$d$$\sim$140 pc,][]{Elias78}. Wide-field mapping observations in $^{12}$CO, $^{13}$CO, C$^{18}$O, N$_2$H$^+$, and SO emission revealed a whole network of filamentary structures in the B211/B213 area \citep{Goldsmith08, Hacar13, Panopoulou14,Tafalla15}.
\citet{Goldsmith08} and \citet{Palmeirim13} found that many low-density striations are elongated parallel to the magnetic field, and that blueshifted and redshifted components in both $^{12}$CO (1--0) and $^{13}$CO (1--0) emission are distributed to the southwest and the northeast of the B211/B213 filament, respectively, as shown in Fig. \ref{fig:3color}. This morphology was suggestive of mass accretion along magnetic field lines into the B211/B213 filament. To quantify mass accretion, \citet{Palmeirim13} assumed cylindrical geometry and used the observed mass per unit length $M_{\rm line}$ to estimate the gravitational acceleration $\phi$($R$) = 2$GM_{\rm line}/R$ of a piece of gas in free-fall toward the filament (where $R$ and $G$ denote radius from filament center and the gravitational constant, respectively). The free-fall velocity of gas initially at rest at a cylindrical radius $R_{\rm init}\sim$2 pc was estimated to reach 1.1 km s$^{-1}$ when the material reached the outer radius $R_{\rm out}\sim$0.4 pc of the B211/B213 filament. This estimation was consistent with the velocity observed in CO, suggesting that the background gas accretes into the B211/B213 filament owing to the gravitational potential of the B211/B213 filament. However, the velocity structure was not investigated in detail.
Investigation of the velocity structure is crucial to confirm this suggested scenario from the kinematic viewpoint. This is the topic of the present paper.
The paper is organized as follows: in Sect. \ref{Sect2}, we describe the $^{12}$CO (1--0) and $^{13}$CO (1--0) data, as well as complementary H${\alpha}$, 857 GHz, and HI data. In Sect. \ref{Sect3}, we estimate the optical depth of the $^{12}$CO (1--0) line and present the $^{12}$CO (1--0) and $^{13}$CO (1--0) velocity structures observed in the B211/B213 cloud. In Sect. \ref{Sect4}, we discuss the cloud structure, whether the surrounding material accretes onto the B211/B213 filament from the kinematic viewpoint, and whether the filament is formed by large-scale compression. In Sect. \ref{Sect5}, we summarize our results.
\begin{figure*}
\centering
\includegraphics[width=155mm, angle=0]{12co_channel.jpg}
\includegraphics[width=155mm, angle=0]{13co_channel.jpg}
\caption{Velocity channel maps in the $^{12}$CO ($J$=1--0, $top$) and $^{13}$CO ($J$=1--0, $bottom$) emission lines in units of K obtained from the \citet{Goldsmith08} data. The line of the sight (LSR) velocities (in km s$^{-1}$) are indicated on the top-left corner of each panel. The velocity width of each channel map is 0.3 km s$^{-1}$. }
\label{fig:channel_co}
\end{figure*}
\section{Observational data}\label{Sect2}
In this paper, we used the $^{12}$CO (1--0) and $^{13}$CO (1--0) data obtained by \citet{Goldsmith08, Narayanan08} with the 14 m diameter millimeter-wave telescope of the Five College Radio Astronomy Observatory (FCRAO). The half-power beam width of the telescope was 45$\arcsec$ for $^{12}$CO (1--0) and 47$\arcsec$ for $^{13}$CO (1--0).
We applied Gaussian spatial smoothing to improve the signal to noise ratio, resulting in an effective beam resolution of $\sim$76$\arcsec$, corresponding to $\sim$0.05 pc at a distance of 140 pc. The velocity resolution of the data is 0.26 km s$^{-1}$ for $^{12}$CO (1--0) and 0.27 km s$^{-1}$ for $^{13}$CO (1--0). The rms noise level is 0.1 K ($T_{\rm A}^*$) for $^{12}$CO (1--0) and 0.05 K ($T_{\rm A}^*$) for $^{13}$CO (1--0), respectively.
As complementary observations of the Taurus cloud region and its large-scale environment, we also used the H${\alpha}$ data\footnote{\url{https://faun.rc.fas.harvard.edu/dfink/skymaps/halpha/data/v1_1/index.html}} of \citet{Finkbeiner03}, as well as $Planck$ 857 GHz\footnote{\url{https://irsa.ipac.caltech.edu/data/Planck/release_1/all-sky-maps/previews/HFI_SkyMap_857_2048_R1.10_survey_2_ZodiCorrected/}} \citep{Planck14} and HI data\footnote{\url{https://www.astro.uni-bonn.de/hisurvey/AllSky_profiles/index.php}} \citep{Kalberla17} from the archive.
\begin{figure*}
\centering
\includegraphics[width=190mm, angle=0]{cloud_structure_model.png}
\caption{($left$) Schematic picture of the structure of the B211/B213 cloud (see Sect. \ref{sect:cloud}), and ($right$) schematic picture describing our toy model of the velocity field (see Sect. \ref{section:model}).}
\label{fig:cloud_structure}
\label{fig:model}
\end{figure*}
\section{Analysis and results}\label{Sect3}
\subsection{$^{12}$CO (1--0) and $^{13}$CO (1--0) optical depths \label{optical_depth}}
The optical depth of the $^{12}$CO (1--0) line was estimated from the FCRAO $^{12}$CO and $^{13}$CO data. Assuming the same excitation temperature for the $^{12}$CO (1--0) and $^{13}$CO (1--0) lines, an isotopic ratio, $R_{\rm i}$ = 62 for $^{12}$C/$^{13}$C \citep{Langer93}, and the same beam filling factor in both lines, we evaluated the optical depth of $^{12}$CO (1--0) using the following equation:
\begin{equation}
\frac{T({\rm ^{13}CO)}}{T({\rm ^{12}CO})}=\frac{1-e^{-\tau({\rm ^{12}CO})/R_{\rm i}}}{1-e^{-\tau(\rm{^{12}CO})}}.
\end{equation}
\noindent Here, $T(\rm{^{12}CO})$ and $\tau(\rm{^{12}CO})$ denote the peak intensity and the optical depth of $^{12}$CO (1--0), respectively.
While the $^{12}$CO (1--0) emission preferentially traces the diffuse extended cloud, the $^{13}$CO (1--0) emission traces the central B211/B213 filament (see Fig. \ref{fig:3color}).
The typical inner width of the filaments observed with {\it Herschel} is $\sim$0.1 pc \citep{Arzoumanian11,Arzoumanian18,Palmeirim13}, which is larger than the 0.05 pc effective beam size of the FCRAO data. Thus, assuming the same beam filling factor in $^{12}$CO (1--0) and $^{13}$CO (1--0) is reasonable.
Figure \ref{fig:tau} displays the resulting map of $^{12}$CO (1--0) optical depth. The optical depth in this map ranges from $\sim$3 to $\sim$300, showing that the $^{12}$CO (1--0) emission is optically thick. In particular, the $^{12}$CO (1--0) optical depth toward the B211/B213 filament itself ($\tau(\rm ^{12}CO)$$\sim$100) is much larger than that found for the surrounding lower density material ($\tau(\rm ^{12}CO)$$\sim$20).
\subsection{$^{12}$CO (1--0) and $^{13}$CO (1--0) velocity channel maps}\label{section:channel}
Figure \ref{fig:channel_co} shows the velocity channel maps observed in $^{12}$CO (1--0) and $^{13}$CO (1--0).
In the maps for $V_{\rm LSR}$ < 3.7 km s$^{-1}$, both $^{12}$CO (1--0) and $^{13}$CO (1--0) emission is seen in the northeastern part of the maps (RA, DEC = 4:24:00, 28:15:00). In the channel maps for 4.0 < $V_{\rm LSR}$ < 7.3 km s$^{-1}$, enhanced emission is seen toward the B211/B213 filament in both $^{12}$CO (1--0) and $^{13}$CO (1--0).
The emission at these velocities is likely to be directly associated with the B211/B213 filament. Furthermore, while the emission at 4 km s$^{-1}$ < $V_{\rm LSR}$ < 6 km s$^{-1}$ is distributed to the southwest of the B211/B213 filament, the emission at 6 km s$^{-1}$ < $V_{\rm LSR}$ < 7 km s$^{-1}$ is distributed to the northeast of the filament.
In the channel maps for $V_{\rm LSR}$ > 7.3 km s$^{-1}$, the distribution of the $^{12}$CO (1--0) and $^{13}$CO (1--0) emission is suggestive of an arc-like structure around L1495.
Figure \ref{fig:3color} ($right$) is a sketch showing the location of each velocity component.
\section{Modeling of the data and discussion}\label{Sect4}
\subsection{3D morphology of the B211/B213 ambient cloud}\label{sect:cloud}
Here, we discuss the 3D morphology of the material surrounding the B211/B213 filament by comparing the extent of the gas in the plane of the sky and its depth along the line of sight. Hereafter, we refer to the system consisting of the B211/B213 filament and its surrounding gas as the B211/B213 cloud (i.e. red, green, and dark blue areas in Fig. \ref{fig:3color} ($right$)).
The projected extent of the B211/B213 cloud in the plane of the sky is more than $\sim$10 pc. Taking the viewing angle into account, the real extent of the cloud may be larger. At the same time, we can estimate the depth of the cloud along the line of sight under the assumption that the surrounding material is filled by gas with density exceeding the critical density of the $^{13}$CO (1--0) line, since $^{13}$CO (1--0) emission is observed over the entire mapped area. The critical density of $^{13}$CO (1--0), $n_{\rm critical}^{\rm ^{13}CO}$, may be estimated as follows:
\begin{equation}
n_{\rm critical}^{\rm ^{13}CO} = \frac{A_{\rm ul}}{\sigma_{\rm cross} \nu} = \frac{A_{\rm ul}}{10^{-15}{\rm cm}^{-2} \times 10^4 \sqrt{T_{\rm ex}}},
\end{equation}
\noindent where $A_{\rm ul}$, $\sigma_{\rm cross}$, $\nu$, and $T_{\rm ex}$ are the Einstein spontaneous emission coefficient, collision cross section, collision velocity, and line excitation temperature. The values of $A_{10}$ and $\sigma_{\rm cross}$ in the LAMDA database\footnote{\url{http://home.strw.leidenuniv.nl/~moldata/datafiles/13co.dat}} are 6.294$\times$10$^{-8}$ s$^{-1}$ and 10$^{-15}$ cm$^{-2}$. The collision velocity can be calculated as $v$=$\sqrt{3k_{\rm B}T_{\rm ex}/m}$ = $10^{4}\sqrt{T_{\rm ex}}$ cm s$^{-1}$, where $k_{\rm B}$ is the Boltzmann constant and $m$ is hydrogen molecular mass. This leads to a value of 1.7 $\times$ 10$^3$ cm$^{-3}$ for the critical density of $^{13}$CO (1--0) assuming $T_{\rm ex}$ $\simeq$ 14 K.
Here, we assumed that the excitation temperature $T_{\rm ex}$ of the $^{13}$CO (1--0) line is the same as the dust temperature $T_{\rm dust}$ $\sim$ 14K derived by \citet{Palmeirim13} from ${\it Herschel}$ data in the ambient cloud around B211/B213 (red and dark blue area in Fig. \ref{fig1} ($right$)). \citet{Palmeirim13} also estimated the mean {\it Herschel} column density in the material surrounding the B211/B213 filament to be $N_{{\rm H}_2}$ $\simeq$ 1.4 $\times$ 10$^{21}$ cm$^{-2}$. Thus, the depth of the cloud (=$N_{{\rm H}_2}/n_{\rm critical}^{\rm ^{13}CO}$) is estimated to be 0.3 pc.
Recently, \citet{Qian15} independently estimated the depth of the whole Taurus molecular cloud and found a value of $\sim$0.7 pc using the core velocity dispersion (CVD) method.
With a projected extent of more than 10 pc and a depth of $\sim$0.3 -- 0.7 pc, we conclude that the 3D morphology of the cloud resembles a sheet-like structure (see Fig. \ref{fig:cloud_structure}).
\textcolor{black}{From HC$_3$N (2--1) and (10--9) observations, \citet{Li12} found that the depth of the dense ($\sim$10$^4$ cm$^{-3}$) portion of the B213 region (i.e. the dense
filament) was $\sim$0.12 pc. This is smaller than our estimate for the depth of the ambient molecular gas, but consistent with the view that the dense inner part of the B213 filament is a cylinder-like structure of $\sim$0.1 pc diameter \citep{Palmeirim13}, embedded in a lower-density sheet-like cloud.
}
\begin{figure}
\centering
\includegraphics[width=80mm, angle=0]{sample_spectrum_A.jpg}
\includegraphics[width=80mm, angle=0]{sample_spectrum_B.jpg}
\caption{Schematic picture of the definition of velocity components associated with the B211/B213 filament. The spectrum in each panel is the $^{13}$CO (1-0) spectrum averaged over a 15$\arcmin$ $\times$ 15$\arcmin$ area with a center position indicated in the top-left corner. The velocity components with a velocity of < 4.0 km s$^{-1}$ or > 7.0 km s$^{-1}$ are regarded as components not associated with the B211/B213 filament. These components are subtracted from the data cube.}
\label{fig:subt}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=190mm, angle=0]{Filament_and_Inflow_fwhm60_n70_s20_pv_fits.jpg}
\caption{Position-Velocity diagram of ($a$) $^{12}$CO (1--0), ($b$) $^{13}$CO (1--0), and ($c$) model and ($d$) velocity offsets between $^{12}$CO (1--0) and $^{13}$CO (1--0) observations and model. Assumed parameters for the accretion model are summarized in Table \ref{Table1}. The cut line of the $PV$ diagrams is indicated in Fig \ref{fig:3color}.
In panels ($a$)-($c$), black squares indicate the peak velocity positions at each offset. In panels ($a$) and ($b$), black crosses are the peak velocity positions at each offset in the model.
In panel ($d$), lines indicate the velocity offset (black) between $^{12}$CO(1-0) and model and (red) between $^{13}$CO(1-0) and model.
In panels ($a$)-($d$), black and grey vertical lines indicate offset = 0$\arcmin$ and |offset| < $R_{\rm out}$.}
\label{fig:pv}
\end{figure*}
\subsection{Accretion of background gas into the B211/B213 filament}\label{sect:accretion}
Here, we compare the velocity pattern seen in $^{12}$CO (1--0) and $^{13}$CO (1--0) emission with the prediction of an accretion gas model, in order to investigate whether the B211/B213 filament accretes ambient cloud gas from a kinematic viewpoint.
\subsubsection{Observed position-velocity diagrams}\label{section:pv_discription}
As mentioned in Sect. \ref{section:channel}, the highly blueshifted and redshifted components at $V_{\rm LSR}$ < 3.7 km s$^{-1}$ and $V_{\rm LSR}$ > 7.3 km s$^{-1}$ do not seem to be directly connected to the B211/B213 cloud/filament. To focus on the velocity field of the gas associated with the B211/B213 filament, we subtracted these two components as follows. We applied Gaussian fitting with $N$ Gaussian components to each pixel, where $N$=1, 2, 3, 4, or 5. Wherever the signal to noise (S/N) ratio of the residual peak intensity was less than 5, the fit was deemed to be acceptable and the corresponding spectrum was assumed to consist of $N$ Gaussian components.
Then, if the peak LSR velocity of a Gaussian component was lower than 4.0 km s$^{-1}$ or higher than 7.0 km s$^{-1}$, the component was not considered to be associated with the B211/B213 filament or cloud and was subtracted from the data cube (see also Fig. \ref{fig:subt} and Fig. \ref{fig:fitting}). Figure \ref{fig:channel_subt} displays the $^{12}$CO (1--0) and $^{13}$CO (1--0) velocity channel maps after subtracting these components. Hereafter, we used these subtracted data cubes.
Figure \ref{fig:pv} shows the resulting position-velocity ($PV$) diagrams in $^{12}$CO (1--0) and $^{13}$CO (1--0) along a direction perpendicular to the B211/B213 filament as indicated in Fig. \ref{fig:3color}.
On these $PV$ diagrams, distinct velocity pattern can be recognized in $^{12}$CO (1--0) and $^{13}$CO (1--0) toward the filament (|offset| < 10$\arcmin$ $\sim$ 0.4 pc). This is probably due to differing optical depths in the two lines. As described in Sect. \ref{optical_depth}, the $^{12}$CO (1--0) optical depth toward the filament is $>$ 50 and much larger than the optical depth toward the outskirts of the filament, suggesting that the $^{12}$CO (1--0) emission only traces the surface of the filament. In the outskirts of the B211/B213 filament (|offset| > 10$\arcmin$), the blueshifted emission is distributed to the southwest (offset > 0$\arcmin$) and the redshifted emission is distributed to the northeast (offset < 0$\arcmin$) of the filament. It can be seen that
the velocities of the blueshifted and redshifted components approach the velocity of the B211/B213 filament as the offset approaches 0 (i.e. the crest of the filament).
\textcolor{black}{Transverse velocity gradients perpendicular to the major axis of filaments have been also observed toward several dense filaments in the Serpens cloud \citep{Dhabal18} as well as the main filament in the northwestern part of the L1495 subregion \citep{Arzoumanian18a}.}
\subsubsection{Gas accretion model}\label{section:model}
\begin{table}
\caption{\textcolor{black}{Properties of the three modeled cloud components}}
\label{Table1}
\centering
\begin{tabular}{c c c c}
\hline\hline
Component & Parameter & \\
\hline
\multirow{4}{*}{Filament$^{\star}$} & $M_{\rm line}$ & 54 $M_{\odot}$ \textcolor{black}{pc$^{-1}$} $^{\dag}$ \\
& $n_{\rm H_2}^0$ & 4.5$\times$10$^{4}$ cm$^{-3}$ $^{\dag}$ \\
& $p$ & 2 $^{\dag}$ \\
& $R_{\rm flat}$ & 0.03 pc $^{\dag}$ \\
& $R_{\rm out}$ & 0.4 pc $^{\dag}$ \\
& $\mathcal{V}_{\rm filament}$ & 6.2 km s$^{-1}$ $^{\ddag}$ \\
\hline \hline
\multirow{3}{*}{Northeastern sheet$^{\star}$} & $\mathcal{V}_{\rm init,N}$ & 6.8 km s$^{-1}$ $^{\ddag}$ \\
& $R_{\rm init,N}'$ & 10 pc \\
& $\theta_{\rm N}$ & 70 deg \\
\hline \hline
\multirow{3}{*}{Southwestern sheet$^{\star}$} & $\mathcal{V}_{\rm init,S}$ & 4.4 km s$^{-1}$ $^{\ddag}$ \\
& $R_{\rm init,S}'$ & 10 pc \\
& $\theta_{\rm S}$ & 20 deg \\
\hline
\hline
\end{tabular}
\tablefoot{(\dag)Adopted from \citet{Palmeirim13}. (\ddag) Peak velocity at $R_{\rm init}$. \textcolor{black}{($\star$) The total masses of the filament, northeastern sheet, and southwestern sheet components are estimated to be $\sim$400 $M_{\odot}$, $\sim$700 $M_{\odot}$, and $\sim$300 $M_{\odot}$, respectively, from the $Herschel$ H$_2$ column density map. }}
\end{table}
The $PV$ diagrams in Fig. \ref{fig:pv} show an asymmetric velocity distribution
on either side of the 0 position (filament crest), suggesting that the sheet-like ambient cloud surrounding the B211/B213 filament has a different inclination to the plane of the sky to the northeast and the southwest of the filament.
To investigate whether the B211/B213 filament accretes gas from the ambient cloud, we thus constructed a 3-component toy model (one filament component and two components for the northeastern and southwestern sheets) under the assumption that the sheet components to the northeast (red-shifted) and the southwest (blues-shifted) lie on the near and far sides of the B211/B213 filament, respectively, as shown in Fig. \ref{fig:model}. Our modeling procedure is summarized in the schematic picture shown in Fig. \ref{fig:modeling_flow}.
\subsubsection*{$\bullet$ Model for the central filament component}
First, we produced a model for the filament.
{\it Herschel} observations of nearby clouds have shown that the radial column density profiles of molecular filaments in the radial direction $R'$ (i.e. perpendicular to the filament crest) can be well described by the following "Plummer-like" function \citep{Arzoumanian11,Palmeirim13}:
\begin{eqnarray}
\begin{tiny}
\begin{split}
\Sigma_p(R')/\mu m_{\rm H} & = \frac{N_{\rm H_2}^0}{[1+(R'/R_{\rm flat})^2]^\frac{p-1}{2}}
&\to
\rho_{p}(R') = \frac{\rho_{\rm c}}{[1+(R'/R_{\rm flat})^2]^\frac{p}{2}} ,
\end{split}
\end{tiny}
\end{eqnarray}
\noindent where $\rho_{\rm c}$, $\Sigma_p$, $\mu$, $m_{\rm H}$, $N_{\rm H_2}^0$, $p$, and $R_{\rm flat}$ are the central density of the filament, the column density as a function of radius $R'$, the mean molecular mass, the hydrogen atom mass, the central column density, the index of the power-law density profile at large radii ($R'$ $\gg$ $R_{\rm flat}$), and the radius of the flat inner region, respectively. For the B211/B213 filament, we adopted $N_{\rm H_2}^0$=1.4$\times$10$^{21}$ cm$^{-2}$, $p$=2.0, and $R_{\rm flat}$=0.03 pc from the fitting results of \citet{Palmeirim13}.
We assumed that the filament itself lies in the plane of the sky and that the shape of the intensity profile of the B211/B213 filament as traced in $^{12}$CO (1--0) and $^{13}$CO (1--0) emission is the same as that found in the {\it Herschel} column density map.
Then, we rescaled the peak integrated intensity to be 2 K km s$^{-1}$ as observed in $^{13}$CO (1--0).
Approximating the Plummer density profile of the filament by a broken power-law,
the gravitational potential in the radial direction $R'$ can be expressed as follows\footnote{Here, $R'$ denotes the radius corrected for inclination to the line of sight, where the relation between the corrected radius $R'$ and radius in the sky plane $R$ is $R' = R/\sin(\theta_{\rm N/S})$ and $\theta_{\rm N/S}$ is the inclination angle of the northeastern/southwestern sheet component to the line of sight.} \citep[cf.][]{Hennebelle13}:
\begin{equation}
\begin{tiny}
\phi(R') =
\begin{cases}
G \rho_{\rm flat} \pi R'^2 & \text{for $R'$ $\le$ $R_{\rm flat}$} \\
G M_{\rm line,flat} \left[1 + 2 \ln\left(\frac{R'}{R_{\rm flat}}\right) + 2 \left(\ln\frac{R'}{R_{\rm flat}}\right)^2 \right] & \text{for $R_{\rm flat}$ < $R'$ $\le$ $R_{\rm out}$} \\
G M_{\rm line,flat} \left[1 + 2 \ln \left(\frac{R_{\rm out}}{R_{\rm flat}}\right) + 2 \left(\ln\frac{R_{\rm out}}{R_{\rm flat}} \right)^2 \right] \\
\qquad +2G M_{\rm line} \ln \left(\frac{R'}{R_{\rm out}}\right) & \text{for $R_{\rm out}$ < $R'$}
\end{cases}
\end{tiny}
\end{equation}
\noindent where $\rho_{\rm flat}$ and $R_{\rm out}$ are the density of the filament at $R'$ $\le$ $R_{\rm flat}$ and outer radius of the filament, respectively. We adopted $n_{\rm H_2}^0$=$\rho_{\rm flat}$/$\mu m_{\rm H}$ = $4.5 \times 10^4 \ {\rm cm}^{-3}$, $R_{\rm flat}$ = 0.03 pc, and $R_{\rm out}$ = 0.4 pc from \citet{Palmeirim13} as summarized in Table \ref{Table1}. In the above equation, $M_{\rm line,flat}$ and $M_{\rm line}$ represent the inner and total masses per unit length of the filament and are given by:
\begin{eqnarray}
M_{\rm line,flat} = \rho_{\rm flat} \pi R_{\rm flat}^2
\end{eqnarray}
\begin{eqnarray}
M_{\rm line} = \rho_{\rm flat} \pi R_{\rm flat}^2 \left[1+2 \ln\left( \frac{R_{\rm out}}{R_{\rm flat}}\right)\right]
\end{eqnarray}
\subsubsection*{$\bullet$ Models for the northeastern and southwestern sheet components}
Second, we produced models for the northeastern and southwestern sheet components assuming that the B211/B213 filament accretes the gas of the sheets as a result of its gravitational potential.
\noindent Taking into account the pressure gradient force, conservation of energy for a parcel of unit mass of the ambient cloud falling onto the central filament may be expressed as follows \citep[cf.][]{Smith94,Smith12}:,
\begin{eqnarray}
\begin{split}
\scriptsize
\frac{1}{2}(\mathcal{V}_{\rm init,N/S}'^0)^2 \mathalpha{+}
& \phi(R_{\rm init,N/S}') \mathalpha{+} C_{\rm s,eff}^2 \ln(\rho_{\rm init}) \\
&\mathalpha{=}
\frac{1}{2}\mathcal{V}(R')^2 \mathalpha{+} \phi(R') \mathalpha{+} C_{\rm s,eff}^2 \ln(\rho(R')),
\end{split}
\end{eqnarray}
\noindent where $\mathcal{V}(R)$ is the projected velocity and $C_{\rm s,eff}$ is the effective sound speed.
The projected velocity $\mathcal{V}(R)$ of the gas flow can thus be expressed as follows:
\begin{eqnarray}
\scriptsize
\mathcal{V}(R)\mathalpha{=}\mathcal{V}_{\rm filament} \mathalpha{\pm} \sqrt{2\left[\frac{1}{2}(\mathcal{V}_{\rm init,N/S}'^0)^2 \mathalpha{+} \phi(R_{\rm init,N/S}')\mathalpha{-}\phi(R') \mathalpha{+} C_{\rm s,eff}^2 \ln \left(\frac{\rho_{\rm init}}{\rho(R')}\right)\right]} \mathalpha{\times} \cos(\theta_{\rm N/S}),
\end{eqnarray}
\noindent where $\mathcal{V}_{\rm filament}$, $\mathcal{V}_{\rm init,N/S}'^0$, $R_{\rm init,N/S}'$, and $\rho_{\rm init}$ are the systemic velocity of the filament, the velocity of the accreting gas at the initial point corrected for inclination, the initial radius of the accreting gas corrected for inclination, and the volume density at the initial point, respectively.
Here, we define ${\mathcal{V}_{\rm init,N/S}'^0}$ as $(\mathcal{V}_{\rm init,N/S} - \mathcal{V}_{\rm filament} )/\cos(\theta_{\rm N/S})$, where $\mathcal{V}_{\rm init,N/S}$ is the projected velocity of the northeastern/southwestern sheet component at $R_{\rm init,N/S}'$.
We adopted $C_{\rm s,eff}$ = 0.9 km s$^{-1}$ from $C_{\rm s,eff} \simeq \delta V_{\rm FWHM}(^{12}{\rm CO})/\sqrt{8\ln2}$, where $\delta V_{\rm FWHM}(^{12}{\rm CO})$ is the $^{12}$CO (1-0) line width (=2.1 km s$^{-1}$) observed toward the B211/B213 filament.
Wherever the value of $C_{\rm s,eff}^2 \ln \left(\frac{\rho_{\rm init}}{\rho(R')}\right)$ was larger than $\frac{1}{2}{(\mathcal{V}_{\rm init,N/S}'^0)^2} + \phi(R_{\rm init,N/S}')-\phi(R')$, we adopted $\mathcal{V}(R) = \mathcal{V}_{\rm filament}$.
The $Herschel$ observations show that the density profile of the B211/B213 filament is proportional to $R'^{-2}$ at $R'$ $\le$ $R'_{\rm out}$ and has a shallower slope at $R'$ $\ge$ $R'_{\rm out}$ \citep{Palmeirim13}. Furthermore, the slope for the southwestern sheet component is slightly steeper than the slope for the northeastern sheet component. At $R'$ > $R'_{\rm init}$, the gas density in the model was assumed to be constant.
To summarize, we assumed the following density distribution as a function of radial direction $R'$ (see Fig. \ref{fig:density}):
For the northeastern sheet component,
\begin{eqnarray}
\footnotesize
\rho(R') =
\begin{cases}
\rho_{\rm flat} &\text{for $R'$ $\le$ $R_{\rm flat}$} \\
\rho_{\rm flat} \left(\frac{R'}{R_{\rm flat}}\right)^{-2} &\text{for $R_{\rm flat}$ < $R'$ $\le$ $R'_{\rm out}$} \\
\rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'}{R_{\rm out}}\right)^{-1.0} &\text{for $R'_{\rm out}$ < $R'$ $\le$ $R'_{\rm init}$} \\
{\rm constant} = \rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'_{\rm init}}{R_{\rm out}}\right)^{-1.0} & \text{for $R'_{\rm init}$ < $R'$} \\
\end{cases}
\end{eqnarray}
For the southwestern sheet component,
\begin{eqnarray}
\footnotesize
\rho(R') =
\begin{cases}
\rho_{\rm flat} &\text{for $R'$ $\le$ $R_{\rm flat}$} \\
\rho_{\rm flat} \left(\frac{R'}{R_{\rm flat}}\right)^{-2} &\text{for $R_{\rm flat}$ < $R'$ $\le$ $R'_{\rm out}$} \\
\rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'}{R_{\rm out}}\right)^{-1.5} &\text{for $R'_{\rm out}$ < $R'$ $\le$ $R'_{\rm init}$} \\
{\rm constant} = \rho_{\rm flat} \left(\frac{R_{\rm out}}{R_{\rm flat}}\right)^{-2}\left(\frac{R'_{\rm init}}{R_{\rm out}}\right)^{-1.5} & \text{for $R'_{\rm init}$ < $R'$} \\
\end{cases}
\end{eqnarray}
\begin{figure}
\centering
\includegraphics[width=95mm, angle=0]{density_profile.jpg}
\caption{Assumed density profile for the 3-component model. Red and blue lines indicate the densities for the northeastern and southwestern sheet components, respectively.}
\label{fig:density}
\end{figure}
We also assumed that both sheet components have integrated intensities of $\sim$1 K km s$^{-1}$ as observed in $^{13}$CO (1-0). To get a good agreement between the models and the observations (see Appendix \ref{appendix:inclination}),
we adopted $\theta_{\rm N}$ = 70$^{\circ}$, $\mathcal{V}_{\rm init,N}$ = 6.8 km s$^{-1}$, and $R_{\rm init,N}'$ = 10 pc for the northeastern sheet component and $\theta_{\rm S}$ = 20$^{\circ}$, $\mathcal{V}_{\rm init,S}$ = 4.4 km s$^{-1}$, and $R_{\rm init,S}'$ = 10 pc for the southwestern sheet component. The parameters of our model are summarized in Table \ref{Table1}. \textcolor{black}{For simplification, we assumed constant inclinations to the line of sight, of 70$^\circ$ for the northeastern sheet component and 20$^\circ$ for the southwestern sheet component, in our model. In reality, however, the inclinations of the two components may vary smoothly with radius from the B211/B213 filament and match on the filament crest.
}
\subsubsection*{$\bullet$ Combined 3-component model}
We first generated an integrated intensity distribution and a peak velocity field for each of the three components with IDL (Interactive Data Language).
Using the MIRIAD task $velimage$\footnote{$velimage$ makes a data cube $output\_cube(x,y,v_{\rm centroid})$ from an input integrated intensity image $input\_intensity(x,y)$, input centroid velocity image $input\_velocity(x,y)$, and dispersion $\sigma$. The $v_{\rm centroid}$-values are the centroid velocity and are input as an ($x,y$) image. The output cube image is produced as $output\_cube(x,y,v_{\rm centroid}) = input\_intensity(x,y) \times \exp(-(v_{\rm centroid}-input\_velocity(x,y))^2/(2\sigma^2)))$.},
we then produced individual data cube components for the filament and the two sheet components
assuming uniform velocity dispersions of 1.3 km s$^{-1}$ for the filament and 0.9 km s$^{-1}$ for the sheet components. The velocity dispersions were obtained from fitting the observed $^{13}$CO (1--0) spectra.
\textcolor{black}{One of the reasons for the larger velocity dispersion observed in $^{13}$CO toward the central filament may be that the B211/B213 filament contains several velocity subcomponents \citep[or "fiber-like" structures, ][]{Hacar13}, possibly
as a result of accretion-driven turbulence \citep[cf. ][]{Hennebelle13,Heitsch13,Andre14}\footnote{In recent numerical simulations of this process,
\citet{Seifried15} did find that the accretion flow increases the velocity dispersion of the central filament, and \citet{Clarke17} suggested that fiber-like structures could be produced as a result of the vorticity generated by an inhomogeneous accretion flow.}.}
Finally, we used IDL to co-add the three individual data-cube components and produce a combined model data cube.
\subsubsection*{$\bullet$ Large-scale kinematic model}
We adopted initial velocities (corrected for inclination) of {$\mathcal{V}_{\rm init,N}'^0$ = 1.8 km s$^{-1}$ $(=[\mathcal{V}_{\rm init,N} - \mathcal{V}_{\rm filament}]/\cos(\theta_{\rm N}))$} for the northeastern sheet component and {$\mathcal{V}_{\rm init,S}'^0$ = $-$1.9 km s$^{-1}$ $(=[\mathcal{V}_{\rm init,S} - \mathcal{V}_{\rm filament}]/\cos(\theta_{\rm S}))$} for the southwestern sheet component.
This almost symmetric initial velocity pattern after correction for inclination is suggestive of gravitational accretion.
If the accreting gas comes from far away positions $R_{\rm far,N/S}'$ ($\gg$ $R_{\rm init}'$) and is accelerated by the gravitational potential of the B211/B213 filament/cloud, the line of sight velocity at $R_{\rm far,N/S}'$ is likely to be similar to $\mathcal{V}_{\rm filament}$. The positions $R_{\rm far,N/S}'$ for the northeastern and southwestern sheet components can be estimated from the equation of
$\mathcal{V}_{\rm init,N/S}'=\sqrt{2[\phi(R_{\rm far,N/S}')-\phi(R_{\rm init,N/S}')]} \times \cos(\theta_{\rm N/S})$ since the pressure density gradient is probably small and can be neglected. We adopted $R_{\rm init,N}'$ = 10 pc, and $R_{\rm init,S}'$ = 10 pc, respectively. Thus, assuming that the initial velocities are entirely generated by gravitational acceleration, the surrounding gas for the northeastern and southwestern sheet components would have to come from $R_{\rm far,N}'$ ($R_{\rm far,N}$) = 270 (260) pc and $R_{\rm far,S}'$ ($R_{\rm far,S}$) = 520 (180) pc (see Fig. \ref{fig:pv_large}). Here, for simplification, we did not include the mass of the sheets when estimating the gravitational potential. Thus, these $R_{\rm far,N/S}'$ values should be considered upper limits. The HI emission observed at $V_{\rm LSR}$ $\sim$ 6 km s$^{-1}$, which corresponds to the systemic velocity of the B211/B213 filament, has an extended emission with an extent of several $\times$ 100 pc which is consistent with the above value of $R_{\rm far,N}'$ (see also Sect. \ref{Sect:compress} and Fig. \ref{fig:channel_hi}). Thus, one of the reasons why the initial velocities at $R_{\rm init,N/S}'$ in the northeastern and southwestern sheet components differ from the velocity of the filament may be the large-scale effect of the gravitational potential of the B211/B213 cloud/filament. We will discuss another possible explanation in Sect. \ref{Sect:compress}.
\subsubsection{Comparing the combined model with the observations}\label{section:comp_model}
The synthetic $PV$ diagram predicted by the model is shown in Fig. \ref{fig:pv} ($c)$ for comparison with the $PV$ diagrams observed in $^{12}$CO (1--0) and $^{13}$CO(1--0) (Figs. \ref{fig:pv}($a$) and \ref{fig:pv}($b$)). A good quantitative agreement especially with the $^{13}$CO (1--0) diagram can be seen.
In particular, in both the model and the observed $PV$ diagrams, the velocity of the gas surrounding the B211/B213 filament (red and dark blue area in Fig. \ref{fig:3color}($right$)) approaches the systemic velocity of the B211/B213 filament as the positional offset approaches 0 (i.e. the filament crest).
While the gas is accelerated by the gravitational potential of the filament/cloud at large scales (several$\times$10 pc), it is decelerated by the pressure gradient force of the dense filament at small scales (several pc) (see Fig. \ref{fig:pv_large} and Fig. \ref{fig:ob2}).
The good agreement between the model and the data indicates that observational kinematic constraints are consistent with the B211/B213 filament accreting background cloud material as a result of its gravitational potential.
This provides strong support to the scenario of mass accretion along magnetic field lines into the filament proposed by \citet{Palmeirim13}.
The mass accretion rate onto the B211/B213 filament was estimated to be 27-50 $M_{\odot}$ pc$^{-1}$ Myr$^{-1}$ by \citet{Palmeirim13}, suggesting that it took $\sim$1--2 Myr to form the filament.
Thus, accretion of gas from the ambient cloud in B211/B213 likely plays a key role in the evolution of the filament.
\begin{figure}
\centering
\includegraphics[width=80mm, angle=0]{large_scale_pv.jpg}
\caption{Position-velocity diagram of the model for the large scale. $\theta_{\rm N}$=70$^{\circ}$ and $\theta_{\rm S}$=20$^{\circ}$ are assumed. Fig. \ref{fig:pv} corresponds to -2.4 pc < offset < 1.6 pc in this figure. The vertical dashed lines mark $R_{\rm init,N}$ = -9.4 pc ($R'_{\rm init,N}$ = 10 pc) and $R_{\rm init,S}$ = 3.4 pc ($R'_{\rm init,S}$ = 10 pc).}
\label{fig:pv_large}
\end{figure}
\subsection{Formation of the B211/B213 filament by large-scale compression}\label{Sect:compress}
\begin{figure}
\centering
\includegraphics[width=80mm, angle=0]{B211-OB_ver2.png}
\caption{
Schematic picture of the relation between the B211/B213 cloud and Per OB2 association. The black arrows indicate the line of sight. The horizontal line indicate the sky plane. Red and blue arrows indicate the direction of the gas accretion in the northeastern and southwestern sheet components, respectively. Green arrows indicate the direction of the compression by Per OB2 association. $\theta_{\rm N}$ and $\theta_{\rm S}$ are the inclination angles of the northeastern and southwestern sheet components to the line of sight. Red and blue arrows of length scaling quantitatively with the magnitude velocity field indicate the direction of the acceleration flow of ambient cloud material.
}
\label{fig:ob2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=90mm, angle=0]{planck_halpha_b2.jpg}
\caption{Distributions of the H${\alpha}$ \citep[color,][]{Finkbeiner03} and 857 GHz dust \citep[grey,][]{Planck14} emission. The units of the H${\alpha}$ and 857 GHz maps are R (Rayleigh, 4$\pi$ $\times$ 10$^{-4}$ photons cm$^{-2}$ s$^{-1}$ sr$^{-1}$) and MJy str$^{-1}$, respectively.
The magenta dashed circle indicates a HI supershell \citep[][]{Lim13}. The diameter of the HI supershell might be $>$ 200 pc since the distances to the Taurus and Perseus clouds are 140 pc and 340 pc, respectively. The distribution of HI emission is shown in Figs. \ref{fig:channel_hi} and \ref{fig:3color_hi}. See also \ref{fig:halpha_appendix}.}
\label{fig:halpha}
\end{figure}
As described in Sect. \ref{sect:accretion}, we adopted different inclinations for the northeastern sheet component ($i$=70$^{\circ}$) and for the southwestern sheet component ($i$=20$^{\circ}$) in our model to get a good agreement with the observations. This suggests that the B211/B213 cloud is actually shaped like a shell (see Fig. \ref{fig:ob2}). One possibility is that this shell-like structure was produced by large-scale compression.
In this section, we try to investigate whether the cloud surrounding the B211/B213 filament is affected by large-scale flow phenomena using wide-field H${\alpha}$ maps tracing gas ionized by massive stars \citep{Finkbeiner03}, the $Planck$ 857 GHz dust continuum map tracing cold dust \citep{Planck14}, and HI map tracing lower density atomic gas \citep{HI4PI16}.
Figure \ref{fig:halpha} (see also Figs \ref{fig:channel_hi} and \ref{fig:3color_hi}) compare the spatial distributions of the H${\alpha}$ and 857 GHz emission in the Taurus-California-Perseus region (e.g. Taurus, Auriga, California, and Perseus). The 857 GHz dust emission traces each molecular cloud and exhibits a hole-like structure. This hole-like structure can also be seen in HI emission as shown in Fig. \ref{fig:channel_hi} and Fig. \ref{fig:3color_hi}.
The H${\alpha}$ emission fills the hole-like structure seen in the 857 GHz dust emission near the center of the field.
The Taurus, California, and Perseus molecular complexes traced by the 857 GHz dust emission are distributed at the edge of the hole-like structure. \citet{Lim13} also found evidence of a shell-like structure using dust extinction and $^{12}$CO (1--0) maps.
The hole-like structure may result from the expansion of a large-scale supershell produced by a supernova in the Per OB2 association that compresses the Taurus cloud from the far side \citep{Olano87, Bally08}.
An H${\alpha}$ absorption feature is detected toward the Taurus cloud (see Fig. \ref{fig:halpha} and Fig. \ref{fig:halpha_appendix}), suggesting that the Taurus cloud lies at the front surface of the large-scale supershell produced by the Per OB2 association.
The distance to the Per OB2 association is estimated to be 340 pc from the Sun \citep{Cernis93}, while the distance to the Taurus cloud is $\sim$140 pc \citep{Elias78}. These distances are consistent with the Taurus cloud lying in front of the Per OB2 association.
The B211/B213 filament also appears to be in front of the HI shell \citep[see Fig. 10 in][]{Chapman11}. This morphology suggests that the B211/B213 filament may have formed as a result of an expanding supershell.
This may provide another reason for the different initial gas velocities differed for the northeastern and southwestern sheet components besides large-scale acceleration by the gravitational potential of the B211/B213 cloud (see Sect. \ref{section:comp_model}).
The Local Bubble surrounding the Sun might also compress the Taurus cloud from the opposite direction.
The Local Bubble surrounding the Sun was produced by supernovae \citep{Snowden98,Sfeir99} and the wall of the Local Bubble is located close to the Taurus cloud \citep{Konyves07,Lallement14}.
Interestingly, the $Planck$ 353 GHz data show variations in the polarization fraction (i.e., polarized intensity/total intensity) across the B211/B213 filament, with lower and higher polarization fractions in the southwestern and northeastern parts of the filament, respectively \citep[see Fig. 10 in ][]{Planck16}. If the gas surrounding the filament is shaped as a shell-like structure with an ordered magnetic field in the plane of each sheet component and if the southwestern sheet component is oriented closer to the line of the sight compared to the northeastern sheet component (cf. Fig. \ref{fig:model}($right$)), the polarization fraction is expected to be lower in the southwestern area (dark blue in Fig. \ref{fig1}($right$)) than in the northeastern area (red in Fig. \ref{fig1}($right$)) assuming uniform dust grain properties. Moreover, both the polarization fraction and the polarization angle show smooth variations across the filament, which is consistent with the northeastern and southwestern sheets being curved (i.e., shell-like). The $Planck$ polarization results are therefore support the present model.
Using magnetic magnetohydrodynamic (MHD) numerical simulations, \citet{Inutsuka15} and \citet{Inoue18} have argued that multiple compressions associated with expanding bubbles can create star-forming filamentary structures within sheet-like molecular cloud. A similar model of anisotropic filament formation in shock compressed layers has been proposed by \citet{Chen14}, also based on MHD simulations. Such anisotropic filament formation model naturally account for transverse velocity gradients across the B211/B213 filament (see Fig. \ref{fig:3color}) and other dense molecular filaments \citep{Dhabal18}, and are good agreement with the observational picture presented.
Based on these considerations, we propose the following scenario for the formation and evolution of the B211/B213 filamentary system:
\begin{enumerate}
\item A large-scale flow associated with the Per OB2 supershell compressed and deformed the cloud centered on the B211/B213 filament and created a bent shell-like structure.
\item Owing to its strong gravitational potential, the B211/B213 filament is growing in mass due to accretion of background gas from the surrounding shell-like structure.
\end{enumerate}
\section{Conclusions}\label{Sect5}
To examine whether the B211/B213 filament is accreting gas from the surrounding cloud, we investigated the velocity patterns observed in the $^{12}$CO (1--0) and $^{13}$CO (1--0) lines. Our main findings may be summarized as follows:
\begin{enumerate}
\item The optical depth of the $^{12}$CO (1--0) line was estimated to be $\sim$3--300. The $^{12}$CO optical depth toward the B211/B213 filament is much larger than that toward the outskirts of the filament. The position-velocity diagrams observed in $^{12}$CO (1--0) and $^{13}$CO (1--0) exhibit different velocity patterns close to the filament, which is likely due to different optical depths.
\item The $^{12}$CO (1--0) and $^{13}$CO (1--0) emission from the B211/B213 filament are seen at an LSR velocity of $\sim$6 km s$^{-1}$. In the northeastern and southwestern parts of the B211/B213 filament, the $^{12}$CO (1--0) and $^{13}$CO (1--0) emission are redshifted and blueshifted, respectively. The line of sight velocities are gradually approaching the systematic velocity of the filament as one gets closer to the filament.
\item The linear extent of the cloud around the B211/B213 filament is more than 10 pc in the plane of the sky. In contrast, the depth of the cloud along the line of sight is estimated to be $\sim$0.3--0.7 pc (=$N_{\rm H_2}$/$n_{\rm critical}^{\rm ^{13}CO}$) under the assumption that the density of the surrounding material is the same as the critical density of $^{13}$CO (1--0). These results suggest that the 3D morphology of the gas cloud surrounding the B211/B213 filament is sheet-like.
\item To investigate whether the B211/B213 filament is in the process of accreting the surrounding gas material, we compared the velocity patterns observed in $^{12}$CO (1--0) and $^{13}$CO (1--0) with our 3-component model. The predictions of the model were found to be in good agreement with the distribution of $^{12}$CO (1--0) and $^{13}$CO (1--0) emission in the observed position-velocity diagrams, supporting the scenario of mass accretion along magnetic field lines into the B211/B213 filament proposed by \citet{Palmeirim13}.
\item From an inspection of the wide-field spatial distributions of H${\alpha}$ and 857 GHz dust emission in the Taurus-California-Perseus region, we concluded that the B211/B213 filament was probably formed as a result of the expansion of a large-scale supershell originated in the Per OB2 association.
This scenario provides a simple explanation for the different inclinations of the northeastern and southwestern sheet components inferred from our modeling analysis.
\item Based on these results, we propose that a) large-scale compression(s) generated by the Per OB2 association initially formed the B211/B213 filament system, and b) accretion of ambient gas material due to the gravitational potential of the filament is now responsible for the growth of the filament.
\end{enumerate}
\begin{acknowledgements}
This work was supported by the ANR-11-BS56-010 project ``STARFICH" and the European Research Council under the European Union's Seventh Framework Programme (ERC Advanced Grant Agreement no. 291294 -- `ORISTARS'). YS also received support from the ANR (project NIKA2SKY, grant agreement ANR-15-CE31-0017). P.~P. acknowledges support from the Funda\c{c}\~ao para a Ci\^encia e a Tecnologia of
Portugal (FCT) through national funds (UID/FIS/04434/2013) and by FEDER through
COMPETE2020 (POCI-01-0145-FEDER-007672) and also by the fellowship SFRH/BPD/110176/2015
funded by FCT (Portugal) and POPH/FSE (EC).
This research has made use of "Aladin sky atlas" developed at CDS, Strasbourg Observatory, France \citep{Bonnarel00, Boch14}.
\end{acknowledgements}
\bibliographystyle{aa}
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{"url":"https:\/\/www.physicsforums.com\/threads\/can-an-ordered-pair-have-identical-elements.897636\/","text":"# I Can an ordered pair have identical elements?\n\nTags:\n1. Dec 18, 2016\n\n### Stoney Pete\n\nHi guys,\n\nHere is a wacky question for you:\n\nSuppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written as the set containing the ordered pair (x, x). But does such an ordered pair, with identical elements, make any sense? After all, order is everything in ordered pairs (and tuples generally), right? But how can one distinguish order when the members of the pair are the same?\n\nI have read somewhere that the Kuratowski definition of an ordered pair (x, y) as {{x}, {x, y}} allows ordered pairs with identical elements, namely as follows: (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}. But how does having the set {{x}} tell us anything about order?\n\nWhat are your thoughts on this? And do you know of any literature dealing with this issue? Thanks for your answers!\n\nStoney\n\n2. Dec 18, 2016\n\n### jambaugh\n\nIf you eliminate ordered pairs with identical elements you couldn't define coordinates for the points on the line y=x. An ordered pair is specifically *not* a set so yes there's nothing odd about the pair having equal elements. Constructionists are fond of defining all mathematical structures in terms of sets to allow everything to have a common axiomatic footing. But there's no need to worry about such foundations when using mathematics. An ordered pair is an ordered pair. (a,b) = (x,y) if and only if a=x and b=y. Nothing more need be said except in specific applications.\n\n3. Dec 18, 2016\n\n### Stoney Pete\n\nI am kinda interested in axiomatic foundations, so for me the set-theoretic construction of ordered n-tuples is important...\n\nWhen you say an ordered pair is specifically not a set, then that is in a sense wrong, since an ordered pair can be defined in set-theoretic terms (e.g. the Kuratowski definition mentioned in my first post).\n\nAnd when you mention the linear function x=y you are missing my point. That linear function indeed gives a set of ordered pairs {(0,0), (1,1), (-1,-1),..., (n, n), (-n,-n)} where the elements in each pair are in a sense identical, but at the same time they specify different geometric values (the first one on the x axis, the second on the y axis). I mean a function where the input is in all respects identical to the output.\n\nAlso, I am not specifically talking about numerical functions but about functions in a more abstract logical manner as mappings from one set to another, no matter what is in the set. For example, causation can be seen as function mapping a set of causes to a set of effects. Now take the old philosophical idea of self-causation, where a thing (e.g. God) causes its own existence. You then have a function where input and output are identical. It is for such cases that I wonder whether the notion of ordered pairs still makes sense. I guess this is not so much mathematics but concerns rather formal logic of metaphysics.... Nevertheless, it specifies a clear question in set theory and the theory of functions: can the elements of an ordered pair (or any n-tuple for that matter) be identical in all respects?\n\n4. Dec 18, 2016\n\n### Stoney Pete\n\nFor a more mathematical example of what I have in mind, consider the function f:N\u2192N where domain and range are the same set N. This function is a set including ordered pairs such as (1, 1), (2, 2) etc. How can we speak of ordered pairs in such cases, where there really is just one element mentioned twice?\n\n5. Dec 18, 2016\n\n### jambaugh\n\nI was not mentioning a function I was mentioning a geometric object, a line, and its coordinate representation, a set of ordered pairs.\n\nI am not sure what is causing you confusion. Remember that the ordered pair is a collection, and not the objects in the collection. If you prefer you may think of the \"slots\" in the pair as representing a pair of variables. You can have distinct variables x and y that may happen to have equal values (x = 3 and y= 3 at the same time).\n\nI would avoid thinking in terms of functions explicitly as the ordered pair concept is more primitive (as in gets defined first).\n\nAnother point: in your comment you seemed to imply you were thinking of the order in the ordered pair as having to do with some intrinsic ordering of the elements. This is not the case. The pair ordering is not a property of the entries, the entries are properties of the pair object. (one is the property of \"first value\", the other is the value of the \"second\" property.)\n\nA pair is just a list of length 2 (with the natural generalization of triples, quadruples, ... n-tuples.) The ordering is simply a matter of the pair NOT being a two element set.\nWhere you ask:\nThe answer is that we can mention the same object twice, \"potato, potato\". And a pair is simply the mathematical semantics of making two references to an object or objects. The two references may refer to the same object or different objects because there's no proscription denying that possibility in the definition of the pair.\n\nIts just like the words we use in alphabetical written language. There's no problem allowing letters to repeat or occur more than once in a word. Think of an ordered pair as a \"two letter\" mathematical \"word\". mm?\n\nOnce you understand this intent in the definition of an ordered pair then you can choose your favorite way to axiomatically construct it from other objects, be they sets, or categories, or functions, or groups.\n\n6. Dec 19, 2016\n\n### Stephen Tashi\n\nPity you! $\\$ But I see your point - you aren't in doubt about the intuitive meaning of \"ordered pair\".\n\nHowever, you aren't asking a precise question. Your question asks how a particular set \"tells us anything about order\". This implies you have a definition of \"order\" in mind that the set should tell us something about. So we don't have a precise question until you offer a definition of what it means to \"tell something about order\".\n\nIf we take for granted that the natural numbers are defined or that ordinal numbers are defined, you could ask how the Kuratowski definition answers some question involving those mathematical concepts - i.e. Does the Kuratowski definition tell us if an element is the \"first\" element of an ordered pair ? However, what definition of order are we using to create the definitions of the natural numbers and ordinal numbers? - e.g. if we ask a question about a \"first\" thing, what definition of \"first\" are we using in our question?\n\nThe Kuratowski definition you quoted doesn't mention the terms \"first member of the ordered pair \" and \"second member of the ordered pair\", so it's fair to say the Kuratowski definition tells us nothing about the meaning of those terms. The game of axiomatics is to begin with certain mathematical concepts and to use those concepts to define more mathematical concepts. (It's a game that can be played in more than one way.) We want our precisely constructed creations to match our intuitive (\"Platonic\") notions of familiar mathematical objects. So, from the point of view of axiomatics the relevant question is not whether the Kuratowski definition contains within it the definitions of \"first\" and \"second\". The relevant question is whether we can use the Kuratowski definition and other already-defined concepts to construct a definition of \"first member \" and \"second member\" that matches our intuitive idea of those concepts.\n\nCan we do that? ( I don't know what Kuratowski did, but I think we can accomplish that task.)\n\n7. Dec 19, 2016\n\n### FactChecker\n\nThe mapping:\n(x,y) \u2194 {{x}, {x,y}} if x \u2260 y\n(x,x) \u2194 {{x}}\n\nSeems well defined and usable for defining ordered pairs. I don't immediately see why one would want to do that, but that might just show my lack of imagination.\n\n8. Dec 19, 2016\n\n### Stephen Tashi\n\nFrom an axiomatic point of view, we need the definition of \"ordered pair\" before we can construct the usual definition for a \"mapping\" as a \"set of ordered pairs such that ...\".\n\n9. Dec 19, 2016\n\n### FactChecker\n\nRight. I guess I should have said it is an association between two alternative (hopefully equivalent) definitions.\n\n10. Dec 19, 2016\n\n### Stephen Tashi\n\nAn interesting technicality in Kuratowski definition of ordered pair is the question of how much human perception of notation is allowed to play a role. If we phrase the definition in the form:\n\n\"The ordered pair (a,b) is defined to be the set {{a},{a,b}}\"\n\nthen we have assumed human perception distinguishes \"(a,b)\" from \"(b,a)\" and thus we assume there is a perceived property of order ( left-to-right) that is utilized in making the definition, but not explicitly explained by the definition itself.\n\nFrom the purely axiomatic point of view, the definition of an ordered pair would be better written in the form like:\n\n\"P is an ordered pair\" means that ....\n\nso that no undefined notion of \"first\" or \"leftmost\" would be assumed.\n\nI wonder how Kuratowski wrote the definition in his original papers.\n\n11. Dec 22, 2016\n\n### Logical Dog\n\nyes it can. :)\n\nin a function, the ordered pair definition usually means that the element on the left is of the domain, and that on the right is the co domain.\nthat is why you see usually a function as an ordered pair defined as\n\nN x R or similar, it is said to be a binary relation\n\nthe cartesian plane R^2 is an example of an ordered pairing where same elements exist,\nfunctions as you said, and even relations\n\nalso as far as I know an ordered pair is an element of a set, and thus, set properties and operations on it apply\n\nthe number of elements inside the ordered pair also reflect the number of sets which have been combined IN ORDER to produce it.\n\nSo {(a, b , c, d) | a in X, b in Y, c in Z, d in Q}.\n\nthe ordered pair (a, b) is different from the ordered pair (b, a) unless a = b.\nordered pairs are also used accodring to book of proof by R HAMMACK. to define relations such as < > =.\n\nSo the = relation on the cartesian plane is the set of all ordered pairs where (a,a). google reflexive relation.\n\nLast edited: Dec 22, 2016\n12. Dec 22, 2016\n\n### FactChecker\n\nYou are correct. Kuratowski's definition is valid. It's just that, as you said, {{a}, {a,b}} = {{a}, {b,a}} represents (a,b) and {{a}, {a,a}} = {{a}} represents (a,a).\n\n13. Dec 22, 2016\n\n### Logical Dog\n\nsorry, I meant the ordered pair (a,b) that is an element of some larger set of ordered pairs, meaning the set of ordered pairs is a set, and (a,b) its element. BUT thats what you understood.\n\n14. Dec 22, 2016\n\n### Logical Dog\n\nIF YOU are having trouble composing ordered pairs, it is easier to LIST out all elements horizontally, and LIST out the other sets elements Vertically, and combine them....ALSO for a function the domain is always the horizontal axis, the co domain the vertical.\n\n{1, 2, 3} x { a, b , 1}\n-----1 2 3\na\nb\n1\n\nedit: HOLD on..I dont understand how (x, x)= {{x}, {x, x}} = {{x}, {x}} = {{x}}..??\n\nfor the set {{x}, {x, x}} you get this ordered pair (assuming the set is \"mutliplied\" by itself):\n(x,x), (x,x,x) and other ugly stuff? but i agree that {x} x {x} = (x,x)\n\nare you taking the power set of (x, x)? bUT (X,x) IS NOT a set. it is a list that is part of a set containing other lists.\n\npower set of {(x,x)} = {{}, {(x,x)}}\n\nLast edited: Dec 22, 2016\n15. Dec 23, 2016\n\n### FactChecker\n\nSorry, in post 12, I was thinking that you were the OP. See the discussion in post 1. This is just his original definition followed by the basic set property that repeated elements can be ignored.\n\nKnow someone interested in this topic? 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Jamón Ibérico FAQ: 15 Most Burning Questions Answered
Jamón is the most famous dry-cured Spanish meat product in the world, possibly even the most famous Spanish product in the world, and jamón Iberico is (rightfully) considered to be its best iteration.
Only accounting for less than 10% of all jamón production in the world, jamón Iberico is praised by experts for its exceptional texture, flavor, and aroma, with even the cheapest varieties thought of as special delicacies.
In the article below, we'll be exploring the most oft-searched questions about jamón Iberico from Google, answering them as succinctly as possible without skipping any vital facts. Are there any questions regarding the topic you wish we had answered? Let us know in the comment section!
What is Jamón Ibérico?
Jamón Iberico is a dry-cured Spanish ham made from the hind leg of a pig with at least 50% of Black Iberian ancestry. What's more, for the pig to be considered of proper ancestry, its Black Iberian ancestor must have been cross-bred with the Duroc pig.
Authentic jamón Iberico can only be produced in Spain or Portugal, as these are native breeding countries of Iberian pigs.
The first step of making jamón Iberico is to clean and cover the hind leg of the Iberian pig with salt. After about two weeks, the salt is washed off, and the meat is left to dry. After around 4-6 weeks, it's moved to another facility for the final curing process. Jamón Iberico needs to be cured for at least twenty-four months, but the process can last for up to four full years.
What is Special About Jamón Ibérico?
Jamón Iberico accounts for around 7-10% of overall jamón production in the world. It can only be made from a specific type of meat (the hind leg of a Black Iberian pig) and has an uncommonly long curing period that lasts between twenty-four and forty-eight months.
The combination of these factors creates a product with a very precise texture and flavor profile. Jamón Iberico is supposed to be chewy and somewhat dry, but at the same time buttery and delicate, with an intense pork flavor and a multitude of undertones that leave a strong aftertaste.
What Does Jamón Ibérico Taste Like?
Jamón Iberico has a very intense and complex flavor. The primary flavor profile is meaty and savory, with a well-expressed pork flavor. But it also has a multi-layered undertone, with robust nuttiness, tender sweetness, smokiness, and even hints of spiciness to it.
Which is the Best Jamón Ibérico?
While all jamón Iberico is considered to be a high-quality delicacy, they are still separated into strict categories, depending on the ancestry, diets of the pigs used in production, and the production method itself.
There are two systems in place categorizing jamón Iberico. The first one utilizes colored labels and is used to denote the ancestry, living conditions, and diets of the pigs:
The black label is used to mark that the pork comes from a pig with 100% Iberian ancestry, was raised "free-range," and has been kept on an acorn diet;
The red label is used to mark that the pork comes from a pig with at least 50% Iberian ancestry, was raised "free-range," and has been kept on an acorn diet;
The green label is used to mark that the pork comes from a pig with at least 50% Iberian ancestry that was raised "free-range," but its diet was supplemented with grains and cereals;
The white label is used to mark that the pork comes from a pig with at least 50% Iberian ancestry but has been "conventionally farmed" (i.e., kept on the farm in captivity) and fed a grain-and-cereal-based diet.
The second categorizing system rises from the first one and has to do with 1) the type of pork used in production; 2) the length of the jamón curing process:
Jamón Ibérico de Cebo is usually marked with a white label. It's cured for at least 24 months. It accounts for around 2/3 of all jamón Iberico produced.
Jamón Ibérico de Cebo de Campo is usually marked with a green label. It's cured for at least 36 months.
Jamón Ibérico de Bellota is made exclusively from pigs on an acorn diet and is thus marked with either a red or black label. The red label jamón Iberico is typically cured for a period between 36 to 48 months, but the process can sometimes last longer and result in a more expensive product. Pata Negra indicates black label jamón Iberico made from 100% Iberian pig (it means black paw in Spanish) since 2014 (the use of the term wasn't regulated before that).
There are also regional varieties of jamón Iberico that have been granted protected designation of origin status and thus cannot be made anywhere else in the world: jamón Ibérico de Guijuelo, jamón Ibérico de Jabugo, jamón Ibérico Dehesa de Extremadura, and jamón Ibérico de Los Pedroches.
However, while they can be harder to find than standard Jamón Ibérico de Bellota Pata Negra, the latter is commonly thought to be the best jamón Iberico variety among all.
What is the Difference Between Jamón Ibérico and Jamón Serrano?
The breed of pig used for jamón production is the main difference between the two. Jamón serrano can be made with most conventional pig breeds (though white Duroc pigs are considered the best), and the production process is overall laxer, as most jamón serrano varieties are not controlled by any regulatory body.
Jamón Iberico typically is a darker, brighter red, and the marbling is more pronounced, with the white streaks of fat more stark. It has a smoother, more tender, and buttery texture compared to jamón serrano due to the higher fat content in Iberian pork. The flavor is also more complex, a bit sweeter but with noticeable notes of spice and smokiness. It's also much nuttier, with the pork flavor more well-balanced with other flavor undertones.
You can read more about the similarities and differences between jamón Ibérico and jamón serrano in our guide that covers this information in-depth.
Is Prosciutto and Jamón Ibérico the Same?
No. While Italian prosciutto is often compared to Spanish jamón as they're both cured meat products made with a pig's hind leg, their textures and flavor profiles significantly differ from one another.
Jamón Ibérico has a more intense meaty aroma and a more complex and robust flavor profile with distinctly nutty and smoky notes. While there are hints of sweetness, the overall flavor profile is more savory and a bit spicy.
In contrast, prosciutto has a more delicate and subtle flavor, with more well-pronounced sweetness.
Also, the long curing process ensures that jamón Iberico, while supple and tender due to the higher-than-average fat content, is still drier and chewier than most cured meat products, including prosciutto.
Can You Eat Jamón Ibérico Raw?
Jamón Iberico is cured meat product, which means that while technically it's uncooked, it's not precisely raw either. Yes, jamón Iberico can be eaten as is, straight out of the package or off the bone, freshly cut. The curing process makes it completely safe for consumption.
Can You Cook Jamón Ibérico?
Yes, you can, though many would argue cooking takes away the flavor complexities that it's prized so highly for in the first place.
The key is not to overcook jamón, even if you're adding it to more complex dishes. Frying it in a pan for a couple of minutes is more than enough to get it crispy if it's the extra crunch you're looking for. The same goes for using it as a base for pasta sauces, a filling in hot paninis, an ingredient in omelets, or a salad topping.
The same goes for the oven: don't cook it for more than a couple of minutes. Say, if you're adding it as a pizza topping, do so at the last minute, with the cheese already melted.
How Do You Eat Jamón Ibérico?
Ideally, jamón Iberico should be thinly sliced and served at room temperature.
Jamón Iberico is best eaten by itself, only accompanied by fresh white bread or tortas, and garnished with a drizzle of extra virgin olive oil. This way, nothing overwhelms its flavor, and you'll get to enjoy the complex flavor profile of this delicacy properly.
What is Good with Jamón Ibérico? What Cheese Goes with Jamón Ibérico?
With jamón Iberico, less is more. While the primary flavor profile is robust and intense, the underlying flavor notes are easy to overwhelm, which means pairing it with other rich ingredients takes away a lot of flavor complexity.
Jamón, be it serrano or Iberico, is traditionally paired with Manchego cheese. Otherwise, in the case of jamón Iberico, mellow, milky cheese makes the best pairing: mozzarella, Emmental, ricotta, chevre, etc. This type of cheese will cut through the saltiness of jamón Iberico without overwhelming the overall flavor.
If you do decide to serve it as a part of the charcuterie board, it's best to pair it with fruits and nuts with a mellow, moderately sweet flavor profile. Apples, pears, and apricots are good choices for fruits, while hazelnuts and almonds make the best pairing among nuts.
Can I Bring Jamón Ibérico from Spain to the USA?
No. Bringing cured meat products like jamón, prosciutto, or salami onto US soil without proper authorization isn't permitted.
If you wish to get a taste of jamón Iberico, you should purchase it from local sellers who have been granted such authorization.
How Long Does Jamón Ibérico Last?
It depends on whether the jamón in question has been commercially packed or is freshly sliced at an artisanal butcher shop.
Commercially packaged jamón Iberico can last anywhere between 9 and 12 months. Pay attention to the expiration date on the label.
Most butcher shops vacuum-seal their jamón after slicing it, which keeps the product fresh for longer. Vacuum-sealed jamón Iberico can last up to 90 days.
As with other dry-cured meat products, once the vacuum is broken, the shelf life of jamón Iberico will start to shorten rapidly. Commercially packaged jamón Iberico will last comparatively longer, for about two months, while the artisanal jamón is best consumed within three weeks.
How Do You Keep Jamón Ibérico Fresh? Does Jamón Ibérico Need to be Refrigerated?
Not necessarily, but it is highly desirable. Similar to other cured meat products, jamón Iberico can last at lower room temperature without damage as long as it's dry and dark. But it's easier to ensure that jamón is appropriately protected from heat, direct light, humidity, and exposure to oxygen if it's kept in the refrigerator.
In other words: will it spoil if it's left out overnight? Not unless you do so with the vacuum unsealed in extreme heat and humidity. Will it last better long-term if kept in the fridge? Absolutely.
Make sure to tightly wrap the slices in plastic wrap and keep them in an air-tight container once you've broken the vacuum.
Can You Freeze Jamón Ibérico?
It's possible in theory, but jamón Iberico is a very delicate product, and defrosting will damage its delicate texture and taste. Unless the only other alternative is leaving it to spoil, you should skip the freezer with this one.
In any case, don't expect it to taste as good thawed as it does fresh.
Why is Jamón Ibérico So Expensive?
Jamón Iberico is so expensive because it's expensive to breed and raise Iberian pigs.
The Iberian pigs haven't been genetically modified to yield like most industrial pigs:
They're smaller in size and yield less meat per head;
They have smaller litters (maximum of 6 in the best conditions, compared to 8-12 an industrialized pig can yield);
They're never administered antibiotics or hormones;
They're slow growers and need about 12-14 (sometimes up to 18) to reach proper weight for slaughter;
They require ample space if they're to be raised "free-range," which means the farmer needs to maintain a larger swath of land per pig.
All of these conditions produce what most experts consider to be the best quality pork in the world, but they make the process of breeding and raising Iberian pigs a costly affair which, in turn, is reflected in products made with Iberian pork, including jamón Iberico.
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// When the user logs out and reaches login page,
// we clear all the history and cache to prevent back link
$scope.$on('$ionicView.enter', function(ev) {
if(ev.targetScope !== $scope){
$ionicHistory.clearHistory();
$ionicHistory.clearCache();
}
});
//Check if user already logged in
firebase.auth().onAuthStateChanged(function(user) {
if (user) {
$ionicHistory.nextViewOptions({
historyRoot: true
});
$ionicSideMenuDelegate.canDragContent(true); // Sets up the sideMenu dragable
$rootScope.extras = true;
sharedUtils.hideLoading();
$state.go('menu2', {}, {location: "replace"});
}
});
$scope.loginEmail = function(formName,cred) {
if(formName.$valid) { // Check if the form data is valid or not
sharedUtils.showLoading();
//Email
firebase.auth().signInWithEmailAndPassword(cred.email,cred.password).then(function(result) {
// You dont need to save the users session as firebase handles it
// You only need to :
// 1. clear the login page history from the history stack so that you cant come back
// 2. Set rootScope.extra;
// 3. Turn off the loading
// 4. Got to menu page
$ionicHistory.nextViewOptions({
historyRoot: true
});
$rootScope.extras = true;
sharedUtils.hideLoading();
$state.go('menu2', {}, {location: "replace"});
},
function(error) {
sharedUtils.hideLoading();
sharedUtils.showAlert("Please note","Authentication Error");
}
);
}else{
sharedUtils.showAlert("Please note","Entered data is not valid");
}
};
$scope.loginFb = function(){
//Facebook Login
};
$scope.loginGmail = function(){
//Gmail Login
};
})
.controller('signupCtrl', function($scope,$rootScope,sharedUtils,$ionicSideMenuDelegate,
$state,fireBaseData,$ionicHistory) {
$rootScope.extras = false; // For hiding the side bar and nav icon
$scope.signupEmail = function (formName, cred) {
if (formName.$valid) { // Check if the form data is valid or not
sharedUtils.showLoading();
//Main Firebase Authentication part
firebase.auth().createUserWithEmailAndPassword(cred.email, cred.password).then(function (result) {
//Add name and default dp to the Autherisation table
result.updateProfile({
displayName: cred.name,
photoURL: "default_dp"
}).then(function() {}, function(error) {});
//Add phone number to the user table
fireBaseData.refUser().child(result.uid).set({
telephone: cred.phone
});
//Registered OK
$ionicHistory.nextViewOptions({
historyRoot: true
});
$ionicSideMenuDelegate.canDragContent(true); // Sets up the sideMenu dragable
$rootScope.extras = true;
sharedUtils.hideLoading();
$state.go('menu2', {}, {location: "replace"});
}, function (error) {
sharedUtils.hideLoading();
sharedUtils.showAlert("Please note","Sign up Error");
});
}else{
sharedUtils.showAlert("Please note","Entered data is not valid");
}
}
})
.controller('menu2Ctrl', function($scope,$rootScope,$ionicSideMenuDelegate,fireBaseData,$state,
$ionicHistory,$firebaseArray,sharedCartService,sharedUtils) {
//Check if user already logged in
firebase.auth().onAuthStateChanged(function(user) {
if (user) {
$scope.user_info=user; //Saves data to user_info
}else {
$ionicSideMenuDelegate.toggleLeft(); //To close the side bar
$ionicSideMenuDelegate.canDragContent(false); // To remove the sidemenu white space
$ionicHistory.nextViewOptions({
historyRoot: true
});
$rootScope.extras = false;
sharedUtils.hideLoading();
$state.go('tabsController.login', {}, {location: "replace"});
}
});
// On Loggin in to menu page, the sideMenu drag state is set to true
$ionicSideMenuDelegate.canDragContent(true);
$rootScope.extras=true;
// When user visits A-> B -> C -> A and clicks back, he will close the app instead of back linking
$scope.$on('$ionicView.enter', function(ev) {
if(ev.targetScope !== $scope){
$ionicHistory.clearHistory();
$ionicHistory.clearCache();
}
});
$scope.loadMenu = function() {
sharedUtils.showLoading();
$scope.menu=$firebaseArray(fireBaseData.refMenu());
sharedUtils.hideLoading();
}
$scope.showProductInfo=function (id) {
};
$scope.addToCart=function(item){
sharedCartService.add(item);
};
})
.controller('offersCtrl', function($scope,$rootScope) {
//We initialise it on all the Main Controllers because, $rootScope.extra has default value false
// So if you happen to refresh the Offer page, you will get $rootScope.extra = false
//We need $ionicSideMenuDelegate.canDragContent(true) only on the menu, ie after login page
$rootScope.extras=true;
})
.controller('indexCtrl', function($scope,$rootScope,sharedUtils,$ionicHistory,$state,$ionicSideMenuDelegate,sharedCartService) {
//Check if user already logged in
firebase.auth().onAuthStateChanged(function(user) {
if (user) {
$scope.user_info=user; //Saves data to user_info
//Only when the user is logged in, the cart qty is shown
//Else it will show unwanted console error till we get the user object
$scope.get_total= function() {
var total_qty=0;
for (var i = 0; i < sharedCartService.cart_items.length; i++) {
total_qty += sharedCartService.cart_items[i].item_qty;
}
return total_qty;
};
}else {
$ionicSideMenuDelegate.toggleLeft(); //To close the side bar
$ionicSideMenuDelegate.canDragContent(false); // To remove the sidemenu white space
$ionicHistory.nextViewOptions({
historyRoot: true
});
$rootScope.extras = false;
sharedUtils.hideLoading();
$state.go('tabsController.login', {}, {location: "replace"});
}
});
$scope.logout=function(){
sharedUtils.showLoading();
// Main Firebase logout
firebase.auth().signOut().then(function() {
$ionicSideMenuDelegate.toggleLeft(); //To close the side bar
$ionicSideMenuDelegate.canDragContent(false); // To remove the sidemenu white space
$ionicHistory.nextViewOptions({
historyRoot: true
});
$rootScope.extras = false;
sharedUtils.hideLoading();
$state.go('tabsController.login', {}, {location: "replace"});
}, function(error) {
sharedUtils.showAlert("Error","Logout Failed")
});
}
})
.controller('myCartCtrl', function($scope,$rootScope,$state,sharedCartService) {
$rootScope.extras=true;
//Check if user already logged in
firebase.auth().onAuthStateChanged(function(user) {
if (user) {
$scope.cart=sharedCartService.cart_items; // Loads users cart
$scope.get_qty = function() {
$scope.total_qty=0;
$scope.total_amount=0;
for (var i = 0; i < sharedCartService.cart_items.length; i++) {
$scope.total_qty += sharedCartService.cart_items[i].item_qty;
$scope.total_amount += (sharedCartService.cart_items[i].item_qty * sharedCartService.cart_items[i].item_price);
}
return $scope.total_qty;
};
}
//We dont need the else part because indexCtrl takes care of it
});
$scope.removeFromCart=function(c_id){
sharedCartService.drop(c_id);
};
$scope.inc=function(c_id){
sharedCartService.increment(c_id);
};
$scope.dec=function(c_id){
sharedCartService.decrement(c_id);
};
$scope.checkout=function(){
$state.go('checkout', {}, {location: "replace"});
};
})
.controller('lastOrdersCtrl', function($scope,$rootScope,fireBaseData,sharedUtils) {
$rootScope.extras = true;
sharedUtils.showLoading();
//Check if user already logged in
firebase.auth().onAuthStateChanged(function (user) {
if (user) {
$scope.user_info = user;
fireBaseData.refOrder()
.orderByChild('user_id')
.startAt($scope.user_info.uid).endAt($scope.user_info.uid)
.once('value', function (snapshot) {
$scope.orders = snapshot.val();
$scope.$apply();
});
sharedUtils.hideLoading();
}
});
})
.controller('favouriteCtrl', function($scope,$rootScope) {
$rootScope.extras=true;
})
.controller('settingsCtrl', function($scope,$rootScope,fireBaseData,$firebaseObject,
$ionicPopup,$state,$window,$firebaseArray,
sharedUtils) {
//Bugs are most prevailing here
$rootScope.extras=true;
//Shows loading bar
sharedUtils.showLoading();
//Check if user already logged in
firebase.auth().onAuthStateChanged(function(user) {
if (user) {
//Accessing an array of objects using firebaseObject, does not give you the $id , so use firebase array to get $id
$scope.addresses= $firebaseArray(fireBaseData.refUser().child(user.uid).child("address"));
// firebaseObject is good for accessing single objects for eg:- telephone. Don't use it for array of objects
$scope.user_extras= $firebaseObject(fireBaseData.refUser().child(user.uid));
$scope.user_info=user; //Saves data to user_info
//NOTE: $scope.user_info is not writable ie you can't use it inside ng-model of <input>
//You have to create a local variable for storing emails
$scope.data_editable={};
$scope.data_editable.email=$scope.user_info.email; // For editing store it in local variable
$scope.data_editable.password="";
$scope.$apply();
sharedUtils.hideLoading();
}
});
$scope.addManipulation = function(edit_val) { // Takes care of address add and edit ie Address Manipulator
if(edit_val!=null) {
$scope.data = edit_val; // For editing address
var title="Edit Address";
var sub_title="Edit your address";
}
else {
$scope.data = {}; // For adding new address
var title="Add Address";
var sub_title="Add your new address";
}
// An elaborate, custom popup
var addressPopup = $ionicPopup.show({
template: '<input type="text" placeholder="Nick Name" ng-model="data.nickname"> <br/> ' +
'<input type="text" placeholder="Address" ng-model="data.address"> <br/> ' +
'<input type="number" placeholder="Pincode" ng-model="data.pin"> <br/> ' +
'<input type="number" placeholder="Phone" ng-model="data.phone">',
title: title,
subTitle: sub_title,
scope: $scope,
buttons: [
{ text: 'Close' },
{
text: '<b>Save</b>',
type: 'button-positive',
onTap: function(e) {
if (!$scope.data.nickname || !$scope.data.address || !$scope.data.pin || !$scope.data.phone ) {
e.preventDefault(); //don't allow the user to close unless he enters full details
} else {
return $scope.data;
}
}
}
]
});
addressPopup.then(function(res) {
if(edit_val!=null) {
//Update address
if(res!=null){ // res ==null => close
fireBaseData.refUser().child($scope.user_info.uid).child("address").child(edit_val.$id).update({ // set
nickname: res.nickname,
address: res.address,
pin: res.pin,
phone: res.phone
});
}
}else{
//Add new address
fireBaseData.refUser().child($scope.user_info.uid).child("address").push({ // set
nickname: res.nickname,
address: res.address,
pin: res.pin,
phone: res.phone
});
}
});
};
// A confirm dialog for deleting address
$scope.deleteAddress = function(del_id) {
var confirmPopup = $ionicPopup.confirm({
title: 'Delete Address',
template: 'Are you sure you want to delete this address',
buttons: [
{ text: 'No' , type: 'button-stable' },
{ text: 'Yes', type: 'button-assertive' , onTap: function(){return del_id;} }
]
});
confirmPopup.then(function(res) {
if(res) {
fireBaseData.refUser().child($scope.user_info.uid).child("address").child(res).remove();
}
});
};
$scope.save= function (extras,editable) {
//1. Edit Telephone doesnt show popup 2. Using extras and editable // Bugs
if(extras.telephone!="" && extras.telephone!=null ){
//Update Telephone
fireBaseData.refUser().child($scope.user_info.uid).update({ // set
telephone: extras.telephone
});
}
//Edit Password
if(editable.password!="" && editable.password!=null ){
//Update Password in UserAuthentication Table
firebase.auth().currentUser.updatePassword(editable.password).then(function(ok) {}, function(error) {});
sharedUtils.showAlert("Account","Password Updated");
}
//Edit Email
if(editable.email!="" && editable.email!=null && editable.email!=$scope.user_info.email){
//Update Email/Username in UserAuthentication Table
firebase.auth().currentUser.updateEmail(editable.email).then(function(ok) {
$window.location.reload(true);
//sharedUtils.showAlert("Account","Email Updated");
}, function(error) {
sharedUtils.showAlert("ERROR",error);
});
}
};
$scope.cancel=function(){
// Simple Reload
$window.location.reload(true);
console.log("CANCEL");
}
})
.controller('supportCtrl', function($scope,$rootScope) {
$rootScope.extras=true;
})
.controller('forgotPasswordCtrl', function($scope,$rootScope) {
$rootScope.extras=false;
})
.controller('checkoutCtrl', function($scope,$rootScope,sharedUtils,$state,$firebaseArray,
$ionicHistory,fireBaseData, $ionicPopup,sharedCartService) {
$rootScope.extras=true;
//Check if user already logged in
firebase.auth().onAuthStateChanged(function(user) {
if (user) {
$scope.addresses= $firebaseArray( fireBaseData.refUser().child(user.uid).child("address") );
$scope.user_info=user;
}
});
$scope.pay=function(address,payment){
// Loop throw all the cart item
for (var i = 0; i < sharedCartService.cart_items.length; i++) {
//Add cart item to order table
fireBaseData.refOrder().push({
//Product data is hardcoded for simplicity
product_name: sharedCartService.cart_items[i].item_name,
product_price: sharedCartService.cart_items[i].item_price,
product_image: sharedCartService.cart_items[i].item_image,
product_id: sharedCartService.cart_items[i].$id,
//item data
item_qty: sharedCartService.cart_items[i].item_qty,
//Order data
user_id: $scope.user_info.uid,
user_name:$scope.user_info.displayName,
status: "Queued"
});
//fireBaseData.refOrder().child($scope.cart_items[i].item_qty).remove();
}
//Remove users cart
fireBaseData.refCart().child($scope.user_info.uid).remove();
sharedUtils.showAlert("Info", "Order Successfull");
// Go to past order page
$ionicHistory.nextViewOptions({
historyRoot: true
});
$state.go('lastOrders', {}, {location: "replace", reload: true});
}
$scope.addManipulation = function(edit_val) { // Takes care of address add and edit ie Address Manipulator
if(edit_val!=null) {
$scope.data = edit_val; // For editing address
var title="Edit Address";
var sub_title="Edit your address";
}
else {
$scope.data = {}; // For adding new address
var title="Add Address";
var sub_title="Add your new address";
}
// An elaborate, custom popup
var addressPopup = $ionicPopup.show({
template: '<input type="text" placeholder="Nick Name" ng-model="data.nickname"> <br/> ' +
'<input type="text" placeholder="Address" ng-model="data.address"> <br/> ' +
'<input type="number" placeholder="Pincode" ng-model="data.pin"> <br/> ' +
'<input type="number" placeholder="Phone" ng-model="data.phone">',
title: title,
subTitle: sub_title,
scope: $scope,
buttons: [
{ text: 'Close' },
{
text: '<b>Save</b>',
type: 'button-positive',
onTap: function(e) {
if (!$scope.data.nickname || !$scope.data.address || !$scope.data.pin || !$scope.data.phone ) {
e.preventDefault(); //don't allow the user to close unless he enters full details
} else {
return $scope.data;
}
}
}
]
});
addressPopup.then(function(res) {
if(edit_val!=null) {
//Update address
fireBaseData.refUser().child($scope.user_info.uid).child("address").child(edit_val.$id).update({ // set
nickname: res.nickname,
address: res.address,
pin: res.pin,
phone: res.phone
});
}else{
//Add new address
fireBaseData.refUser().child($scope.user_info.uid).child("address").push({ // set
nickname: res.nickname,
address: res.address,
pin: res.pin,
phone: res.phone
});
}
});
};
})
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,119 |
Q: Pythonic way to convert paragraph to sentence with associated entities Data is coming from a dataset used to train a ner and it looks like this :
'This is my text. It has multiple sentence. It is great'
[{'text': 'is', 'start': 5, 'end': 7, 'type': 'ENTITY'}, {'text' :'has', 'start':20, 'end':23, type:'ENTITY}, {'text': 'is', 'start': 46, 'end': 48, 'type': 'ENTITY'},
It's composed of a text with its associated entities. Entities got a start and end values that determine the position of the first and last character of the word/sequence of word from the beginning of the text.
My goal is to split the text into sentence with the associated entities, this mean to check whether the entity is within the sentence. It also mean to get the correct start and end position (from start of sentence and not of text).
Here is for example, the desired output for the second sentence of the paragraph. It contains every entities of the sentence with updated start and end values :
('This is my text.',
[{'text': 'is',
'start': 5,
'end': 7,
'type': 'ENTITY'}]),
('It has multiple sentence.',
[{'text': 'has',
'start': 3,
'end': 6,
'type': 'ENTITY'}]),
('It is great',
[{'text': 'is',
'start': 3,
'end': 5,
'type': 'ENTITY'}]),
I managed to do it with horrible multiples loops and if, it's quite disguting. If anyone could tell me what's the most pythonic/efficient way to do so ?
If you're curious to see, here's how I did it with loops :
def split_sentences(st):
st = st.strip() + '. '
sentences = re.split(r'[.?!][.?!\s]+', st)
return sentences[:-1]
sentences = split_sentences(text)
start_sentence = 0
final_res = []
for text in sentences:
sentence_ent = []
for ent in full_line['entities']:
if(ent['start']-start_sentence < len(text) and ent['start'] - start_sentence >0):
if(ent['type'] in labels):
ent['start'] -= start_sentence
ent['end'] -= start_sentence
sentence_ent.append(ent)
if(len(sentence_ent)>0):
final_res.append((text, sentence_ent))
start_sentence += len(text)+2
Thanks
A: I would have used set for this
import re
sentence = 'Epidemiology of clinical feline herpesvirus infection in zoo-housed cheetahs (Acinonyx jubatus) OBJECTIVE: To determine the incidence of and risk factors for clinical feline herpesvirus (FHV) infection in zoo-housed cheetahs and determine whether dam infection was associated with offspring infection. DESIGN: Retrospective cohort study. ANIMALS: 144 cheetah cubs born in 6 zoos from 1988 through 2007.'
entities = [{'text': 'feline', 'start': 25, 'end': 31, 'type': 'EUKARYOTE'}, {'text': 'herpesvirus', 'start': 32, 'end': 43, 'type': 'VIRUS'}, {'text': 'zoo', 'start': 57, 'end': 60, 'type': 'ORGANISM'}, {'text': 'cheetahs', 'start': 68, 'end': 76, 'type': 'EUKARYOTE'}, {'text': 'feline', 'start': 167, 'end': 173, 'type': 'EUKARYOTE'}, {'text': 'herpesvirus', 'start': 174, 'end': 185, 'type': 'VIRUS'}, {'text': 'infection', 'start': 192, 'end': 201, 'type': 'DISEASE_OR_SYNDROME'}, {'text': 'zoo', 'start': 205, 'end': 208, 'type': 'ORGANISM'}, {'text': 'cheetahs', 'start': 216, 'end': 224, 'type': 'EUKARYOTE'}, {'text': 'dam infection', 'start': 247, 'end': 260, 'type': 'DISEASE_OR_SYNDROME'}, {'text': 'infection', 'start': 291, 'end': 300, 'type': 'DISEASE_OR_SYNDROME'}, {'text': 'Retrospective cohort study', 'start': 310, 'end': 336, 'type': 'GENE_OR_GENOME'}, {'text': 'ANIMALS', 'start': 338, 'end': 345, 'type': 'CHEMICAL'}, {'text': '144', 'start': 347, 'end': 350, 'type': 'CARDINAL'}, {'text': 'cheetah', 'start': 351, 'end': 358, 'type': 'CHEMICAL'}, {'text': 'cubs', 'start': 359, 'end': 363, 'type': 'ORGANISM'}, {'text': '1988 through', 'start': 384, 'end': 396, 'type': 'DATE'}, {'text': '2007', 'start': 397, 'end': 401, 'type': 'DATE'}]
def split_sentences(st):
st = st.strip() + '. '
sentences = re.split(r'[.?!][.?!\s]+', st)
return sentences[:-1]
splitted_sentence = split_sentences(sentence)
modified_entity = {}
for i in entities:
modified_entity[i['text'].lower()] = i
output = []
for each_sentence in splitted_sentence:
temp_entity = []
words = set((i.lower() for i in re.findall(r'\w+', each_sentence)))
match = words.intersection(modified_entity.keys())
for each_match in match:
#start = each_sentence.lower().find(each_match)
start_end = [i.span() for i in re.finditer(each_match, each_sentence.lower())]
for start, end in start_end:
modified_entity[each_match].update({
'start': start,
'end': end
})
temp_entity.append(modified_entity[each_match].copy())
output.append([each_sentence, temp_entity])
print(output)
[['Epidemiology of clinical feline herpesvirus infection in zoo-housed cheetahs (Acinonyx jubatus) OBJECTIVE: To determine the incidence of and risk factors for clinical feline herpesvirus (FHV) infection in zoo-housed cheetahs and determine whether dam infection was associated with offspring infection',
[{'text': 'infection',
'start': 44,
'end': 53,
'type': 'DISEASE_OR_SYNDROME'},
{'text': 'infection',
'start': 192,
'end': 201,
'type': 'DISEASE_OR_SYNDROME'},
{'text': 'infection',
'start': 251,
'end': 260,
'type': 'DISEASE_OR_SYNDROME'},
{'text': 'infection',
'start': 291,
'end': 300,
'type': 'DISEASE_OR_SYNDROME'},
{'text': 'cheetahs', 'start': 68, 'end': 76, 'type': 'EUKARYOTE'},
{'text': 'cheetahs', 'start': 216, 'end': 224, 'type': 'EUKARYOTE'},
{'text': 'feline', 'start': 25, 'end': 31, 'type': 'EUKARYOTE'},
{'text': 'feline', 'start': 167, 'end': 173, 'type': 'EUKARYOTE'},
{'text': 'zoo', 'start': 57, 'end': 60, 'type': 'ORGANISM'},
{'text': 'zoo', 'start': 205, 'end': 208, 'type': 'ORGANISM'},
{'text': 'herpesvirus', 'start': 32, 'end': 43, 'type': 'VIRUS'},
{'text': 'herpesvirus', 'start': 174, 'end': 185, 'type': 'VIRUS'}]],
['DESIGN: Retrospective cohort study', []],
['ANIMALS: 144 cheetah cubs born in 6 zoos from 1988 through 2007',
[{'text': '144', 'start': 9, 'end': 12, 'type': 'CARDINAL'},
{'text': 'cheetah', 'start': 13, 'end': 20, 'type': 'CHEMICAL'},
{'text': '2007', 'start': 59, 'end': 63, 'type': 'DATE'},
{'text': 'cubs', 'start': 21, 'end': 25, 'type': 'ORGANISM'},
{'text': 'ANIMALS', 'start': 0, 'end': 7, 'type': 'CHEMICAL'}]]]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,453 |
Conotyla elpenor är en mångfotingart som beskrevs av Shear 1971. Conotyla elpenor ingår i släktet Conotyla och familjen Conotylidae. Inga underarter finns listade i Catalogue of Life.
Källor
Vinterdubbelfotingar
elpenor | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,287 |
Q: A Couple Obj-C Questions EDIT: My internet went out last night ._.
Well I'm new to the language. I got some basics down but:
*
*-(XYPoint *)origin
In this, why does the return value for this method look like a pointer? I'm confused. I know what void, id, double, etc are but I don't get why this has a pointer.
*I was going through Kochans book, and I got to a program.
myRect.origin = myPoint
NSLog(@"origin:(%i,%i)",myRect.origin.x,myRect.origin.y)
Or something like that.
But after the NsLog I put In a release. Then called the origin again, but it still got printed. Shouldn't it have gave an error?
Later, I printed another NSLog calling another variable, then after, I called the origin again, but this time I was given an error, though I was not when I tried calling it after the release. Sorry if this seems vague, but I will elaborate if needed.
A: *
*Simply because the return type is a pointer type, so it's designated as returning a pointer.
Note that anything can be turned into a pointer type where the pointer is returned rather than the object it points to in memory, but that's probably something more advanced than just Objective-C classes.
*Release doesn't always mean the object gets deallocated right away. It can be instantaneous, or in a second, or a few, that it actually happens. Or, if the object was retained elsewhere, then it doesn't get deallocated yet at all.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,963 |
Daniel Seiter (nebo Saiter) (asi 1642 nebo 1647 Vídeň- 1705 Turín) byl italský barokní malíř.
Byl žákem Johanna Karla Lotha v Benátkách, pak se přestěhoval do Říma, aby pracoval v ateliéru
Carlo Maratta. Cestoval do Turína, kde pomáhal zdobit fresky v kupoli kaple v Ospedale Maggiore.
Také maloval v Brunšviku a Drážďanech.
Reference
Externí odkazy
Narození v 17. století
Úmrtí v roce 1705
Italští barokní malíři
Rakouští malíři
Muži
Bentvueghels | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,579 |
Q: A min-heap with better than O(logn) increase key? I'm using a priority queue that initially bases the priority of its elements on a heuristic. As elements are dequeued the heuristic is updated and elements currently in the queue may have their keys increased.
I know there are heaps (Fibonacci heaps specifically) that have amortized O(1) decrease key operations, but are there any heap structures that have similar bounds on the increase key operations?
For my application this is far from a performance issue (a binary heap works fine) it's really just about academic curiosity.
Edit: to clarify, I'm looking for a data structure that has a faster than O(logn) time for the increase key operation, not decrease key. My application never decreases the key as the heuristic over-estimates the priority.
A: Binary heaps are too unflexible to beat logarithmic complexity.
Binomial heaps just allow a more efficient join-operation.
Other heaps with good decrease-key performance are pairing heaps and 2-3 heaps
A: Binomial heaps take o(log n) time for decrease key operations! Isn't this slower than fibonacci heaps?
A: Actually, with Fibonacci heaps, the increase key operation is the same as decrease key. IMHO, it is only a tradition to name the operation "decrease key", because it is used in some algorithms. But the Fibonacci heap implementation allows both.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,047 |
Q: Can we change "You have 1 new answer; 1 new comment" to something grammatically correct? When you get new answers and comments, Stack Overflow tells you "You have 1 new answer; 1 new comment." As far as I understand, that's not how semicolons work. Here are a couple suggestions for what could be used instead:
*
*You have 1 new answer, 1 new
comment
*You have 1 new answer and 1 new
comment
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,034 |
Currency converter result page of conversion 560 Australian Dollar in Albanian Lek. Exchange rate of this pair updated every day. FXConvert.net is free, fast and easy to use online tool which give latest rates of pair AUD-ALL.
Price of five hundred and sixty Australian Dollar, cost 44241.88 Albanian Lek and converted with today exchange rate.
This graph show how much is 560 Australian Dollars in Albanian Lek - 44241.88085 ALL, according to actual pair rate equal 1 AUD = 79.0034 ALL. Yesterday this currency exchange rate plummeted on -0.05052 and was lek 78.95284 Albanian Lek for AU$ 1. On the last week currencies rate was on lek0.31457 ALL higher. Last month was lower on - lek 0.86493. Price for 1 AU dollar was 78.13843 Lek, so 560 Australian Dollar was worth 43757.522319974 in Albanian Lek. On this graph you can see trend of change 560 AUD to ALL. And average currency exchange rate for the last week was lek 78.83181 ALL for AU$1 AUD. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,534 |
Jasper Hale is een van de hoofdpersonen uit de film en de boekenreeks Twilight, geschreven door Stephenie Meyer. Hij is de adoptiezoon van Carlisle en Esme Cullen en de adoptiebroer van Edward, Emmett en Alice Cullen en Rosalie Hale. Alice Cullen is tegelijkertijd ook zijn echtgenote.
Hoewel Jasper en Rosalie dezelfde achternaam dragen, zijn ze geen echte familie van elkaar. Jasper heeft de naam Hale aangenomen om het voor de buitenwereld, die immers niets mag weten van het feit dat het gezin Cullen uit vampiers bestaat, te laten lijken of hij en Rosalie een tweeling zijn. Dit kan omdat de twee qua uiterlijk nogal op elkaar lijken.
In de film wordt de rol van Jasper vertolkt door Jackson Rathbone.
Achtergrond
Jasper Hale werd geboren als Jasper Whitlock in 1843 in Texas. Hij ging in 1861 het leger in, om mee te vechten in de Amerikaanse Burgeroorlog. Vanwege zijn charismatische persoonlijkheid en zijn strategisch inzicht klom hij al snel op tot de rang van majoor.
In 1863 kwam hij een vampier tegen, genaamd Maria. Maria realiseerde zich dat Jasper met zijn leger-achtergrond voor haar van nut kon zijn, omdat ze de macht wilde hebben in Monterrey en hiervoor een leger van vampiers wilde creëren. Daarom doodde ze hem niet, maar veranderde ze hem in een vampier en zorgde ze dat hij haar hielp.
Jasper trainde de nieuwe vampieren, om te zorgen dat zij goede vechters werden. Als ze niet langer meer nodig waren (ongeveer een jaar na hun transformatie, waarna hun kracht afnam), moest hij hen doden. Na een paar decennia kreeg Jasper er genoeg van en besloot dat hij dit niet langer meer wilde doen. Hij verliet Maria en leefde een tijdje bij twee oude vrienden, Peter en Charlotte.
In diezelfde periode besloot Jasper dat hij zich niet langer wilde voeden met het bloed van mensen. Na zijn transformatie tot vampier had hij namelijk een gave gekregen: hij kon de emoties van mensen om hem heen voelen en beïnvloeden. Dit hield o.a. in, dat hij de angst en wanhoop van zijn prooi voelde vlak voor hij hen doodde. Hier kon hij niet langer meer tegen en daarom verliet hij ook Peter en Charlotte. Hij verschool zich hopeloos in Philadelphia en had geen idee wat hij nu moest doen.
In deze toestand werd hij gevonden door Alice Cullen, die hem in een van haar visioenen had gezien en vervolgens naar hem op zoek was gegaan. Samen kwamen ze uiteindelijk na een lange tocht bij het gezin Cullen terecht. Rond diezelfde tijd trouwde Jasper met Alice.
Karakter
Jasper wordt omschreven als lang, mysterieus, gespierd en met honingblond haar. Zijn lichaam is bedekt met littekens, als gevolg van de vele gevechten met de nieuwe vampiers die hij trainde. Hij heeft dezelfde achternaam als Rosalie, Hale, aangenomen zodat het lijkt of ze een tweeling zijn in het adoptiegezin Cullen.
Jasper is de 'jongste' van het gezin Cullen - wat ook inhoudt dat hij zijn instincten nog niet altijd goed kan bedwingen. Hij is altijd gewend geweest zich te kunnen voeden met mensenbloed wanneer hij maar wilde, en de nieuwe "vegetarische" levensstijl valt hem soms zwaar. Hoe moeilijk het voor Jasper is om menselijk bloed te weerstaan wordt duidelijk in New Moon, de tweede film. Tijdens Bella's verjaardagsfeestje bij de Cullens thuis, door Alice georganiseerd, snijdt Bella zich per ongeluk, en bloedt maar een beetje aan haar vingers, waardoor Jasper de verleiding niet kan weerstaan en Bella aanvalt, hetgeen door Edward vermeden wordt.
Externe links
IMDB-profiel van Jasper Hale
Twilightpagina van Jasper Hale op Wikia
Personage uit Twilight | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,747 |
Noto gentleman driver, fu assieme a Paco Godia fu il primo pilota spagnolo in Formula 1.
Rimase gravemente ferito prima della 24 Ore di Le Mans 1953 e dovette abbandonare l'automobilismo ad alto livello. Morì in un incidente stradale nel 1960.
Carriera
Gli inizi
Juan Jover nacque in una ricca famiglia spagnola nel 1903. Grazie al denaro a sua disposizione poté permettersi di iniziare la sua carriera di pilota, concentrandosi prima sulle due ruote e successivamente sull'automobilismo. Esordì in una gara locale nel 1923 come motociclista e, dopo la seconda guerra mondiale, prese parte a diverse corse automobilistiche in tutto il continente europeo. Nel 1949 giunse secondo alla 24 Ore di Le Mans in coppia con Henri Louveau, risultato che ripeté pochi mesi più tardi alla 24 Ore di Spa.
Formula 1
Nel 1951 fece il proprio esordio in Formula 1 prendendo parte al Gran Premio di Spagna. Dopo essersi qualificato diciottesimo non riuscì a prendere il via della corsa, molto probabilmente per un guasto al motore. Questa rappresentò l'unica apparizione di Jover nella massima serie automobilistica.
Risultati completi
Dopo la Formula 1
Dopo il 1951 Jover si dedicò soprattutto a gare con vetture sport o di Formula 2. Nel 1953 ottenne un secondo posto in una gara in salita in Spagna, ma alcuni mesi dopo fu protagonista di un grave incidente durante le prove della 24 Ore di Le Mans. A oltre 200 chilometri orari, a causa di un suo errore di valutazione, colpì una barriera e venne sbalzato fuori dalla sua vettura. Riportò fratture multiple alla gamba sinistra di cui rischio l'amputazione, che venne evitata grazie all'intervento del medico Soler-Roig. Questo incidente pose di fatto termine alla sua carriera di pilota ad alto livello. Passò infatti un anno a svolgere riabilitazione, tornò solo alla fine del 1954 alle corse. Negli anni seguenti vinse alcuni eventi locali, tra cui La Rabassada (gara in salita spagnola) nel 1958.
Morì in un incidente stradale, avvenuto per cause non ancora del tutto chiare, nel 1960. In suo onore, dal 1963 al 1968, venne disputato il Trofeo Juan Jover, gara valida per il campionato di Formula 2.
Note
Altri progetti
Piloti di Formula 1 spagnoli | {
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} | 8,033 |
Justin Bieber will be given the Champ Of Charity award at the upcoming 2014 Young Hollywood Awards this Monday, July 28.
Justin Bieber is set to receive the Champ of Charity Award at the 2014 Young Hollywood Awards on Monday, in recognition of his five years of outstanding work with the Make-A-Wish Foundation.
The 20-year-old, who is oft in headlines for less-than-stellar activities, will get the chance to show his lesser known side at the Kelly Osbourne-hosted Annual Young Hollywood Awards.
The event honors and celebrates the accomplishments of young talent in the entertainment industry, and will air at 8/7c on The CW Network July 28.
Say what you will about Bieber, but he's a worthy recipient of the Champ of Charity award.
The "Baby" singer has worked with the Make-A-Wish Foundation since 2009.
On August 10, 2013, Bieber set a record as the largest recording artist contributor at Make-A-Wish. He granted his 200th "Wish" when he met then eight-year-old Annalysha Brown-Rafanan before his Believe tour concert in Atlanta, Georgia. Annalysha still suffers from a life-threatening liver condition.
Justin will be joined at the YHAs by stars Vanessa Hudgens, Ashley Tisdale, Bella Thorne, Kat Graham and Ansel Elgort.
Other nominees for awards include songstress and Bieber-duet collaborator Ariana Grande, the boyband One Direction, Nina Dobrev, Austin Mahone, Lea Michele, Jennifer Lawrence, Shailene Woodley, and Emma Stone.
In addition to being honored at the YHAs, there will be another highlight for the Biebs on the night. The Canadian star will be presented with his award by Oz teen singer Cody Simpson. The pair are currently collaborating on a special musical project and hit the studio twice this week.
"As the recipient of last year's Champ of Charity Award, it's an honor and a privilege to present my friend Justin Bieber with the award this year for his exemplary work with Make-A-Wish at The Young Hollywood Awards on Monday night," Simpson told Hollywood Life.
In 2013, Make-A-Wish granted over 226,000 children in the US alone their one expressed wish, along with thousands more in the 47 countries it operates in around the world.
On Thursday, Bieber also took to Twitter to express thanks to the Young Hollywood Awards organization, adding a note of praise to the children he has met through Make-A-Wish.
Will you be tuning in to see Justin Bieber receive his Champ of Charity award at the YHAs?
Head to Young Hollywood Awards' website for more information on the big night to come. | {
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<?xml version="1.0" encoding="UTF-8"?>
<!--
MIT License
Copyright (c) 2015 Rob Terpilowski
Permission is hereby granted, free of charge, to any person obtaining a copy of this software
and associated documentation files (the "Software"), to deal in the Software without restriction,
including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense,
and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so,
subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING
BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY,
WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE
OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
-->
<project xmlns="http://maven.apache.org/POM/4.0.0" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:schemaLocation="http://maven.apache.org/POM/4.0.0 http://maven.apache.org/maven-v4_0_0.xsd">
<modelVersion>4.0.0</modelVersion>
<parent>
<groupId>com.sumzerotrading</groupId>
<artifactId>SumZeroParent</artifactId>
<version>0.1.7-SNAPSHOT</version>
<relativePath>../SumZeroParent</relativePath>
</parent>
<artifactId>sumzero-api</artifactId>
<name>sumzero-api</name>
<description>sumzero APIs</description>
<packaging>pom</packaging>
<modules>
<module>sumzero-broker-api</module>
<module>sumzero-market-data-api</module>
<module>sumzero-historical-data-api</module>
<module>sumzero-real-time-bar-api</module>
<module>sumzero-strategy-api</module>
<module>sumzero-reporting-api</module>
</modules>
</project>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,275 |
/**
*/
package CIM15.IEC61970.Informative.InfAssets;
import CIM15.IEC61968.Assets.Asset;
import CIM15.IEC61970.Core.Curve;
import java.util.Collection;
import org.eclipse.emf.common.notify.NotificationChain;
import org.eclipse.emf.common.util.EList;
import org.eclipse.emf.ecore.EClass;
import org.eclipse.emf.ecore.InternalEObject;
import org.eclipse.emf.ecore.util.BasicInternalEList;
import org.eclipse.emf.ecore.util.InternalEList;
/**
* <!-- begin-user-doc -->
* A representation of the model object '<em><b>Asset Property Curve</b></em>'.
* <!-- end-user-doc -->
*
* <p>
* The following features are supported:
* <ul>
* <li>{@link CIM15.IEC61970.Informative.InfAssets.AssetPropertyCurve#getSpecification <em>Specification</em>}</li>
* <li>{@link CIM15.IEC61970.Informative.InfAssets.AssetPropertyCurve#getAssets <em>Assets</em>}</li>
* </ul>
* </p>
*
* @generated
*/
public class AssetPropertyCurve extends Curve {
/**
* The cached value of the '{@link #getSpecification() <em>Specification</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getSpecification()
* @generated
* @ordered
*/
protected Specification specification;
/**
* The cached value of the '{@link #getAssets() <em>Assets</em>}' reference list.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @see #getAssets()
* @generated
* @ordered
*/
protected EList<Asset> assets;
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
protected AssetPropertyCurve() {
super();
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
protected EClass eStaticClass() {
return InfAssetsPackage.eINSTANCE.getAssetPropertyCurve();
}
/**
* Returns the value of the '<em><b>Specification</b></em>' reference.
* It is bidirectional and its opposite is '{@link CIM15.IEC61970.Informative.InfAssets.Specification#getAssetPropertyCurves <em>Asset Property Curves</em>}'.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Specification</em>' reference isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Specification</em>' reference.
* @see #setSpecification(Specification)
* @see CIM15.IEC61970.Informative.InfAssets.Specification#getAssetPropertyCurves
* @generated
*/
public Specification getSpecification() {
if (specification != null && specification.eIsProxy()) {
InternalEObject oldSpecification = (InternalEObject)specification;
specification = (Specification)eResolveProxy(oldSpecification);
if (specification != oldSpecification) {
}
}
return specification;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public Specification basicGetSpecification() {
return specification;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
public NotificationChain basicSetSpecification(Specification newSpecification, NotificationChain msgs) {
Specification oldSpecification = specification;
specification = newSpecification;
return msgs;
}
/**
* Sets the value of the '{@link CIM15.IEC61970.Informative.InfAssets.AssetPropertyCurve#getSpecification <em>Specification</em>}' reference.
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @param value the new value of the '<em>Specification</em>' reference.
* @see #getSpecification()
* @generated
*/
public void setSpecification(Specification newSpecification) {
if (newSpecification != specification) {
NotificationChain msgs = null;
if (specification != null)
msgs = ((InternalEObject)specification).eInverseRemove(this, InfAssetsPackage.SPECIFICATION__ASSET_PROPERTY_CURVES, Specification.class, msgs);
if (newSpecification != null)
msgs = ((InternalEObject)newSpecification).eInverseAdd(this, InfAssetsPackage.SPECIFICATION__ASSET_PROPERTY_CURVES, Specification.class, msgs);
msgs = basicSetSpecification(newSpecification, msgs);
if (msgs != null) msgs.dispatch();
}
}
/**
* Returns the value of the '<em><b>Assets</b></em>' reference list.
* The list contents are of type {@link CIM15.IEC61968.Assets.Asset}.
* It is bidirectional and its opposite is '{@link CIM15.IEC61968.Assets.Asset#getAssetPropertyCurves <em>Asset Property Curves</em>}'.
* <!-- begin-user-doc -->
* <p>
* If the meaning of the '<em>Assets</em>' reference list isn't clear,
* there really should be more of a description here...
* </p>
* <!-- end-user-doc -->
* @return the value of the '<em>Assets</em>' reference list.
* @see CIM15.IEC61968.Assets.Asset#getAssetPropertyCurves
* @generated
*/
public EList<Asset> getAssets() {
if (assets == null) {
assets = new BasicInternalEList<Asset>(Asset.class);
}
return assets;
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@SuppressWarnings("unchecked")
@Override
public NotificationChain eInverseAdd(InternalEObject otherEnd, int featureID, NotificationChain msgs) {
switch (featureID) {
case InfAssetsPackage.ASSET_PROPERTY_CURVE__SPECIFICATION:
if (specification != null)
msgs = ((InternalEObject)specification).eInverseRemove(this, InfAssetsPackage.SPECIFICATION__ASSET_PROPERTY_CURVES, Specification.class, msgs);
return basicSetSpecification((Specification)otherEnd, msgs);
case InfAssetsPackage.ASSET_PROPERTY_CURVE__ASSETS:
return ((InternalEList<InternalEObject>)(InternalEList<?>)getAssets()).basicAdd(otherEnd, msgs);
}
return super.eInverseAdd(otherEnd, featureID, msgs);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public NotificationChain eInverseRemove(InternalEObject otherEnd, int featureID, NotificationChain msgs) {
switch (featureID) {
case InfAssetsPackage.ASSET_PROPERTY_CURVE__SPECIFICATION:
return basicSetSpecification(null, msgs);
case InfAssetsPackage.ASSET_PROPERTY_CURVE__ASSETS:
return ((InternalEList<?>)getAssets()).basicRemove(otherEnd, msgs);
}
return super.eInverseRemove(otherEnd, featureID, msgs);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public Object eGet(int featureID, boolean resolve, boolean coreType) {
switch (featureID) {
case InfAssetsPackage.ASSET_PROPERTY_CURVE__SPECIFICATION:
if (resolve) return getSpecification();
return basicGetSpecification();
case InfAssetsPackage.ASSET_PROPERTY_CURVE__ASSETS:
return getAssets();
}
return super.eGet(featureID, resolve, coreType);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@SuppressWarnings("unchecked")
@Override
public void eSet(int featureID, Object newValue) {
switch (featureID) {
case InfAssetsPackage.ASSET_PROPERTY_CURVE__SPECIFICATION:
setSpecification((Specification)newValue);
return;
case InfAssetsPackage.ASSET_PROPERTY_CURVE__ASSETS:
getAssets().clear();
getAssets().addAll((Collection<? extends Asset>)newValue);
return;
}
super.eSet(featureID, newValue);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public void eUnset(int featureID) {
switch (featureID) {
case InfAssetsPackage.ASSET_PROPERTY_CURVE__SPECIFICATION:
setSpecification((Specification)null);
return;
case InfAssetsPackage.ASSET_PROPERTY_CURVE__ASSETS:
getAssets().clear();
return;
}
super.eUnset(featureID);
}
/**
* <!-- begin-user-doc -->
* <!-- end-user-doc -->
* @generated
*/
@Override
public boolean eIsSet(int featureID) {
switch (featureID) {
case InfAssetsPackage.ASSET_PROPERTY_CURVE__SPECIFICATION:
return specification != null;
case InfAssetsPackage.ASSET_PROPERTY_CURVE__ASSETS:
return assets != null && !assets.isEmpty();
}
return super.eIsSet(featureID);
}
} // AssetPropertyCurve
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,073 |
{"url":"https:\/\/byjus.com\/question-answer\/a-full-length-image-of-a-distant-tall-building-can-definitely-beseen-by-using\/","text":"Question\n\n# A full-length image of a distant tall building can definitely be seen by using?\n\nOpen in App\nSolution\n\n## Convex mirrorAs we all know, a convex mirror's image is always smaller than the object's size, resulting in an erect and fictitious image of the thing. A diverging mirror is another name for it. It is also utilized as a rear view mirror in vehicles because it covers a large area behind the mirror. Demonstrating with the help of a diagramA full-length image of a distant tall building using a convex mirrorA ray of light $AD$ running parallel to the major axis is reflected along with $DX$ when a tall building $AB$ is at a distance from the mirror. When $DX$ is traced backwards, it appears to come from $F$. Another ray, $AE$, is reflected back along with $EA$ as it travels towards the centre of curvature ($C$). When created backwards, these two diverging rays $DX$ and $EA$ appear to cross at $A\\text{'}$. As a result, the image of the building generated is imaginary, upright, decreased and behind the mirror. Using a convex mirror, the full-length image of the distant tall building may be clearly viewed.Hence, a full-length image of a distant tall building can definitely be seen by using a convex mirror.\n\nSuggest Corrections\n1","date":"2023-01-30 21:16:25","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 11, \"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\": 0.5324060916900635, \"perplexity\": 625.4029624313803}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"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-2023-06\/segments\/1674764499829.29\/warc\/CC-MAIN-20230130201044-20230130231044-00490.warc.gz\"}"} | null | null |
Outpatient only, no call or hospital work, 39 hours per week. Competitive salary, benefits, CME, malpractice and all practice costs paid. This clinic is a new, upscale facility with strong demographic and traffic location. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,733 |
I'm loving these pregnancy photos at Land's End in San Francisco! The couple said they wanted nature photos and loved the idea of incorporating a few shots with the ocean bluffs as a nod to starting and raising their family here in San Francisco, Bay Area.
It was foggy all over San Francisco this day. In fact, there was such thick fog I texted them mid-afternoon asking if they could arrive a bit earlier to Land's End. They did and as luck would have it, the fog completely cleared right in our little nook! It was kinda crazy actually! Because an hour earlier it was so thick I was worried that our light would disappear too quick for a golden hour shoot. That's fickle San Francisco for you!
This couple was just so cool and laidback – I had a blast working with them. I'm looking forward to meeting their newborn when she arrives!
Contact me to schedule your maternity + newborn session. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,194 |
Q: TypeError: unsupported operand type(s) for -: 'float' and 'str' I am new to Python and am stuck with what to do right now because I keep getting this error.I am trying to change the following equation like:
z = np.power(((float(X) * theta.T)-float(Y)), 2)
but I can't seem to get it to work.
My code:
# convert from data frames to numpy matrices
X = np.matrix(x.values)
Y = np.matrix(y.values)
theta = np.matrix(np.array([0,0]))
# cost functionstrong text
def computeCost(X, Y, theta):
z = np.power(((X * theta.T)-Y), 2)
return np.sum(z) / (2 * len(X))
print('computeCost(X, y, theta) = ' , computeCost(X, Y, theta))
the matrix is full of float numbers:
X = [[ 1. 6.1101] [ 1. 5.5277] [ 1. 8.5186] ...
The error message :
TypeError: unsupported operand type(s) for -: 'float' and 'str'
I appreciate all the help I can get. Thanks a lot
A: do check datatypes of your dataframe
df.info()
if there's 'object' datatype, that's your string
you may want all datatypes in your dataframe as float, you can do it
df.astype('float')
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,953 |
Same God
Home » Same God
A film by Linda Midgett
Wheaton College is considered to be the "Harvard" of evangelical schools, according to director Linda Midgett, a college founded by abolitionists dedicated to rooting out slavery and racism. Midgett herself is a graduate of the school, which she says historically has been the "keeper" of evangelical culture and a bellwether of its future.
Last December, Wheaton terminated the employment of Dr. Larycia Hawkins, a tenured, African-American professor (the first in the school's 150-year history), after she donned a hijab as an act of solidarity with Muslim women. Dr. Hawkins decided to wear a hijab during the Christian season of Advent to bring attention to discrimination faced by Muslim women. She posted a photo of herself on Facebook with the explanation: "I love my Muslim neighbor because s/he deserves love by virtue of her/his human dignity. I stand in human solidarity with my Muslim neighbor…We worship the same God."
Within hours of posting, parents called Wheaton demanding that Dr. Hawkins be fired. Faculty were divided. She was put on academic leave.
The controversy continues. Midgett will explore the fallout and unresolved issues as production of the film goes forward.
AUBURN UPDATE
Currently in development
Linda Midgett
All donations for the production and distribution of Same God are tax-deductible to the fullest extent of the law.
If you would like to make a donation online, please click here:
Download Donation Form PDF
For more information, you can contact Laura Healy, Program Administrator at Auburn at:
Website: http://www.auburnseminary.org
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As Sara advocates for LGBTQ+ justice in Wyoming, we're there to help
Sharing Gratitude from Auburn's Board Chair
Reproductive Justice with Adaku Utah and Emma Jordan-Simpson
Philly District Attorney | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,234 |
Q: Scale and keep ratio after rotating bitmap by using matrix I got a picture of a sun, 860x860 pixels.
I want to rotate the sun with the anchor-point being center of the screen.
This is what I got so far:
class GraphicalMenu extends View{
int screenH;
int screenW;
int angle;
Bitmap sun, monster;
public GraphicalMenu(Context context){
super(context);
BitmapFactory.Options options = new BitmapFactory.Options();
options.inScaled = false;
sun = BitmapFactory.decodeResource(getResources(),R.drawable.sun,options);
monster = BitmapFactory.decodeResource(getResources(),R.drawable.monster,options);
}
@Override
public void onSizeChanged (int w, int h, int oldw, int oldh) {
super.onSizeChanged(w, h, oldw, oldh);
screenH = h;
screenW = w;
sun = Bitmap.createScaledBitmap(sun, w, h, true);
monster = Bitmap.createScaledBitmap(monster, w, h, true);
}
@Override
public void onDraw(Canvas canvas) {
super.onDraw(canvas);
//Increase rotating angle.
if (angle++ >360)
angle =0;
Matrix matrix = new Matrix();
matrix.setRotate(angle , getWidth()/2, getHeight()/2);
canvas.drawBitmap(sun, matrix, new Paint());
//Call the next frame.
canvas.drawBitmap(monster,0 , 0, null);
invalidate();
}
}
I've tried to change this line:
sun = Bitmap.createScaledBitmap(sun, w, h, true);
to:
sun = Bitmap.createScaledBitmap(sun, h, h, true);
but then the sun leaves the center of the screen and rotates far out to the right.
How can I fit the sun to the screen?
And how can I make it keep its ratio?
edit
Screenshot from running it on my N5 and the picture of the sun.
A: If I understand correctly, you have several problems with your code:
*
*If you're making multiple calls to onSizeChanged() then you'll want to keep the original bitmap (pre-scaled) otherwise when you upscale the bitmaps again, they will look pixelated because every time you downscale them, you're losing information.
*When you provide a matrix through which to transform your bitmap for drawing onto the canvas, that matrix will apply to the bitmap itself, not the screen as a whole. So when you make the drawBitmap() call, what you are actually requesting is for the bitmap to be rotated around the local point (getWidth()/2, getHeight()/2), not the screen point.
*You're not actually rotating around the screen centre, you're rotating around the View's centre. Unless the View takes up the entire screen, you won't get the intended effect.
I can't pinpoint exactly what your problem is, but uploading some images would likely clarify what you're trying to achieve and what your current situation is.
| {
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} | 4,715 |
Mary Elizabeth « Beth » Hughes est une actrice américaine, née le à Alton (Illinois), morte le à Los Angeles (Californie).
Biographie
Au cinéma, Mary Beth Hughes contribue à cinquante-huit films américains de 1939 à 1957 (notamment pour MGM, Fox et Warner), avant deux derniers sortis en 1971 et 1974. Mentionnons le western L'Étrange Incident de William A. Wellman (1943, avec Henry Fonda et Dana Andrews), ainsi que les drames La Cible vivante d'Anthony Mann (1945, avec Erich von Stroheim et Dan Duryea) et La Femme aux chimères de Michael Curtiz (1950, avec Kirk Douglas, Lauren Bacall et Doris Day).
À la télévision, elle apparaît dans trente-six séries, entre 1950 et 1970, dont Monsieur et Madame détective (deux épisodes, 1958-1959) et Rawhide (deux épisodes, 1959-1963).
Filmographie partielle
Au cinéma
1939 : Emporte mon cœur (Broadway Serenade) de Robert Z. Leonard
1939 : de S. Sylvan Simon
1939 : Femmes (The Women) de George Cukor
1939 : Dancing Co-Ed de S. Sylvan Simon
1939 : Mon mari court encore (Fast and Furious) de Busby Berkeley
1940 : d'H. Bruce Humberstone
1940 : La Rançon de la gloire (Star Dust) de Walter Lang
1940 : d'Archie Mayo
1941 : (Ride on Vaquero) d'Herbert I. Leeds
1941 : Design for Scandal de Norman Taurog
1941 : Sleepers West d'Eugene Forde
1941 : Dressed to Kill d'Eugene Forde
1942 : Ce que femme veut (Orchestra Wives) d'Archie Mayo
1942 : The Night Before the Divorce de Robert Siodmak
1942 : Blue, White and Perfect d'Herbert I. Leeds
1943 : L'Étrange Incident (The Ox-Bow Incident) de William A. Wellman
1943 : de Jean Yarbrough
1944 : de Sam Newfield
1944 : de Frank McDonald
1945 : The Lady Confesses de Sam Newfield
1945 : La Cible vivante (The Great Flamarion) d'Anthony Mann
1948 : Furie sauvage (The Return of Wildfire) de Ray Taylor
1948 : Inner Sanctum de Lew Landers
1949 : El Paso, ville sans loi (El Paso) de Lewis R. Foster
1950 : La Femme aux chimères (Young Man with a Horn) de Michael Curtiz
1951 : Close to My Heart de William Keighley
1951 : La Caravane des évadés (Passage West) de Lewis R. Foster
1954 : La Tueuse de Las Vegas (Highway Dragnet) de Nathan Juran
1974 : The Working Girls de Stephanie Rothman : Mrs. Borden
À la télévision
1956 : Badge 714 (Dragnet), première série
Saison 6, épisode 6 The Big Limp 1958-1959 : Monsieur et Madame détective (The Thin Man)
Saison 1, épisode 28 The Departed Doctor (1958) de Bretaigne Windust
Saison 2, épisode 21 Mayhem to Music (1959) de Don Weis
1959 : Rintintin (The Adventures of Rin Tin Tin)
Saison 5, épisode 16 Stagecoach to Phoenix 1959 : Au nom de la loi (Wanted : Dead or Alive)
Saison 1, épisode 24 Campagne électorale (Secret Ballot) de Don McDougall
1959-1963 : Rawhide Saison 1, épisode 21 Incident in No Man's Land (1959) de Jack Arnold
Saison 5, épisode 16 Incident at Spider Rock (1963)
1961 : Denis la petite peste (Dennis the Menace)
Saison 2, épisode 21 Dennis Goes to Camp'' de William D. Russell
Liens externes
.
Actrice américaine
Naissance à Alton (Illinois)
Naissance en novembre 1919
Décès en août 1995
Décès à Los Angeles
Décès à 75 ans | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,720 |
Q: How to show the actual data of [object Object] in kendo grid I'm trying to implement kendo grid with dynamic columns and the data showing [object Object].
How to show the address[object,Object] data in my kendo grid and i would like the sample output to be like the table below:-
Name | Phone | Address
------------------------------------------------
John Smith | (519) 420-2391 | Address 1: London
| | Address 2: 123
var peoples = [],
address = [];
peoples = [{
id: 1,
name: "John Smith",
phone: "(519) 420-2391",
address: [{
address1: "london",
address2: "123"
}]
}];
$("#grid").kendoGrid({
dataSource: {
data: peoples,
},
scrollable: true,
sortable: true,
resizable: true,
pageable: true,
columnMenu: true,
columns: [{
field: "name",
title: "Name"
}, {
field: "phone",
title: "Phone number"
}, {
field: "address",
title: "Address"
}],
});
<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<title>Untitled</title>
<link rel="stylesheet" href="http://kendo.cdn.telerik.com/2016.2.714/styles/kendo.common.min.css">
<link rel="stylesheet" href="http://kendo.cdn.telerik.com/2016.2.714/styles/kendo.rtl.min.css">
<link rel="stylesheet" href="http://kendo.cdn.telerik.com/2016.2.714/styles/kendo.default.min.css">
<link rel="stylesheet" href="http://kendo.cdn.telerik.com/2016.2.714/styles/kendo.mobile.all.min.css">
<script src="http://code.jquery.com/jquery-1.12.3.min.js"></script>
<script src="http://kendo.cdn.telerik.com/2016.2.714/js/angular.min.js"></script>
<script src="http://kendo.cdn.telerik.com/2016.2.714/js/jszip.min.js"></script>
<script src="http://kendo.cdn.telerik.com/2016.2.714/js/kendo.all.min.js"></script>
</head>
<body>
<div id="example">
<div id="grid"></div>
</div>
</body>
</html>
A: You can utilize the columns.template functionality to achieve the desired result. (Reference)
Depending on your data-structure you could use a template similar to this:
Address 1: <span>#: address[0].address1 # </span>
<br/>
Address 2: <span>#: address[0].address2 # </span>
I've also create a Dojo showing an example.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,411 |
Q: Pig - JsonMetadata - Could not find schema file There has been a major lag time when Pig evaluates LOAD statements on certain versions of pig.
Upon switching versions of Pig (during a cluster upgrade), Pig's Grunt (and via file script as well) took 30+ seconds on each LOAD statement. This was in stark contrast to the usual <1 second needed to process each LOAD statement. Pig's debug showed the LOAD statement was evaluated in about the same manner between different versions, but what differed was especially interesting.
In Version 0.9.2 (w/ Java 8), the LOAD statement was processed in <1 second. However in Version 0.11.1 and 0.12.0, the load statement was processed in 30+ seconds with the main line differing being this one:
[main] DEBUG org.apache.pig.builtin.JsonMetadata - Could not find schema file for /logs/visits/*/*visits_v15*.lzo
That debug message was not displayed until the command finished being processed which leads me to suspect the loading of the schema data is what's hanging up the entire process. The LOAD statement completes in <1 second on 0.11.1 and 0.12.0 when I specify an exact file, but still shows the warning:
[main] DEBUG org.apache.pig.builtin.JsonMetadata - Could not find schema file for /logs/visits/2014-08-01/2014-08-01-23-45-07.PDT.visits_v15.server.log.lzo
Example Pig Script:
SET debug 'on';
REGISTER s3://path/to/elephant-bird.jar;
v15_data = LOAD '/logs/visits/2014-08-01/2014-08-01-23-45-07.PDT.visits_v15.server.log.lzo';
It doesn't matter if I use Elephant Bird or not as the same debug message comes up and the performance characteristics are the same.
There are ~60 columns in each file and there are thousands of files written per day. Oddly, running the same style script above with just a "DUMP v15_data;" often gives a failed job with this style of error:
java.io.IOException: Deserialization error: invalid stream header:
Pig Versions tested:
0.11.1
0.12.0
0.13.0 - I believe I tested this one previously, but not as thoroughly as 0.11.1 or 0.12.0
A: After struggling with this issue for a while, I found a solution was to use the "-noschema" flag. So USING PigStorage('-noschema')
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,443 |
\section{Introduction}
If light is treated as an electromagnetic wave in vacuum, it is straightforward to derive the formula for the Doppler effect by using, for instance, the transformation equations of the four~-~wavevector $(\omega/c, k_i)$:
\begin{equation}\label{omegaf}
\omega = \omega\,' {{\sqrt{1-B^2}}\over{1-B\cos \theta}}; \qquad B=\frac{v}{c}
\end{equation}
As usual, we are dealing with two inertial frames ($O,O'$) whose axis are parallel and oriented along the same directions: $O'$ is considered in motion with respect to $O$ with velocity $v$ along the positive direction of the common $x\equiv x'$ axis; $\theta$ is the angle that the wavevector $\vec k$ forms with the $x$ axis.
\par
If we introduce the concept of photon and write for its energy $E_p=\hbar \omega$, equation (\ref{omegaf}) becomes valid also for the photon energy:
\begin{equation}\label{pho}
E_p = E_p' {{\sqrt{1-B^2}}\over{1-B\cos \theta}}; \qquad B=\frac{v}{c}
\end{equation}
Alternatively, equation (\ref{pho}) can be derived by treating photons as relativistic particles whose energy~-~momentum four~-~vector is $(E/c,\vec p)$, with $E=\hbar \omega$ and $p=\hbar\omega/c$.
\par
The light source and its physical state do not enter equations (\ref{omegaf}, \ref{pho}). Therefore, the interpretation of equation (\ref{omegaf}) reads: if $\omega '$ is the angular frequency of the light wave measured in the reference system $O'$, then the angular frequency of the same wave measured by $O$ is given by equation (\ref{omegaf}). A similar interpretation holds for equation (\ref{pho}). Notice that the source needs not to be at rest in $O'$.
\par
Equations (\ref{omegaf}, \ref{pho}), are used for describing the Doppler effect also when experiments deal with photons emitted or absorbed by atoms/nuclei. Moreover,
since around 1905 till nowadays, the (transverse) Doppler effect for light has been interpreted as an experimental corroboration of time dilation both in research papers and textbooks.
\par
This paper discusses these issues in the frameworks of the wave and the corpuscular descriptions of light; it discusses also some historical passages bringing out the underlying epistemological aspects. Therefore, it may be of some interest for researchers and for teachers at university or high school levels.
\section{Doppler effect for photons\label{dopplersec}}
\subsection{Stark and Einstein around 1905: atoms as clocks}
Within the theoretical framework around 1900, the emission of light by matter was due to the { harmonic vibrations of point electrical charges (electrons). On the basis of this model, Stark concluded that lines emitted by atoms in flight should have been Doppler shifted. In 1905, he succeeded in detecting this shift (to the first order in $v/c$) for light emitted by hydrogen atoms \cite{stark1}. This achievement, together with the later discovery of the Stark effect, earned him the Nobel prize in physics in 1919 \footnote{Stark discovery of the effect holding his name has been the result of a research program. As discussed in \cite{matteo}, the contemporaneous and independent discovery of the same effect by Lo Surdo, was an accidental one.}.
\par
Einstein commented the results obtained by Stark with these words:
\begin{quote}
In an important paper published last year, Mr. J. Stark ({J. Stark, {\em
Ann. d. Phys.} 21 (1906), 401}) demonstrated that the moving positive ions of
canal rays emit line spectra by identifying the Doppler effect and following
it quantitatively. He also undertook experiments with the intention of
detecting and measuring an effect of the second order (proportional to
$(v/c)^2$); however, the experimental arrangement, which was not set up specifically
for this purpose, was not adequate for achieving reliable results.
\par
I will show here briefly that the principle of relativity
in conjunction with the principle of the constancy of the velocity of light
makes it possible to predict the above effect.
As I showed in an earlier paper, it follows from these principles that
a uniformly moving clock runs at a slower rate as judged from a
``stationary'' system than as judged by a co~-~moving observer.
If $\nu$ denotes the number of the clock's strokes per unit time for
the observer at rest, and $\nu_0$ the corresponding number for the
co~-~moving observer, then
\begin{equation}\label{fre}
{{\nu}\over{\nu_0}} =\sqrt { 1 - \left[{{v}\over{c}}\right]^2 }
\end{equation}
or to first approximation
\begin{equation}\label{freappr}
{{\nu-\nu_0}\over{\nu_0}} = -{{1}\over{2}} \left[{{v}\over{c}}\right]^2
\end{equation}
The atom ion of the canal rays that emits and absorbs radiation of certain
frequencies is thus to be conceived as a fast~-~moving clock, and the
relation just indicated can therefore be applied to
it \cite[p. 232]{ein07}.
\end{quote}
Both Stark and Einstein assumed that the {\em atom is a clock}. Clearly, this assumption is justified only if the atom is the seat of some periodic motion. This motion could be, for instance, the supposed harmonic motion of bound electrons; or the supposed elliptical motion of the electron in Bohr's model of hydrogen atom: however, in this case, the frequency of the light emitted by the hydrogen atom differs from the frequency of the electronic motion. Anyway, the advent of quantum mechanics forbids any description of atoms as seats of periodic motion of electrons: {\em atoms are not clocks} \footnote{The fact that we now use {\em atomic} clocks should not confuse us: in these clocks, a quantum transition between two atomic levels is used as a locking parameter of the resonant frequency of a quartz oscillator.}.
\par
Nevertheless, let us come back to Einstein.
After having assumed that
atoms are clocks, he
applied to them the time dilation formula. He thus obtained equation (\ref{fre}) that yields the ratio between the ``proper'' frequency $\nu_0$ of an atom~-~clock and the frequency $\nu$ of the same atom~-~clock for an observer that sees the atom~-~clock in uniform motion with velocity $v$.
\par
Einstein did not refer to the formula for the Doppler effect (\ref{omegaf}) derived two years before in his theory of special relativity by treating light as an electromagnetic wave \cite[p. 161]{ein05}. In this paper, Einstein proved also that the energy of a ``light complex'', i.e. the light energy contained in a sphere moving at the speed of light with respect to an inertial frame, in passing from this frame to another, changes in exactly the same way as the light frequency. Of course, this means that the light energy and its frequency are connected by a relativistic invariant: $U = (Nh)\nu$. However, Einstein did not take this step: as stressed and commented by Pais, Einstein kept well separated special relativity from the light quanta hypothesis \cite[p. 909]{pais}.
\subsection{Schr\"odinger, 1922: atoms emit light quanta endowed with energy and momentum \label{erwinsec}}
In 1922 Schr\"odinger dealt with the radiation emitted
by atoms in motion in terms of light
quanta \cite{erwin}.
Schr\"odinger makes it clear from the beginning that, once we accept Einstein's idea
that the quantum $h\nu$ `always carries' a linear momentum $h\nu/c$, we have to
recognize that the emission of a quantum $h\nu$ by an atom produces a ``jump''
in its velocity and that {\em this jump is responsible
for the Doppler shift}.
\par
The problem posed by Schr\"odinger is illustrated in fig. \ref{erwinfig} (not used by Schr\"odinger).
In the reference frame of the measuring apparatus, it is solved by writing down the conservation equations for energy:
\begin{equation}\label{energia2}
E_p=\gamma_1 E_1-\gamma_2 E_2
\end{equation}
and momentum:
\begin{eqnarray}
\gamma_1 {{E_1}\over{c^2}}v_1 \cos\theta_1 &= & \gamma_2
{{E_2}\over{c^2}}v_2 \cos\theta_2 +{{E_p}\over{c}}
\label{qmx}\\
\gamma_1 {{E_1}\over{c^2}}v_1 \sin\theta_1 &= & \gamma_2
{{E_2}\over{c^2}}v_2 \sin\theta_2\label{qmy}
\end{eqnarray}
\begin{figure}[htb]
\centerline{
\includegraphics[width=5cm]{fig_1.eps}
}
\caption{\label{erwinfig}
Emission of a light quantum by the atom $A$ in motion. The light quantum is emitted along the direction $A\rightarrow
O$: $O$ is the entrance slit of the spectrograph. The subscript $1$ denotes the quantities before the emission; the subscript $2$ the quantities after the emission.
}
\end{figure}
\par\noindent
$E_p$ is the energy of the light quantum; $E_1$ and $E_2$ are the rest energies of the atom before and after the emission, respectively; $\gamma_1,\,\gamma_2$ are the relativistic factors before and after the emission.
\par
After a somewhat lengthy analytical manipulation we get:
\begin{equation}\label{uguale}
E_p =E^0_p {{\sqrt{1-v_1^2/c^2}}\over{1 -(v_1/c)\cos \theta_1 }}
\end{equation}
with
\begin{equation}\label{zero}
E^0_p= \Delta E \left( 1 - {{\Delta
E}\over{2E_1}}\right),\: (v_1=0)
\end{equation}
where $\Delta E= (E_1-E_2)$; then, $\Delta E$ is the energy difference between the two states of the atomic transition. $E^0_p$ is the measured light quantum energy when the atom is at rest before the emission. {\em Both $\Delta E$ and $E_p^0$ are relativistic invariants since they depend only on rest energies}. The term $\Delta E/2E_1$ is in general negligible, unless we are dealing with $\gamma$ photons emitted by free nuclei.
\par
Schr\"odinger's did not write equation (\ref{uguale}). He stopped at an intermediate formula containing both atoms' velocities, before and after the emission \cite[p. 303]{erwin}:
\begin{equation}\label{finale2}
\nu = \nu^* {{1}\over
{\sqrt{\gamma_1 \left[1 - (v_1/c) \cos \theta_1 \right]\times
\gamma_2 \left[1 - (v_2/c) \cos \theta_2 \right] }}}
\end{equation}
where
\begin{equation}\label{fstar}
\nu^* = { {E_1^2 -E_2^2}\over {2h\sqrt{E_1E_2}} }
\end{equation}
A simple calculation suggested by the fact that the atom's velocity after the emission is determined by its velocity before the emission and by the energy~-~momentum conservation leads to the more significant equations (\ref{uguale}) and (\ref{zero}) \cite[p. 197-203]{erwingg}. Obviously, Schr\"odinger was well aware of the relation between the two velocities; this renders more intriguing the fact that he stopped at equation (\ref{finale2}).
\par
Schr\"odinger's treatment can be applied
also to the case of a photon absorbed by an atom in flight with respect to the
laboratory reference frame \cite[p. 201-202]{erwingg}. The energy $E_p$ that a photon must have for being absorbed by the atom
is again given by equation (\ref{uguale}), where, in this case:
\begin{equation}\label{nu03}
E_p^0 = \Delta E\left( 1 + {{\Delta E}\over{2E_1}}\right)
\end{equation}
is the energy of the photon absorbed by an atom at rest
before the absorption and, of course, now $\Delta E=E_2-E_1$.
\par
Equation (\ref{uguale}) is formally identical to equation (\ref{pho}) or, since $E_p=\hbar \omega$, to equation (\ref{omegaf}): however, the meanings of these formulas are quite different. While equations (\ref{omegaf}, \ref{pho}) ignore the emitting/absorbing particle and its physical state, equation (\ref{uguale}), through equation (\ref{zero}) for emission or equation (\ref{nu03}) for absorption, focuses on the energy difference $\Delta E$ between the two quantum levels of the emitting/absorbing particle and the energy $E_p^0$ emitted/absorbed when the particle is at rest before emission/absorption (equations \ref{zero}, \ref{nu03}). Furthermore: while equations (\ref{omegaf}, \ref{pho}) connect quantities measured in two reference frames, Schr\"odinger's treatment uses only the reference frame of the experimental apparatus owing to the use of the relativistic invariants $\Delta E$ and $E_p^0$.
\par
In order to further clarify the physical meaning of Schr\"odinger's treatment, let us illustrate the features of the emission/absorption process as described by equations (\ref{uguale}) and (\ref{zero}, \ref{nu03}).
In the case of photons emitted/absorbed,
the increase/decrease of their energy with respect to the quantity $\Delta E$ is due to an energy~-~momentum exchange with the emitting/absorbing particle. For instance, let us consider an atom that emits a photon. If the photon is emitted in the forward direction, its energy is increased, with respect to $\Delta E$, by exactly the same amount by which the kinetic energy of the atom is decreased; if the photon is emitted in the backward direction, its energy is decreased, with respect to $\Delta E$, by exactly the same amount by which the kinetic energy of the atom is increased. In the case of absorption, the photon energy required for exciting the atom, when the photon flights against the atom, is decreased, with respect to $\Delta E$, by exactly the same amount by which the kinetic energy of the atom is decreased; when the photon is chasing the atom, the photon energy required for exciting the atom, is increased, with respect to $\Delta E$, by exactly the same amount by which the kinetic energy of the atom is increased.
\par
Nowadays, this energy--momentum exchange is basically exploited by those who laser--cool the atoms or use saturation spectroscopy.
\par
Schr\"odinger's approach can be generalized by taking into account the dependence of the energy of the photon emitted/absorbed on the gravitational potential. As shown, for instance by M{\o}ller \cite[pp. 401~-~407]{moller}, it is sufficient to rewrite, for the emission case, equation (\ref{zero}) as follows:
\begin{equation}\label{nu+-approx}
E^0_p \approx \Delta E\left( 1 - {{\Delta E}\over{2E_1}}\right)
\left(1+ {{\phi}\over{c^2}} \right)
\end{equation}
where $\phi$ is the gravitational potential and the $\approx$ sign is due to the approximation for small gravitational potential.
An analogous correction must be made for equation (\ref{nu03}) (absorption case).
\par
Schr\"odinger's paper has been rapidly forgotten; its derivation has been rediscovered by Davisson \cite{dav}; more recent ones can be found, for instance, in the books by French \cite[pp. 197~-~199]{french} and M{\o}ller \cite[pp. 401~-~407]{moller}. No one quotes Schr\"odinger's paper.
The idea of describing the interaction of photons with particles by writing down the conservation equations for energy and linear momentum was used some years later by Compton \cite{compton} and Debye \cite{debye}, without quoting Schr\"odinger, for explaining the Compton effect.
The reasons why Schr\"odinger's paper has been forgotten are not clear: we can only guess some plausible ones. First of all, around 1920, the concept of light quanta had not yet been accepted by the scientific community; secondly, the fact that Schr\"odinger stopped at equations (\ref{finale2}) and (\ref{fstar}) might have obscured the relevance of the paper.
\subsection{Ives and Stilwell, 1938: atoms as clocks, again\label{ivessec}}
In late Thirties of past century, Ives and Stilwell set up an apparatus for the
realization of the experiment devised by Einstein in 1907 \cite{ives}.
The experiment was fully financed
by the Bell Telephone Laboratories where Ives was then
working.
It was a weird twist of fate that
a test of relativity suggested by its founder has been taken up,
performed and interpreted by an anti~-~relativist.
\begin{quote}\small
Ives (1882~-~1953) has been a staunch opponent
of relativity. He
builded his views, characterized by an ether~-~based theory, through
a series of papers written in about fifteen years: they have been collected in a
volume \cite{iveslibro}. Unfortunately, the grossly anti~-~relativist preface by one of the editors throws an unfavorable shadow on Ives' papers. A review of this book by Arthur Miller can be found in \cite{ivesrev}.
Ives' theory is a very intricate one and lies on a procedure for clocks synchronization based on the use of two pairs of rods and clocks, one pair being, by assumption, not affected by their motion in the ether. By assumption,
the velocity of light is isotropic and equal to $c$ only in
a reference frame at rest in the ether.
Ives' coordinates transformations converge to Lorentz's as the velocities of rods and clocks used for synchronization go to zero. Thus, in principle, the two coordinate transformations could never coincide. According to Miller, ``Ives's Larmor~-~Lorentz theory was never developed to the point where it could
be seriously considered as an alternative to the special relativity theory''.
\end{quote}
The article's title is unequivocal: ``An experimental study of the rate of a moving atomic clock'. As in Einstein's paper \cite{ein07}, the atom is considered as a clock. Ives and Stilwell observe that, as far as the transverse Doppler effect is concerned
\begin{quote}
\dots it would be extremely difficult to be sure that observation was made exactly
at right angles to the direction of the rays, and very small deviations from this
direction would introduce shifts of the order of magnitude of
the expected effect \cite[p. 215]{ives}.
\end{quote}
However, this difficulty
\begin{quote}
\dots can be avoided by observing not at right angles, but in two directions, with
and against the motion of the particles; the observation being made simultaneously
by the use of a mirror in the tube. Under these conditions the displaced Doppler lines
are observed corresponding to motion toward and away from the observer, and the
effect to be observed is a shift of the center of gravity of the displaced lines with
respect to the undisplaced line. As shown in an earlier paper of this series this
shift of center of gravity is expressed by the equation $\lambda =\lambda_0 (1-v^2/c^2)
^{1/2}$ where $v$ is the observed or measured velocity of the positive
particles \cite[p. 216]{ives}.
\end{quote}
Since the experimental setup used by Ives and Stilwell has inspired many experimenters up to nowadays, it is worth discussing it in some details: see the Appendix.
The experiment allowed to measure the transverse Doppler effect in the approximation of small velocities: of course, in the conclusions, Ives and Stilwell interpret their
results as a confirmation of Ives's ether based theory.
\par
An year later, Robert Clark Jones, he too at Bell Laboratories, interpreted Ives and Stilwell
results in a special relativity approach, assuming, of course, that
atoms can be treated as clocks.
Jones did not attack Ives' standpoint; he wrote instead:
\begin{quote}
The conceptual background of these theories [Larmor's and Lorentz's] is not the
one which is most popular with physicists today, however, and for this reason
it seemed worth while to obtain the theoretical predictions from the point of
view of the special theory of relativity, particularly {\em since the relativistic point
of view yields the results in so simple a manner}. The theoretical
predictions we shall obtain here are identical with those obtained by Ives from electron
theory \cite[p. 337]{jones} (my italics).
\end{quote}
Some years later, a similar experiment with canal rays has been carried out by Gerhard
Otting \cite{otting}.
The contrast with Ives and Stilwell's paper
is striking. Otting does not comment
either the formulas or their interpretation: he is interested only
in the correspondence between formulas and experimental data.
\par
The experiment has been repeated again in the sixties by Mandelberg and
Witten \cite{mand}.
The authors recall that
\begin{quote}
An analysis of the experiments of Ives and Stilwell and of Otting indicates that although their
reported experimental points seem to fit the curve with
an accuracy of about $2$ to $3\%$, the experimental uncertainty
is more nearly $10~-~15\%$ \cite[p. 529]{mand}.
\end{quote}
Hence, the necessity of repeating the experiment.
The basic experimental setup was the same as that of the older
experiment: only the precision was improved. According to Mandelberg and Witten ``The experimental
result is that the exponent in the quadratic
expression for the Doppler shift, $(1-B^2)^{1/2}$, is found to
be $0.498\pm 0.025$ \cite[p. 529]{mand}.'' ``This implies an over-all precision in this experiment of $5\%$ the limit on the accuracy being imposed by the width of the beam lines \cite[p. 536]{mand}.''
\section{Doppler effect for photons as a {\em direct} consequence of Lorentz transformations\label{dopplerlorsec}}
Starting about 1970,
the Doppler shift of radiation emitted/absorbed by atoms/nuclei in flight has been viewed as a direct consequence of Lorentz transformations: the idea that the atoms are clocks has gone, at least as an explicit statement.
\subsection{Experiments with $\gamma$ photons}
Olin et al. used
an experimental set up similar to that of Ives and Stilwell \cite{olin}:
the Doppler shift of $8.64\, MeV\,\gamma$ photons emitted by $^{20}Ne$ nuclei was studied. The velocity of the emitting nuclei was $0.012\, c$ or $0.049\, c$. The detector was an annular $Ge(Li)$ junction that measures the energy of $\gamma$ photons.
In order to test special relativity through the transverse Doppler effect, the authors discuss the formula:
\begin{equation}\label{olin}
E(\theta)=E_0\frac{F(\beta)}{1-\beta\cos\theta}
\end{equation}
where the significant quantity is the photon energy and the function $F(\beta)$ is equal to $(1-\beta^2)^{1/2}$ if special relativity is correct ($\beta=v/c$).
They write:
\begin{quote}
This phenomenon [Doppler shift] is a
geometrical property of space~-~time, and
is intimately connected with the problem of
synchronization of clocks in different frames
of reference \cite[p. 1633]{olin}.
\end{quote}
\subsection{Direct observation of the transverse Doppler shift in Hydrogen}
The transverse Doppler shift of the $H_\alpha$ hydrogen line has been observed {\em directly} (i.e. perpendicularly with respect the direction of the atoms' motion) by Hasselkamp et al. \cite{hass}. The detector was a photomultiplier used in the single photon counting mode.
According to the authors:
\begin{quote}
Equation (\ref{omegaf}) is a consequence of the Lorentz transformation of time. The experimental confirmation of the validity of (\ref{omegaf}) is therefore a verification of time dilation \cite[p. 152]{hass}.
\end{quote}
\subsection{Lasers enter the scene}
While, from Ives and Stilwell's experiment, the Doppler shift has been studied by measuring the energy of the emitted photons, a major change occurred with the appearance of lasers: the measured quantity became the energy of the photons absorbed by the atoms in flight.
The measuring techniques varied from the simple use of lasers for exciting a quantum transition of the atoms in flight \cite{mac}, to the utilization of two photons absorption \cite{kai} or saturation spectroscopy \cite{saat}; collinear (probing laser beam parallel/antiparallel to the atoms' motion), orthogonal (probing laser beam perpendicular to the atoms' motion) and variable (different angles between the directions of the probing laser beam and the atoms' motion) geometries have been used.
For a rather recent review, see \cite{gwinner}.
\par
These papers consider the Doppler shift as a direct consequence of Lorentz transformations and compare the predictions of special relativity with those of the Mansouri~-~Sexl kinematic test theory of special relativity \cite{sexl}.
This theory is based on the assumption that the speed of light is isotropic only in a hypothetical preferred reference frame and use generalized coordinates transformations that take into account also the possibility of different clocks synchronization procedures.
\par
Very recently, Chou et al. have studied the transverse Doppler effect and the gravitational red shift using an optical clock \cite{chou}.
The unique feature of
optical clocks consists in the use a {\em single} ion at rest in an electromagnetic trap. As far as the transverse Doppler shift is concerned,
Chou et al. have compared the frequency of the probing laser when the ion is at rest (with respect to the laboratory) with that of the same probing laser when the ion is set in a harmonic motion along a direction approximately perpendicular to the direction of the laser beam: as in the other experiments with lasers, the experiment checks the absorption of photons by the ion in motion. Chou et al. interpret their results on the basis of the equation:
\begin{equation}\label{chou}
\frac{\delta f}{f_0}=\frac{1}{<\gamma (1-v_\|/c)>}-1\approx -\frac{1}{2}\frac{<v^2>}{c^2}
\end{equation}
where $v_\|$ denotes the component of the ion speed along the direction of the probing laser beam; equation (\ref{chou}) is, of course, another way of writing equation
(\ref{omegaf}) by averaging over time. $<v_\|>=0$ because the ion's motion is harmonic; $1/\gamma\approx -(1/2)<v^2>/c^2$ because $v\approx 10\, m s ^{-1}\ll c$.
Since the experimental data fit equation (\ref{chou}), Chou et al. conclude that they have experimentally tested time dilation.
\section{Experiments, formulas and theories: what are we measuring? \label{what}}
Sections (\ref{dopplersec}, \ref{dopplerlorsec}) show that all the papers written after Schr\"odinger's seminal article completely ignore its treatment; instead, they rely on equations (\ref{omegaf}, \ref{pho}) that compare the angular frequencies of a light wave or the photon energies in two distinct reference frames. Let us begin with equation (\ref{omegaf}).
It is well known that we can derive a relativistic Doppler formula valid for both acoustic or light signals \cite{doppleracot, doppleracot2}:
\begin{equation}\label{dacot}
\frac{\omega_a}{\omega_e}=\frac{1-(v_a/V)\cos (\vec V, \vec v_a)}{1-(v_e/V)\cos (\vec V, \vec v_e)} \frac{\sqrt{1-v_e^2/c^2}}{\sqrt{1-v_a^2/c^2}}
\end{equation}
The reference system is the one in which the medium is at rest; $V$ is the signal velocity; $v_e$ and $v_a$ the emitter and the absorber velocity. For light in vacuum, this formula reduces to formula (\ref{omegaf}).
Equation (\ref{dacot}) is obtained by assuming that either the source emits signals of ideally null duration at a specified time interval or a periodic wave. In the first case, the phenomenon's period is the time interval between two consecutive signals; in the latter, it is the wave period.
\par
However, atoms and nuclei do not emit waves or null duration signals, but photons endowed with energy and linear momentum: therefore, equation (\ref{omegaf}) can not be applied to atoms or nuclei \footnote{Photons or even single photons can be used as light signal of ideally null duration; however, in this case, the frequency measured is that of the signal period, not the one given by the relation $\nu=E_p/h$ which is the object of this paper.}. Nevertheless, equation (\ref{omegaf}) describes the experimental data: this is due to the fact that equation (\ref{uguale}) reduces to (\ref{omegaf}) by assuming $\omega'=E_p^0/\hbar$, where $E_p^0$ is given by equation (\ref{zero}) for emission or by equation (\ref{nu03}) for absorption and that the velocity entering equation (\ref{omegaf}) is the atom/nucleus velocity before emission/absorption. Then, equation (\ref{omegaf}) can be used only through a series of conceptual shifts in passing from a treatment of the emission/absorption process in term of photons, based on relativistic invariants ($\Delta E, E_p^0$) and one reference system (that of the experimental apparatus), to a formula belonging to the wave theory of light and connecting two quantities (the angular frequencies) in two distinct reference frames. The use of equation (\ref{pho}) does not involve the wave theory of light: however, also in this case, the various conceptual and approximation steps should be made explicit.
\par
The experiments discussed in sections (\ref{dopplersec}, \ref{dopplerlorsec}) are easily explained using Sch\"odinger's approach within the reference frame of the experimental apparatus.
Let us first consider the experiments in which the flying particle emits photons.
In the laboratory reference frame, given an atom/nucleus and the two quantum levels of the transition (i.e. $\Delta E$), equation (\ref{zero}) yields $E_p^0$; alternatively, $E_p^0$ can be measured directly when the particle, before emission, is at rest. Then equation (\ref{uguale}) predicts the energy of the photon emitted by the flying particle in terms of the measured velocity $v_1$ and the angle $\theta_1$. If the experimental test is positive, it corroborates a prediction of the joint use of relativistic dynamics and quantum mechanics.
\par
The experiments with lasers deserve a separate discussion because they deal with {\em absorption} of photons by atoms/nuclei.
In the laboratory reference frame, given an atom/nucleus and the quantum levels of the transition (i.e. $\Delta E$), equation (\ref{nu03}) yields $E_p^0$; alternatively, $E_p^0$ can be measured directly
when the particle, before absorption, is at rest. Then, given the laser photons energy $E_p^0$, the particle velocity $v_1$ and the angle $\theta_1$, equation (\ref{uguale}) yields the energy that the laser photon must have in order to be absorbed by the flying particle. If the experimental test is positive, i.e. if the flying particle absorbs the laser photon, then it corroborates a prediction obtained by the joint use of relativistic dynamics and quantum physics.
\par
The measurement of the gravitational red shift by Chou et al. \cite[p. 1632]{chou}, calls for a final remark. The generalization of Schrodinger's treatment that takes into account the gravitational potential yields:
\begin{equation}\label{dopplergrav}
E_p\approx \Delta E\left( 1 + {{\Delta E}\over{2E_1}}\right)
\left(1+ {{\phi}\over{c^2}} \right) {{\sqrt{1-v_1^2/c^2}}\over{1 -(v_1/c)\cos \theta_1 }}
\end{equation}
where, as already pointed out in section \ref{erwinsec}, the $\approx$ sign comes in for small gravitational potentials.
Equation (\ref{dopplergrav}) describes contemporaneously both effects: the Doppler effect due to the velocity of the absorbing particle and the gravitational one. Instead, in the quoted paper, the two effects seem to derive from two distinct theoretical backgrounds.
\par
Why Schr\"odinger treatment keeps being neglected?
The first and basic answer is a pragmatic one: as pointed out, formulas (\ref{omegaf}, \ref{pho}) describe the experimental data. However, physics is not a game between formulas and experiments: formulas belong to theories with a well defined application domain. Therefore, formulas can not be extrapolated from a theory and applied to phenomena belonging to other theoretical frameworks. As Heinrich Hertz put it some time ago:
\begin{quote}
The very fact that different modes of representation contain
what is substantially the same thing, renders the proper
understanding of any one of them all the more difficult. Ideas
and conceptions which are akin and yet different may be
symbolised in the same way in the different modes of representation.
Hence for a proper comprehension of any one of
these, the {\em first essential is that we should endeavour to understand
each representation by itself without introducing into it
the ideas which belong to another} \cite[p. 21]{hertz_ew} (my italics).
\end{quote}
Applied to our case, this means that we should not mix ideas and formulas coming from two different ``representations'' as the undulatory and the corpuscular theory of light. We should instead define their domains of applications and find out when, how and why their predictions coincide.
\par
But there is another reason:
the nineteenth century has deeply embedded the undulatory description of light in the background knowledge of physicists. Within this knowledge, the Doppler effect has been constantly viewed as a wave phenomenon: the emergence of the light quantum did not change this rooted habit.
The prevailing influence of the undulatory description emerges also in the language: for instance, the locution `photon frequency' is often used instead of `photon energy' and, in general, when dealing with photon the equations are written in terms of frequencies instead of energies.
\par
There are two questions left: a) the Doppler effect as a direct consequence of Lorentz transformations; b) the statement according to which the experimental corroboration of the Doppler effect for photons is a corroboration of time dilation.
\par
About a). As we have seen, the Doppler effect for photons is a consequence of the relativistic conservation laws for energy and momentum and the concept of photon endowed with energy and linear momentum: the term ``direct'' is clearly inappropriate since we must add to Lorentz transformations the laws of relativistic dynamics and the quantum concept of atoms/nuclei energy levels.
\par
Instead, b) is a sound statement. With a specification: it is an indirect corroboration. The relativistic factor $\sqrt{1-v^2/c^2}$, basically a time dilation factor, enters equation (\ref{uguale}) through the relativistic conservation equations; therefore, an experimental corroboration of equation (\ref{uguale}) is, primarily, a corroboration of relativistic dynamics and some quantum hypothesis. That the factor $\sqrt{1-v^2/c^2}$ is a time dilation factor, can be easily shown by ideal experiments with light signals of null duration. This kind of approach has been firstly outlined by Bondi \cite{bondi}, who, however, uses also geometric considerations. Derivations of time dilation, length contraction and Lorentz transformations based only on ideal experiments with light signals of null duration can be found in, for instance, \cite{uga} and \cite[p. 29-42]{erwingg}.
\par
Then, every corroboration of a formula containing the relativistic factor $\sqrt{1-v^2/c^2}$ can be considered as an indirect corroboration of time dilation.
For example, a typical formula of this kind is the one giving the radius $R$ of the circular trajectory of a point electrical charge that enters an uniform magnetic field perpendicularly to it.
With obvious notations, we have (neglecting irradiation):
\begin{equation}\label{raggio}
R= \frac{1}{\sqrt{1-v^2/c^2}}\frac{mv}{qB}
\end{equation}
Since equation (\ref{raggio}) is derived by the joint use of the relativistic force law and the expression of Lorentz force, its experimental verification constitutes a corroboration of both laws and, {\em indirectly}, of time dilation.
\section{Conclusions}
A survey of the experiments on the Doppler effect for photons (through about a century) shows that the explanation of these experiments has been initially given by considering the atoms as clocks; then by using the Doppler formula of the wave theory of light (or its purely formal reformulation in terms of photon energies). The corpuscular treatment of the Doppler effect put forward by Schr\"odinger in 1922 has been completely ignored in spite of the fact that, in other physical contexts, it is commonplace that during the emission/absorption of a photon by an atom/nucleus the emitting/absorbing particle exchanges energy and momentum with the photon: for instance, in the emission/arborption of photons by free nuclei, in the laser cooling of atoms and in saturation spectroscopy.
\par
The origins of this omission are of different kind: pragmatic (agreement between formulas, wherever coming from, and experiments), historical (deep rooting of the wave theory of light) and epistemological (neglect of basic epistemological rules).
\par
Physics is not simply a game between formulas, from wherever they are, and experiments: one should not use formulas from one theory (the wave theory of light) to describe phenomena concerning photons which belong to another theory (the corpuscular theory of light). This epistemological criterium, put forward in its general form more than a century ago by Heinrich Hertz, should not be forgotten, particularly in teaching; however, also in research, the neglect of basic epistemological criteria may, in the long run, lead astray.
\section{Appendix: Ives and Stilwell experiment}\small
The formulas appearing below from (i) to (iv) are relativistic formulas, not Ives'. However, the central result given by equation (\ref{dt}) is the same also for Ives.
\begin{enumerate}
\item The wavelength $\lambda_B$ of the light emitted by the
incoming atoms
at a small angle $\theta$ to the beam direction is given by:
\begin{equation}
\lambda_B=\lambda_0 {{1-B\cos \theta}\over{(1-B^2)^{1/2}}}\approx \lambda_0\left(1-B\cos \theta+
{{1}\over{2}}B^2\right)
\end{equation}
where $\lambda_0$ is the natural wavelength, $B=v/c\ll 1$ and
$v$ is the velocity of the
atoms.
\item The wavelength $\lambda_R$ of the light emitted by the
receding atoms at the angle $(\pi-\theta)$ to the beam direction is instead:
\begin{equation}
\lambda_R=\lambda_0 {{1+B\cos \theta}\over{(1-B^2)^{1/2}}}\approx \lambda_0\left(1+B\cos \theta+
{{1}\over{2}}B^2\right)
\end{equation}
\item The average of the two wavelengths is:
\begin{equation}\label{dt}
\lambda_Q={{1}\over{2}}(\lambda_B+\lambda_R)=
\lambda_0\left(1+
{{1}\over{2}}B^2\right)
\end{equation}
and equals the wavelength which would be observed at right angles to the beam.
\item The difference between the two wavelengths is:
\begin{equation}\label{lamd}
2\lambda_D=\lambda_R-\lambda_B=2\lambda_0 B \cos \theta
\end{equation}
\end{enumerate}
where $\lambda_D$ represents the first order Doppler effect.
\par
The emitting particles where hydrogen atoms. The observed
line, on a photographic plate, was the $H_\beta$ line.
Ives and Stilwell:
\begin{description}
\item[a)] derived the experimental value of the atoms velocity $v$ from (\ref{lamd}), using for $\lambda_0$ the wavelength of the ``undisplaced line'', i.e. the central line appearing on the plate together with the ``displaced lines'' $\lambda_B$ and $\lambda_R$;
\item[b)] predicted the value of the transverse Doppler shift by using this value
of the atoms velocity;
\item[c)] compared the calculated value of the transverse Doppler shift with
that measured according to (\ref{dt});
\item[d)] concluded by stating that the theoretical predictions agree with experimental results within
measurements precision \footnote{As a matter of fact, Ives and Stilwell used also an a') step (instead of a)) in which
the velocity of the emitting atom was assumed to be the velocity of the incoming
$H_2^+$ or $H_3^+$ ion calculated through the relation $eV=(1/2) M v^2$, where
$e$ is the charge of the ion, $M$ its mass, $v$ its velocity and $V$ the accelerating
potential. Ives and Stilwell found that both procedures for calculating the velocity
of the emitting atoms resulted in a predicted transverse Doppler shift
(point b) above) in agreement with the measured one (point c) above).}.
\end{description}
\normalsize
\vskip5mm
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Aisha Saleem Khan
Medicinally Important Trees
Aisha Saleem Khan
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
ISBN 978-3-319-56776-1e-ISBN 978-3-319-56777-8
DOI 10.1007/978-3-319-56777-8
Library of Congress Control Number: 2017939121
© Springer International Publishing AG 2017
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The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Dedicated to my parents as
without their prayers, I would not
have been able to work on this book
Acknowledgments
My sincere thanks are to my students, my colleagues, my friends, and my family members who supported and helped me in writing this book.
With profound gratitude, I also pay special thanks to HRH Prince Khalid Bin Sultan and the entire team of Makshaff Service Ltd., Riyadh, where my father worked nearly two decades in the role of Senior Accountant Payroll.
Without the help of Makshaff Service Ltd., it would not have been possible for my father to bear the educational costs of my siblings who got their education from the USA, Australia, and the UK universities, and I would not have been able to publish with Springer US. I am indeed grateful for the generosity of HRH Khalid Bin Sultan, General Ayed Al Jeaid (CEO), Mr. Ali Al Shaibany (CFO), and the entire Makshaff team.
I am thankful to my younger brother, Muntaha, who first gave me the idea of compiling all the information I had in the form of a book, as previously I was doing this research as a passion to make a diary and to keep my knowledge updated, which might also help me in teaching. I have always had the desire to have an updated publication on the following subject but had no idea that I would be able to compile this information in the form of a book.
I also express my thanks to my nephews Saad and Huzaifa and my friends who always went with me to remote areas in order to collect data and to take photographs of trees.
Finally, I am also thankful to Springer's editorial team, especially Eric Stannard and Jeffrey Taub, for helping me and giving me the opportunity to write a book on the following subject, which I believe will be very valuable for readers.
Contents
1 An Introduction to Medicinally Important Trees 1
1.1 Introduction 1
1.2 Phytochemicals of Medicinal Importance of Woody Plants 2
1.3 Solvents and Techniques for Extraction and Isolation of Medicinal Compounds 4
1.4 Important Medicinal Activities of Woody Plants 4
1.4.1 Antidiabetic Activity of Important Trees 5
1.4.2 Trees with Anticancer Activity 5
1.4.3 An Account of Antimicrobial Activity of Important Woody Plants 8
1.4.4 Antiviral and Possible Anti-HIV Activity of Important Woody Plants 9
1.4.5 Cardioprotective and Hepatoprotective Activities 12
1.4.6 Analgesic and Antipyretic Activities 13
1.4.7 Trees with Aphrodisiac and Antifertile Activities 14
1.4.8 Medicinal Properties of Important Leguminous Trees 17
1.4.9 Medicinally Important Figs, Nuts, and Edible Fruits 18
2 Important Trees with Antidiabetic Activities 21
2.1 Introduction 21
2.2 Important Trees with Antidiabetic Activities 22
2.3 Achras sapota L. (Syn: Manilkara zapota ) 23
2.4 Bombax ceiba L. 24
2.5 Barringtonia acutangula (L.) Gaertn. 27
2.6 Casuarina equisitifolia L. 28
2.7 Conocarpus lancifolius Eng. 29
2.8 Eriobotrya japonica Lindl. 32
2.9 Euphorbia pulcherrima Willd. 34
2.10 Jasminum sambac L. 34
2.11 Kigelia pinnata Jacq. (Syn: Bignonia africana ) 36
2.12 Lagerstroemia indica rosea 38
2.13 Hibiscus rosa sinensis L. 41
2.14 Morus alba L. 42
2.15 Murraya koenigii L. 44
2.16 Opuntia ficus-indica L. 45
References 48
3 Trees with Anticancer Activities 55
3.1 Introduction 55
3.2 Important Trees with Anticancer Activities 58
3.3 Bauhinia variegata L. 58
3.4 Callistemon citrinus L. 60
3.5 Carica papaya L. 62
3.6 Cycas revoluta Thunb. 65
3.7 Dillenia indica L. 66
3.8 Jacaranda mimosifolia D. Don 67
3.9 Jasminum officinale L. 69
3.10 Magnolia grandiflora L. 70
3.11 Plumeria obtusa L. 72
3.12 Plumeria rubra L. 74
3.13 Sapium sebiferum L. (syn: Triadica sebifera ) 75
3.14 Schleichera oleosa Lour. 77
3.15 Thuja occidentalis L. 78
References 80
4 Trees with Antimicrobial Activities 85
4.1 Introduction 85
4.2 An Account of Woody Plants with Antimicrobial Activities 87
4.3 Ailanthus altissima (Mill.) Swingle 87
4.4 Bougainvillea spectabilis Willd. 89
4.5 Cedrus deodera Roxb. (Syn Pinus deodara Roxb. ex D. Don) 91
4.6 Chukrasia velutina Roxb. (Syn: Chukrasia tabularis A. Juss) 92
4.7 Madhuca longifolia L. 93
4.8 Melia azerdarach L. 95
4.9 Podocarpus macrophyllus Thunb 98
4.10 Polyalthia longifolia Sonn. 99
4.11 Toona ciliata M. Roem (Syn: Cedrela toona ) 100
4.12 Tecoma stans L Juss. Ex Kunth Syn (Bigonia Stans) 102
4.13 Terminalia mantaly L. 104
References 105
5 Woody Plants with Possible Anti-HIV Activity 109
5.1 Introduction 109
5.2 Anti-HIV Compounds 110
5.3 Trees with Possible Anti-HIV Potential 110
5.4 Artocarpus integrifolia L. (Syn: Artocarpus heterophyllus Lam.) 111
5.5 Aegle marmelos L. 113
5.6 Caesalpaeina pulcherrima L. 114
5.7 Gleditsia triacanthos Linn. 115
5.8 Euphorbia royleana Boiss 117
5.9 Jatropha curcas L. 119
5.10 Heterophragma adenophyllum Seem. 120
5.11 Mimusops elengi L. 122
5.12 Platanus orientalis L. 123
5.13 Syzygium cumini L. 124
5.14 Tamarix aphylla L. 126
References 127
6 Trees with Hepatoprotective and Cardioprotective Activities 133
6.1 Introduction 133
6.2 An Account of Some Trees with Hepatoprotective and Cardioactive Activities 134
6.3 Alstonia scholaris L. R. Br (Syn: Echites scholaris ) 134
6.4 Anogeissus acuminata (Roxb. ex DC.) 137
6.5 Crataeva religiosa Forst f. 138
6.6 Carissa carandas L. 139
6.7 Cupress es sempervirens L. 143
6.8 Diospyros Spp. 144
6.8.1 Diospyros cordifolia Roxb. 147
6.9 Nerium oleander L. 147
6.10 Terminalia arjuna Roxb. 149
6.11 Thevetia peruviana Pers. (Syn: Cascabela thevetia ) 152
References 154
7 Antipyretic and Analgesic Activities of Some Economically Important Woody Plants 159
7.1 Introduction 159
7.2 An Account of Important Trees 159
7.3 Brachychiton populneus (Schott & Endl.) R. Br 161
7.4 Ceiba speciosa A. St.-Hill (Syn: Chorisia speciosa ) 162
7.5 Eucalyptus citriodora Hook. (Syn: Corymbia citriodora ) 166
7.6 Murraya exotica L. (Syn: Murraya paniculata ) 168
7.7 Pinus roxbrghii Sarg. 169
7.8 Pterospermum acerifolium L. 172
7.9 Putranjiva roxburghii Wall. 174
7.10 Salix babylonica L. (Syn. Salix japonica Thunb.) 175
7.11 Salix tetrasperma Roxb. 177
7.12 Tectona grandis L. 178
7.13 Zizyphus mauritiana Lam. 180
References 182
8 Aphrodisiac and Abortifacient Activities of Important Trees 187
8.1 Introduction 187
8.2 An Account of Aphrodisiac and Abortifacient Activities of Economically Important Woody Plants 189
8.3 Albizia lebbeck (L.) Benth. 189
8.4 Broussonetia papyrifera L. (Syn: Morus papyrifera ) 192
8.5 Butea monosperma Lam. (Syn B. frondosa Koenig Ex Roxb.) 193
8.6 Dombeya rotundifolia Hocsht. 195
8.7 Lantana camara L. 196
8.8 Myrtus communis L. 200
8.9 Ricinus communis L. 201
8.10 Saraca indica L. (Syn: Saraca asoca Roxb., De. Wild) 204
References 205
9 Leguminous Trees and Their Medicinal Properties 211
9.1 Introduction 211
9.2 An Account of Medicinally Important Leguminous Trees 213
9.3 Acacia catechu L. (Syn: Senegalia catechu ) 214
9.4 Acacia modesta Wall. (Syn: Senegalia modesta ) 214
9.5 Albizia procera Roxb. 216
9.6 Cassia fistula L. 218
9.7 Dalbergia sissoo Roxb. 218
9.8 Delonix regia Raf. 221
9.9 Erythrina suberosa Roxb. 221
9.10 Millettia ovalifolia (Syn: M. peguensis ) 224
9.11 Parkinsonia aculeata L. 225
9.12 Prosopis juliflora Swart. 227
9.13 P. spicigera L. (Syn: P. cineraria ) 227
References 230
10 Figs and Their Medicinal Value 235
10.1 Introduction 235
10.2 Summary of Trees That Produce Medicinally Important Figs 235
10.3 Ficus benghalensis L. 236
10.4 F. benjamina L. 237
10.5 F. carica L. 239
10.6 F. elastica (Roxb.) 241
10.7 F. glomerata Roxb. (Syn: F. racemose L.) 241
10.8 F. infectoria Miq. (Syn: F. virens ) 242
10.9 F. lyrata Warb. (Syn: F. sycomorus ) 245
10.10 F. macrophylla L. 245
10.11 F. religiosa L. 247
10.12 F. retusa L. (Syn F. microcarpa ) 249
References 251
11 Nuts and Their Nutritive and Medicinal Value 255
11.1 Introduction 255
11.2 An Account of Medicinal Properties of Some Nuts 255
11.3 Anacardium occidentale L. 256
11.4 Juglans regia L. 258
11.5 Pistacia vera L. 259
11.6 Prunus amygdalus L. Batsch 260
References 263
12 Medicinally Important Edible Fruits 267
12.1 Introduction 267
12.2 Medicinal Properties of Important Fruits 267
12.3 Citrus x sinensis L. 270
12.4 Citrus x limon L. 272
12.5 Malus domestica L. 276
12.6 Mangifera indica L. 279
12.7 Prunus persica L. Batsch 280
12.8 Psidium guajava L. 283
12.9 Punica granatum L. 286
12.10 Phoenix dactylifera L. 289
References 291
Index297
© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_1
# 1. An Introduction to Medicinally Important Trees
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
This chapter gives an introduction to medicinally important trees with reference to their ecological distribution and economic and medicinal importance. It also includes a brief introduction to the natural products of plants, as medicinal activities of plants are mainly due to the wide variety of these natural products, which include alkaloids, glycosides, terpenoids, tannins, and flavonoids. Important medicinal activities of trees are briefly described, which include antidiabetic, anticancer, antimicrobial, antiviral, hepatoprotective, and cardioprotective activities. Aphrodisiac, abortifacient, antipyretic, and analgesic properties of trees are also briefly introduced.
## 1.1 Introduction
Plants have been used for treating different diseases and for alleviating pain since ancient times. However, how plants have been used for curing a particular disease or symptoms differs in different cultures. The Ayurvedic concept of using plants for healing purposes originated around 500 B.C. in India. Engravings of medicinal plants on ancient buildings indicate that they have been used for treating different diseases for a long time. Assyrian clay tablets almost 4000 years old revealed the use of 250 medicinal and poisonous plants, including Atropa belladonna, Mandragora officinarum, and Papaver somniferum. Emperor Shen Nung described the use of 350 drugs in 3000 B.C. Later in India, Ayurveda mentioned the use of traditional medicine in about 900 B.C.
Almost 3000 plants are known to possess medicinal potential but more than 6000 plants are used by traditional herbal practitioners. The World Health Organization has reported 20,000 plant species studied for medicinal purposes, as more than 80% of the population of developing countries is facing difficulties due to synthetic drugs and therefore relies on traditional medicines to maintain their health. Discovery of phytochemicals in making drugs and their use in dietary supplements are expected to increase in future, as through advancement in analytical techniques, more knowledge is becoming available about the phytochemistry and metabolomics of medicinally important plants, which is not only helpful in the identification of these compounds but many phytochemicals of medicinal importance can be identified and used for various therapeutic purposes.
Many phytochemicals with medicinal importance include molecules which are synthesized during secondary metabolic activities of plants and are therefore commonly known as plant secondary metabolites or plant natural or functional products. They include many alkaloids, glycosides, flavonoids, terpenoids, and tannins. They also form an important criterion in plant classification. Identification and isolation of these bioactive compounds is crucial in the development of modern plant-based drugs. Medicinal activities of these bioactive compounds are mostly evaluated on model animals like rats, and many of the known compounds are still not applied on humans. Therefore, there is sufficient need to evaluate medicinal activities of plant-based phytochemicals on humans and to develop drugs which do not have any side effects like antibiotics.
## 1.2 Phytochemicals of Medicinal Importance of Woody Plants
Alkaloids are important medicinal compounds with nitrogen-containing molecules and their presence gives a bitter taste to plants. So far 12,000 alkaloids have been elucidated from plants which are part of the defensive system of plants. Due to their bitter taste, many herbivores and pests cannot consume them as they are difficult to digest. Many Acacia spp. are a source of important alkaloids of medicinal importance. Aztecs used alkaloid drugs for hallucination, divination, and magic-religious purposes. Alkaloid extracts from plants like Hyoscyamus, Atropa, and Datura were used as aphrodisiacs. Many alkaloids which have been known to be involved in execution cases throughout history are aconitine, atropine, colchicine, mescaline, morphine, and strychnine. Alkaloids like benzophenanthridine, protoberberine, psychotrines, and michellamine B have anti-HIV activities.
Terpenoids are another class of medicinal compounds which are carbon- and hydrogen-containing units also known as isoprenoid compounds. Over 40,000 terpenoids are reported from different parts of economically important plants. Many monoterpenoids make fragrances of plants which plants biosynthesize in their plastids in order to attract their pollinators, and many of these essential oils act as insect and mosquito repellents and also inhibit microbial growth and infections. Many of these essential oils, which are mostly monoterpenoids, have pleasant fragrances and are therefore used in food, most commonly as herbs due to their aroma and taste and also due to their antimicrobial properties. Essential oils like eucalyptol, menthol, linalool, nerolidol, limonene, eugenol, isoeugenol, myrcene, and pinene exhibit many antimicrobial properties against a wide range of microorganisms. Betulinic acid is a triterpenoid which is known to possess anti-HIV activities and to cause membrane disruption by lipophilic compounds. Terpenoids from the bark of the tree Pteleopsis suberosa in Mali are known to inhibit gastric ulcer. The stem bark of Ekebergia capensis contains a mixture of approximately ten triterpenoids which possess antimalarial activities.
Flavonoids are phenol-containing compounds which are important antioxidant, anticancer, cardioprotective, and neuroprotective compounds. Over 6000 flavonoids are reported in plants where they form colored pigments in plants, mostly in flowers and fruits, and provide protection against ultraviolet (UV) light. They are synthesized from phenylalanine and classified as chalcones, flavones, flavonols, flavandiols, anthocyanins, proanthocyanidins, and aurones. Many flavonoids like anthocyanins are antioxidants and are medicinally important compounds. Berries like bilberries, blueberries, cranberries, raspberries, mulberries, and strawberries and many citrus fruits are rich sources of flavonols of medicinal value including quercetin, which possesses anticarcinogenic and antiatherosclerosis activities. Their role as a natural modulator of cancer multidrug resistance is also important. Flavonoids like biochanin and daidzein and green tea polyphenols like epigallocatechin-3-gallate (EGCG) can inhibit multidrug resistance activities in many cancer cells by inhibiting P glycoprotein transporters.
Polyphenols of blueberries and teas provide protection against Parkinson's disease and Alzheimer's disease. Kaempferol and its derivatives are also found in many fruits which are also antioxidant, anti-inflammatory, and antiulcer agents. Many edible fruits which contain anthocyanins like edible berries have high nutritive value and their intake provides protection against many diseases. They are also commonly used in many nutraceutical products and their use is being recommended by physicians due to their antioxidant nature, nutritive value, and ability to stimulate some hormones. The role of flavonoids and their derivatives is also being explored as possible anti-HIV agents as they are reported to inhibit HIV-1 protease, integrase, and reverse transcriptase. Isoflavones can inhibit transcription by repressing HIV-1 promoter activity. Coumarins reported from more than 1300 species are phenolic compounds which possess anti-inflammatory, antimicrobial, and antiviral activities. Hydroxycinnamic acids and phytoalexins, which are coumarin derivatives, are also known to have antimicrobial and antifungal activities.
Glycosides are also important medicinal compounds which contain sugar molecules attached to a nonsugar group, while saponins are glycosides like glycosidic triterpenoids, which possess antimicrobial, insecticidal, and allelopathic activities. Cardioactive glycosides from plants like Nerium oleander are effective on heart muscles, while cyanogenic glycosides are found to occur in more than 2600 plants like almonds, peaches, and cherries and release hydrogen cyanide (HCN) due to the breakdown of cell walls.
Tannins are a group of polymeric astringent phenolic compounds commonly present in different parts of plants which can cause leather tanning and can precipitate gelatin. They are either hydrolyzed tannins, which are gallic acid derivatives, or condensed tannins, which are flavonoid derivatives. Many tannins present in plants used for making green teas and red wines are medicinally important compounds with antimicrobial, anticancer, and antioxidant properties. Condensed tannins are capable of binding with the cell wall of ruminal bacteria and prevent protease activities. Methanolic extracts from the bark of trees like Terminalia alata possess tannins responsible for antibiotic activities.
## 1.3 Solvents and Techniques for Extraction and Isolation of Medicinal Compounds
Various traditional techniques which are employed for extraction of medicinal compounds from plants include soaking or maceration of plant parts in suitable solvents or decoction, which is widely used in Chinese and Ayurvedic medicinal practices which involve heating or boiling of plant materials. Extraction of desired compounds is also achieved through percolation using methanol or ethanol or other solvents like acetone, petroleum ether, or hexane. Water is the best solvent to extract phytochemicals. Traditionally, the leaves of trees with known medicinal value are ingested as tea which is steeped in hot water or tinctures are prepared using alcohols, or steam containing boiling suspensions of leaves can also be inhaled. Poultices are also used traditionally, which are made from concentrated teas or tinctures. The dried parts of plants are also applied externally, possibly with the addition of a small amount of oil.
However, initial screening is achieved through alcoholic extraction followed by organic extraction including methanol, ethanol, dichloromethane, ether, or chloroform extraction. Alcoholic extractions include grinding the dried parts of plants, soaking in methanol or ethanol, followed by filtration and washing and then drying under reduced pressure or dissolving again in alcohol with determined concentration. However, extraction in water involves initial soaking of plants, drying, blending, and filtrate formation, which is then cleared through multiple centrifuges and screened for antimicrobial, antifungal, or antiviral activities in different assays.
Further chemical analysis is done through many techniques which include extraction and identification of medicinal phytochemicals including different chromatographic methods like high-performance liquid chromatography (HPLC), droplet counter-current chromatography (DCCC), capillary zone electrophoresis, bioautography, radioimmunoassays, fast atom bombardment mass spectroscopy (MS), Fourier transform infrared spectroscopy (FTIR), near infrared Fourier transform (NIR-FT), Raman microspectroscopic techniques, electrospray ionization mass spectrometry (ESI-MS), tandem mass spectroscopy, infrared, 1-D or 2-D (dimensional) nuclear magnetic resonance (NMR), ion exchange, silica gel column chromatography, and x-ray crystallography.
## 1.4 Important Medicinal Activities of Woody Plants
Although many trees have more than one medicinal activity, in this book they are characterized on the basis of their main and unique medicinal activities. With billions of medicinal trees in the world, it is not possible to include the medicinal properties of so many trees within one book, so only a few species are included at this stage. Important medicinal activities included are antidiabetic, anticancer, antipyretic, analgesic, anti-HIV, hepatoprotective, cardioprotective, aphrodisiac, and nutritive value of many legumes, figs, nuts, and edible fruits.
### 1.4.1 Antidiabetic Activity of Important Trees
Diabetes mellitus is one of the most rapidly increasing diseases, affecting approximately 2.8% of the global population, and is expected to increase to 5.4% by 2025. It is one of the most common endocrine disorders, which results in deficiency of insulin-causing high blood glucose level known as type 1 diabetes. In type 2 diabetes, the body does not produce enough insulin or does not use it properly.
Plants which can reduce the level of glucose in blood and stimulate the secretion of insulin by the pancreas have antidiabetic activities. Many trees are reported to have antidiabetic potential. More than 50% of plants with antidiabetic potential are distributed in Asia, and more than 35% of compounds for diabetic treatments are extracted from leaves. However, the fruits of many plants with antidiabetic potential can be consumed orally in the form of juices.
Many plants from families like Fabaceae, Lamiaceae, Liliaceae, Moraceae, and Rosaceae are known to possess phytochemicals like alkaloids, carotenoids, polyphenols, flavonoids, terpenoids, coumarins, and other metabolites which possess antidiabetic activities and can maintain glucose level in blood. Some of the important trees with antidiabetic activity which are discussed in this book include Achras sapota, Aegle marmelos, Bauhinia variegata, Bombax ceiba, Hibiscus rosa sinensis, Eriobotrya japonica, Euphorbia pulcherrima, Ficus spp., Jasminum sambac, Kigelia pinnata, Lagerstroemia indica rosea, Murraya koenigii, and Roystonea regia (Fig. 1.1a–c). Active constituents include aegeline from Aegle marmelos, vindoline and vleurosine from Catharanthus roseus, strictinin and isostrictinin from Psidium guajava, trigonelline from Trigonella foenum-gracuem, arboran and aleosin from Aloe barbadensis, nimbidin from Azadirachta indica, pinitol from Bougainvillea spectabilis, bengalinoside from Ficus benghalensis, moran A from Morus alba, bisindole alkaloid from Murraya koenigii, amygdallin from Prunus persica, and techomine from Tecoma stans.
Fig. 1.1
(a–c) Trees with antidiabetic activities. (a) Bombax ceiba (red cotton tree) is an important medicinal tree with antidiabetic and antimicrobial activities due to the presence of compounds like shamimin (flavonol-C-glycoside). (b) Pods of Cassia fistula (golden shower tree) are used traditionally for the treatment of diabetes and for antiseptic purposes. (c) Ficus species are medicinally important figs. F. religiosa (sacred fig) is an important antidiabetic fig with β-sitosteryl glycoside as an active compound
### 1.4.2 Trees with Anticancer Activity
Many trees are known to have anticancer potential. Cancer is one of the important leading causes of death and is responsible for one in eight deaths worldwide. Chemotherapy is widely used for cancer treatment, but the use of chemotherapeutic drugs causes many toxic effects. Plant-based anticancer agents in the market include vinca alkaloids like vinblastine, vincristine, and vindesine, and epipodophyllotoxins (etoposide and teniposide), taxanes (paclitaxel and docetexal), and camptothecin derivatives like camptothecin and irinotecan. They interact with methyltransferases or histone deacetylases and inhibit their activities or act as DNA damage preventers or interrupt with mitotic activities of proliferating cells. Many sulforaphanes from vegetables like broccoli and cabbage, isoflavones from edible legumes, and pomiferins from Osage oranges are histone deacytylases inhibitors. Taxanes like paclitaxel drug from Taxus sp. can reduce the rate of replication of cancer cells by stabilizing or polymerizing microtubules in the cells. Anticancer compounds in combination like Taxus diterpenes, Podophyllum lignans, and alkaloids from Camptotheca can be effectively used to reduce the growth of cancer cells.
The National Cancer Institute has cataloged approximately 35,000 plants species with potential anticancer activities. Over 50 plants are reported to have anticancer (tumor-inhibiting) properties due to the presence of their bioactive compounds (anticarcinogens). However, anticancer compounds are mostly antioxidants or they have antitumor activity like polyphenols, isoflavones, taxols, and brassinosteroids. Polyphenols include tannins, flavonoids, curcumin, resveratrol, and gallocatechins. They interfere with carcinogenic compounds through binding with proteins or through regulation of acetylation, methylation, or phosphorylation.
Curcumin is known to suppress expression of tumor necrosis factor through interaction with various stimuli. Flavonoids from many plants have shown anticancer activities against many cancer cell lines including hepatoma, cervical carcinoma, and breast cancer. Genistein, isoflavone from soy beans, is known to inhibit the growth of many cancer cell lines including leukemia, lymphoma, prostate, breast, and lung cancer. It is reported as a protein tyrosine kinases inhibitor. It is capable of arresting the G2/M cell cycle in breast cells, gastric adenocarcinoma cells, and human melanoma cells. Biochanin is another anticancer isoflavone from soy which is cytotoxic on cell growth in the mammary carcinoma cell line MCF-7, myeloid leukemia, and pancreatic tumor cells. Flavonoids from plants like Erythrina suberosa have cytotoxic effects against human leukemia. Flavonoids from citrus fruits like tangeretin and nobiletin can inhibit human breast cancer cell lines by blocking the progression of G1 phase of the cell cycle.
Popular trees with anticancer activities include Abrus precatorius, Albizia lebbeck, Dillenia indica, Delonix regia, Erythrina suberosa, Magnolia grandiflora, Podocarpus macrophyllus, Murraya koenigii, Nasturtium officinale, Prunus persica, Pterospermum acerifolium, Thuja occidentalis, Vinca rosea, and Ziziphus nummularia. Important edible fruits with anticancer properties are Achras sapota, Bauhinia variegata, Eriobotrya japonica, Mangifera indica, Prunus persica, and Syzygium cumini (Fig. 1.2a–d).
Fig. 1.2
(a–d) Some trees with anticancer activities. (a) Prunus persica (peach) is an important tree with anticancer and antidiabetic activities (due to the presence of amygdallin, an active constituent). (b) Thuja occidentalis (white cedar) is an important medicinal and ornamental plant with antidiabetic and anticancer activities due to its bioactive compound, thujone. (c) Erythrina suberosa (red coral tree) is found to be effective against various cancer cell lines and also possesses antiplasmodial and anxiolytic activities. In some experiments, flavonoids extracted from stem bark (4′-Methoxy licoflavanone [MLF] and Alpinum isoflavone [AIF]) showed cytotoxic effects in HL-60 cells (human leukemia). Further, extract from all parts of the plant can treat chronic dysmenorrhea and can maintain menstrual flow by reducing abdominal fats. (d) Murraya koenigii (curry tree) is an important anticancer, antidiabetic, and antimicrobial tree
Brassinosteroids are naturally occurring compounds in plants which are significant against cancer and are known to bind with proteins and can inhibit initiation of hormone-sensitive and hormone- insensitive cancer cells. 28-Homocastasterone (28-homoCS) and 24-epibrassinolide (24-epiBL) are reported to be important anticancer molecules, including against breast cancer, prostate cancer, T-lymphoblastic leukemia Carboplatin, Etoposide Phosphate, Melphalan Hydrochloride (CEM), multiple myeloma Roswell Park Memorial Institute (RPMI) 8226, cervical carcinoma HeLa, lung carcinoma A-549, and human osteosarcoma (HOS) cell lines, which are effective even at micromolar concentrations against many cancer cell lines.
However, many medicines prepared from herbs are hardly absorbed due to their complex structure and large size, so research is being conducted through nanobiotechnology which is aimed at reducing the particle size and increasing the dissolution rate of bioactive compounds. Bromelain is an anticancer compound isolated from Ananas comosus and more effective when formulated with nanoparticles by triggering the apoptosis of benign cells. Gold nanoparticles of Antigonon leptopus and Acalypha indica have shown anticancer activities against breast cancer cell lines. Quercetin is applied through superparamagnetic magnetite against breast cancer cell lines.
### 1.4.3 An Account of Antimicrobial Activity of Important Woody Plants
Antimicrobial properties of many trees are associated with alkaloids, tannins, terpenoids, polyamines, isothiocyanates, thiosulfinates, glucosides, and polyacetylenes. Flavonoids and acetylene compounds are also effective against malaria and liver disorders. Many herbs also contain compounds with antimicrobial activities against a wide range of microorganisms and fungal spores including quinones, catechols, catechins, eugenols, isoeugenol, capsaicin, curcumin, piperine, hypericin, chrysin, coumarin, menthol, berberine, warfarin, and harmane.
Flavonoids like robinetin, myricetin, apigenin, rutin, galangin, 2,4,2′-trihydroxy-5′-methylchalcone, and lonchocarpol A also possess antimicrobial activities. Their antimicrobial activity is related to the position and number of hydroxyl groups attached to the ring, as higher toxicity is reported against many microorganisms with increase in number of hydroxyl groups. Catechol and pyrogallol are hydroxyl-containing flavonoids and contain toxic effects against many microorganisms. Many catechins present in teas are antimicrobial against many bacteria like Vibrio cholerae, Streptococcus mutans, and Shigella.
Trees like Pterocarpus santalinus possess a broad spectrum of antimicrobial activities due to isoflavone glycosides. Many antimicrobial compounds are being extracted from plants to be used for therapeutic purposes due to problems such as increase in antibiotics resistance (Fig. 1.3a–e). Antimicrobial activities of tannins are associated with their potential to inhibit microbial adhesion, enzymes, and cell envelope transport proteins. They also possess many insecticidal, antiviral, and antifungal activities.
Fig. 1.3
(a–e) Antimicrobial properties of trees are due to alkaloids, glycosides, and other phenolic compounds. (a) Millettia ovalifolia (rosewood tree) possesses flavonoids and rotenone which are known to have antibacterial and anticancer activities. (b) Melia azedarach (white cedar) is well known due to its antimicrobial and other medicinal properties. Azadirachtin, nimbin, and nimbidin have antiviral and antifungal properties. (c) Ethyl acetate extract of Toona ciliata (red cedar) is reported to have antibacterial activity against Salmonella typhi and Staphylococcus epidermidis, while methanolic extract is significant against Klebsiella pneumoniae, S. typhi, Staphylococcus aureus, and S. epidermidis. (d) Spreading crown of Kigelia pinnata (sausage tree) is an important medicinal plant with antimalarial compound (lapachol) and antibacterial compound (kigelinone). (e) Essential oil of Pinus roxburghii (chir pine) is antimicrobial and possesses insecticidal properties
Fungicidal activities of approximately 1281 species are reported from 184 families. Members of the families Anacardiaceae, Compositae, Cruciferae (Brassicaceae), Labiatae (Lamiaceae), Liliaceae, Ranunculaceae, Rosaceae, and Solanaceae exhibit antifungal activities.
Antimicrobial activities are screened commonly through either broth dilution assays and disk or agar well diffusion assay. Inoculated plates are also exposed to UV light to check for the presence of light-sensitizing phytochemicals. Antifungal activities are measured through spore germination assays in which phytochemicals are added to fungal spores and then observed microscopically to check spore germination and then check for antibiotic effects.
### 1.4.4 Antiviral and Possible Anti-HIV Activity of Important Woody Plants
Acquired immunodeficiency syndrome (AIDS) is an immunosuppressive disease caused by the human immunodeficiency virus (HIV) which causes life-threatening opportunistic infections. It is one of the leading causes of death in Africa and the number of people infected with HIV is increasing worldwide.
Many trees are reported to be effective against viruses and for the treatment of AIDS due to their possible anti-HIV potential. Crude extract of balsam from many trees, which is also known as bee glue, is a mixture of complex terpenoids, flavonoids, benzoic acids, esters, and phenolic acids. It is effective against herpes simplex virus (HSV-1), adenovirus type 2, vesicular stomatitis virus, and poliovirus (Fig. 1.4a–c).
Fig. 1.4
(a–c) Trees with antiviral properties. (a) Artocarpus integrifolia (jackfruit) is an important plant with antibacterial, antidiabetic, and possible anti-HIV potential due to the presence of lectins like jacalin which interact with the lymphocyte cell-surface-molecule CD4, a receptor of HIV-1. (b) Fruits and seeds of Mimusops elengi (bullet wood tree) are a rich source of triterpenoid saponins like mimusopsides A and B, mimusopins and mimusopsins, and mimusopic acids which have possible anti-HIV potential. (c) Fruit and seeds of Syzygium cumini (jambul) are rich in antioxidants with anticancer and anti-HIV properties
Many flavonoid compounds exhibit antiviral activities against multiple viruses which include swertifrancheside, glycyrrhizin, and chryrsin, which can inhibit HIV replication. Ellagitannins, hydroxymaprounic acid, betulinic acid, catechins, quercetins, flavones, cornusins, ursolic acids, and lectins are associated with anti-HIV properties.
Polyphenols in tea like epigallocatechin-3-gallate (EGCG) are reported to interfere with the envelope protein of HIV-1. Scutellarin can inhibit many stages of HIV replication by inhibiting HIV-1 reverse transcriptase activity, HIV-1 particle attachment, or cell fusion. Chrysin is also known to be a potent inhibitor of HIV-1 transcription in infected cells, and calanolide isolated from Calophyllum lanigerum has the potential to be used for drug development against HIV.
Alkaloids like papaverine from Papaver somniferum, buchapine from Evodia roxburghiana, nitidine from Toddalia asiatica, piperidine from Buchenavia capitata, and harmine from Symplocos setchuensis have anti-HIV potential.
Phenolic compounds like lithospermic acid isolated from Salvia miltiorrhiza; punicalagin, gallic acid, and chebulagic acid from Terminalia chebula; repandusinic acid from Phyllanthus niruri; and mallatojaponin from Mallotus japonicus are also reported to have anti-HIV activities.
### 1.4.5 Cardioprotective and Hepatoprotective Activities
Flavonoids and polyphenols found in many trees are known to possess cardioprotective activities, as consumption of edible fruit containing flavonoids reduces oxidative stress which causes change in lipid peroxidation in arterial macrophage and in lipoproteins and is therefore beneficial for reducing cardiovascular disorders. Hypochlorite scavenging activity of some flavonoids is also reported, which can cause atherosclerosis. Many flavonoids in tea possess lipid-lowering activities, while polyphenols like diverin can reduce low-density lipoprotein (LDL) while increasing high-density lipoprotein (HDL) in coronary heart disease patients.
Chrysin, curcumin, catechins, chrysoeriol, eugenol, frederine, gingerol, diosgenin, hesperidin, hydroxytyrosol (from Olea europaea), kaempferol, lycopene, resveratrol, sesamin, mangiferin, and periplogenin from Aegle marmelos are reported to have cardioprotective activities against doxorubicin (DOX). Arjunolic acid is a triterpenoid saponin from Terminalia arjuna that possesses cardioprotective properties and reduces the effects of cytotoxic antibiotics like doxorubicin and reduces myocardial toxicity when administrated in rats (Fig. 1.5a–c).
Fig. 1.5
(a–c) Trees with cardioactive and hepatoprotective properties. (a) Fruit and leaf extracts of Carissa carandas (Bengal currant) possess cardioprotective and hepatoprotective properties. (b) Cardioactive properties of Nerium oleander (rose-bay) are due to cardioactive glycosides which are extracted from leaves including gentiobiosyl-oleandrin, gentiobiosyl-nerigoside and gentiobiosyl-beaumontoside and also due to alkaloids like neriin and oleandrin. Oleandrin and its aglycone oleandrigenin are shown to have anticancer properties. (c) Terminalia arjuna (arjun tree) possesses cardioprotective properties against chronic stable angina, endothelial dysfunction, heart failure, and ischemic mitral regurgitation
Natural products from plants possess healing properties for recovery of intoxicated liver. A number of plants are known to have antihepatitis and anticirrhosis activities. Flavonoids like naringenin from Citrus spp. are useful in the treatment of hepatic fibrosis, while naringin promotes the hepatic antioxidant defense system. Proanthocyanidins and anthocyanins, which are widely present in blueberries, protect hepatocytes from oxidative stress and maintain normal functions of the liver. Extracts from cladode of Opuntia ficus-indica are known to protect liver health by scavenging free radical species and by enhancing antioxidant activities. They also reduce hepatic toxicity of organophosphorus insecticide chlorpyrifos.
### 1.4.6 Analgesic and Antipyretic Activities
Many trees are known to inhibit the activities of pyrogens (disease-causing agents) and are therefore known as "antipyretic." They inhibit the activity of prostaglandin synthase with low selectivity without causing any side effects. Many plants with antipyretic activities also relieve pain which is mostly due to fever and therefore also possess analgesic effects, i.e., pain-relieving activities. Therefore, there is a need to search for plants with antipyretic activities which can be used as a substitute for paracetamol and other synthetic drugs.
More than 500 species of Salix, commonly known as willows, are reported worldwide and have analgesic and antipyretic activities (Fig. 1.6a–b). They are used traditionally and in drugs due to their pain- and fever-relieving properties. The main active compound is salicin from tree bark along with phenolic glycosides like salicortin, fragilin, and tremulacin. Aspirin is one of the nonsteroidal anti-inflammatory drugs from willow leaves which has long been used due to its analgesic and antipyretic effects due to its inhibitory effects on cyclooxygenases, which can form prostanoids. However, discovery of cyclooxygenases-2 has increased the demands of plant phytochemicals with anti-inflammatory activities.
Fig. 1.6
(a–b) Antipyretic, anti-inflammatory, and analgesic properties of many Salix spp. (Indian willow) are due to the main compound salicin. (c) Analgesic, antimicrobial, anticancer, antiplasmodial, and antidiabetic activities of Ziziphus mauritiana (jujube) are due to cyclopeptides and jujubosides
### 1.4.7 Trees with Aphrodisiac and Antifertile Activities
Plants and plant-based products have long been usedto improve sexual behavior. Aphrodisiac properties of many trees are reported due to the presence of compounds which can maintain levels of the sex hormone testosterone and can also improve blood flow. Many aphrodisiacs work by improving the testosterone level or by controlling the central nervous system, or by crossing the blood-brain barrier and stimulating the area of sexual arousal or through giving nutrients required for sexual health.
Aphrodisiac properties of plants like Moringa oleifera are due to saponins and due to their high nutritive value. Other plants with aphrodisiac activities are discussed in Chap. (Fig. 1.7a–d). Date palm pollens contain estradiol and flavonoids, which are reported to improve the reproductive system of adult male rats. Administration of 50% ethanolic extract of nutmeg, clove, and penegra improved mating behavior of mice.
Fig. 1.7
(a–d) Some trees and shrubs with aphrodisiac and abortifacient activities. (a) Methanolic extract of Albizia lebbeck (flea tree) pods possesses antifertile activities. (b) Tea prepared from Jacaranda mimosifolia (fern tree) leaves and bark decoction is aphrodisiac and is used to regulate fertility and lactation. (c) Mangifera indica (mango) is an important edible tree in many Asian countries which possesses aphrodisiac, anticancer, and antidiabetic activities. (d) Seed extract of Ricinus communis (castor oil plant), commonly known as castor oil plant, possesses anticonceptive activity due to the presence of ricinoleic acid which has spermicidal effects. (e) Dombeya rotundifolia (wild pear) possesses abortifacient activity
Terminalia catappa seeds at a dose of 1500 mg/kg exhibited improvement of aphrodisiac activities in rats for 7 days but a higher dose of 3000 mg/kg inhibited sexual behavior. Fruit, leaves, and bark extracts of Ficus religiosa are traditionally cooked with milk and sugar and used as an aphrodisiac. Yohimbe acts as a stimulant for pelvic nerve ganglia and also boosts adrenaline supply to nerve endings, which improves sexual sensation. Some herbal aphrodisiac products available in the market include VigRX OilTM, MaxodermTM, Virility Pills, ProEnhanceTM, Virility Patch, and RXTM.
Many plants are known to induce abortion or to have antifertile activities and are used as emmenagogues. They have been used traditionally worldwide for inducing abortion. Extracts of such plants can inhibit fertility,cause abortion, or stimulate uterine contractions. They are either directly swallowed as abortifacients or a drink or tea is made from these plants' parts. Many of them cause disruption and desynchronizing of preovulatory and preimplanting events. Antifertile agents which prevent ovulation or fertilization are known as contraceptives while those which affect after implantation are known as abortifacients. Oral contraceptives like Depo-Provera and Norplant© have been used successfully in many areas of the Pacific.
Plants like Albizia lebbeck, Carica papaya, Crocus sativus, Dombeya rotundifolia, Ricinus communis, Myristica fragrans, Tamarindus indica, and Trichosanthes kirilowii are known to have anticonceptive activities. Interestingly, many plants which act as aphrodisiacs are also known to be abortifacients; the actual difference is due to the concentration of extract of plants, which can either promote fertility or can cause infertility.
However, the majority of these plants and their aphrodisiac and abortifacient efficacy are tested on rats, which showed improvement in sexual behavior through administration of aqueous, ethanolic, methanolic, chloroform, hexane, and petroleum ether extracts of different parts of these plants, and this has not yet been tested experimentally on humans and their mechanism of action on human tissues is still unknown and needs to be investigated further.
### 1.4.8 Medicinal Properties of Important Leguminous Trees
Edible legumes make up an important part of seed vegetables and are a rich source of proteins. They include lentils, peas, and chickpeas and beans like fava beans, kidney beans, pinto beans, and soybeans. Soymilk, tofu (bean curd), and miso are important products of soybeans. Soymilk is rich in many phytoestrogens which reduce osteoporosis, with a nutritive value equal to cow's milk. Many trees of the legume family are a source of food, fodder, and fuel and are medicinally important because they possess important plants with antibacterial, antifungal, antimicrobial, antiallergic, antidiabetic, and anticancer activities due to the presence of flavonoids (isoflavones), furanocoumarins, terpenoids, quinones, and xanthones.
Many trees of the legume family also serve as a source of biofuel, which can be derived from recently dead organic or biological material, whereas fossil fuel is derived from long dead material. Many plants are being evaluated to explore their potential to be used as biofuel. Some trees are a source of timber, fuel, cordage, and paper making. Well-known plants are Acacia and Albizia spp., Butea monosperma, Dalbergia sissoo, Delonix regia, and Erythrina suberosa (Fig. 1.8a–d).
Fig. 1.8
(a–c) Medicinally important legumes. (a) B. variegata (orchid tree) is a leguminous tree with a wide range of medicinal activities like anticancer, antimicrobial, anti-inflammatory, nephroprotective, and hepatoprotective properties which are attributed to many flavonoids, kaempferols, and cytokines present in leaves. (b) Delonix regia (royal poinciana) also known as flame of forest is an important medicinal leguminous tree due to the presence of many flavonoids many of which are antioxidants with anticancer and radioprotective properties. (c) Saraca indica (asoka) is a legume with antibacterial, anticancer, and antitumor potential
### 1.4.9 Medicinally Important Figs, Nuts, and Edible Fruits
Figs belong to the Ficus species, which is a large genus comprising over 800 medicinally important species distributed worldwide. However, some of the important species are discussed in Chap. with their important phytochemicals and medicinal value. Although many species of Ficus are not edible, they possess many antidiabetic, anticancer, and antimicrobial activities (Fig. 1.9a, b).
Fig. 1.9
(a) Figs like Ficus benghalensis (banyan) are antidiabetic due to the presence of leucoanthocyanidins. (b) Ficus macrophylla (strangler fig) is commonly known as strangler fig and possesses important antimicrobial and antidiabetic properties
Edible fruits, figs, and nuts are rich sources of micronutrients, fibers, vitamins, and minerals and are also important as they possess numerous medicinal activities. Thousands of different types of fruits are grown around the world and are also collected from wild plants. Various citrus fruits, pome fruits, stone fruits, dried fruits with hard shells like almonds, cashew nuts, walnut, pecans, macadamia, Brazil nuts, hazelnuts, pistachios, and pine nuts, and fruits growing in the tropics like bananas, papayas, mangoes, and pineapples are widely consumed due to their nutritive and medicinal value (Fig. 1.10a–e). Many edible dried fruits maintain cardiovascular health and provide protection against coronary heart diseases.
Fig. 1.10
(a–e) Medicinally important edible fruits (a–b) Prunus sp. (peach), (c) Litchi chinensis (d–e), and Malus domestica (apple)
© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_2
# 2. Important Trees with Antidiabetic Activities
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
This chapter explains antidiabetic activities of woody plants like Achras sapota, Bombax ceiba, Barringtonia acutangula, Casuarina equisitifolia, Conocarpus lancifolius, Eriobotrya japonica, Euphorbia pulcherrima, Jasminum sambac, Kigelia pinnata, Lagerstroemia indica rosea, Hibiscus rosa sinensis, Morus alba, Murraya koenigii, and Roystonea regia. More than 400 plants are reported to have antidiabetic activities due to their effects on lowering blood cholesterol, decreasing lipid peroxidation, and increasing the renewal of parietal cells in the pancreas and thereby stimulating the secretion of pancreatic insulin. The medicinal properties of many such trees are described in this section.
## 2.1 Introduction
Diabetes mellitus, a serious health problem which is caused due to increase in plasma glucose concentration resulting from insufficient insulin, results in many metabolic abnormalities and disturbed metabolism of proteins, carbohydrates, and lipids. Insulin is released by pancreatic β-cells and regulates glucose homeostasis. It stimulates hepatocytes, myocytes, and adipocytes to uptake glucose from the circulatory system which can be further catabolized for metabolic needs or stored as glycogen. When insulin cannot be utilized effectively by cells, it results in diabetes.
There are almost 150 million people worldwide suffering from diabetes (Moller and Filler 1991). However, type 2 diabetes (T2DM) is the most encountered form of diabetes, which accounts for more than 80% of total cases (Mlinar et al. 2007). Diabetes is also known to cause hyperlipidemia by disturbing metabolic pathways. Insulin deficiency stimulates lipolysis in adipose tissues and gives rise to hyperlipidemia.
Management of diabetes without any side effects is still a challenge for medical science, and research is being conducted on medicinal plants with antidiabetic activities in order to identify potential compounds and facilitate their therapeutic application. More than 800 plants are used in different ways to treat diabetes worldwide, particularly those which belong to the families Asteraceae, Cucurbitaceae, Euphorbiaceae, Leguminosae, Lamiaceae, Moraceae, and Rosaceae. Many of them have proved to be effective in lowering blood glucose level in alloxan-induced and streptozotocin-induced diabetic mice, Swiss rats, or Wistar rats. However, some plants medicate diabetic nephropathy by lowering blood cholesterol, decreasing lipid peroxidation, and increasing the renewal of parietal cells in the pancreas and thereby stimulating the secretion of pancreatic insulin.
Many saponins are antidiabetic compounds which stimulate insulin release (Keller et al. 2011), improve insulin sensitivity (Kwon et al. 2009), or promote glucose uptake (Wang and Zhang 2012). Plants with antihyperlipidemic activities also reduce blood glucose, improve lipid profile, increase insulin secretion from the pancreas, and inhibit intestinal absorption of glucose (Fig. 2.1a–e).
Fig. 2.1
(a–e) (a) Cassia javanica (pink shower) is an important tree with antidiabetic, anticancer and antifungal properties which are due to important phytochemicals like anthraquinones and sterols (b) Fruits (red) of F. benghalensis (Indian banyan) are source of leucopelargonins (an active antidiabetic constituent) (c) Hydrochloric acid extract of Sapindus mukorossi (Chinese soapberry) fruit possesses antihyperglycemic and antihyperlipidemic properties (d) Sterculia alata (Buddha coconut) seeds are used traditionally for treating diabetes (e) Seeds and leaves of Roystonea regia (Cuban royal palm) are used traditionally for treatment of diabetes
## 2.2 Important Trees with Antidiabetic Activities
Antidiabetic activity of important trees is discussed in the following section.
## 2.3 Achras sapota L. (Syn: Manilkara zapota)
* Vernacular name: Sapodilla
* Family: Sapotaceae
Ecological Distribution and Morphological Characteristics
A. sapota is an evergreen, latex-producing tree which is native to tropical America but is also grown in many countries in Southeast Asia.
It can grow up to a height of 30 m. The glossy leaves are medium sized, alternately arranged, and elliptic to ovate in shape. The flowers are monoecious, white, and look like bells. The fruit is a berry, which is edible (Fig. 2.2a, b).
Fig. 2.2
(a, b) Antihyperglycemic activities of Achras sapota, sapodilla are due to the presence of sterols and triterpenoids.Significant hypoglycemic effects of seed extracts are reported in streptozotocin-induced diabetic rats at 400 mg of aqueous extract and 200 mg of ethanolic extract (a) Spiral leaves containing fruits (b) Fruit
Important Phytochemicals and Medicinal Value
The main constituents are alkaloids, steroids, flavonoids, saponins, tannins, amino acids, anthraquinone, deoxy sugars, and phenolic compounds. Important phytochemicals in seeds are saponin, sapotin, achrassaponin, and sapotinine. The juice is a rich source of sugars, proteins, ascorbic acid, phenolics, carotenoids, and minerals like iron, copper, zinc, calcium, and potassium. The radical-scavenging potential of sapota juice is due to its nutraceutical metabolites, like phenolics, carotenoids, and ascorbic acid (Monalisha et al. 2010).
Antidiabetic activity of the seeds is due to the presence of saponins as significant hypoglycemic effects are reported in streptozotocin-induced diabetic rats at 400 mg of aqueous extract and 200 mg of ethanolic extract (Zheng et al. 2012; Ruchmani et al. 2014). The roots and leaves are also reported to have hypoglycemic activity (Muthadi et al. 2000; Fayek et al. 2012).
Traditional Uses
Bark of the plant is antibiotic, astringent, and febrifuge. The leaves are used to treat cold and cough. They have antimicrobial and antioxidant activities (Nair and Chanda 2008; Kaneria et al. 2009; Kaneria and Chanda 2012). The latex of sapota is used for filling tooth cavities. A paste of the seeds is applied to cure bites of poisonous animals. The fruit is edible, aromatic, astringent, and sweet (Balerdi and Shaw 1998).
## 2.4 Bombax ceiba L.
* Vernacular names: Red cotton, Silk tree
* Family : Bombacaceae
Ecological Distribution and Morphological Characteristics
It is also known as the red cotton tree due to the color of the flowers. It is native to Pakistan, India, and Nepal. The leaves are large, petiolate, compound, divided into 5–7 leaflets, digitate, and lanceolate. Large red flowers appear in early spring, and the fruit is a capsule having fibers like the cotton plant (Fig. 2.3a–h). The seeds are smooth, black, or gray embedded in long white wool.
Fig. 2.3
(a–h) (a) Leaves and fruits of Bombax ceiba, red cotton tree possess antidiabetic activity. Main antidiabetic compound is C-flavonol glucoside shamimin. Bark extract showed significant hypoglycemic and hypolipidemic effects in streptozotocin-induced diabetic rats at dose of 600 mg/kg/bw (b) Digitate leaves (c) Flowers (d) Buds (e) Branch containing fruits (f) Cotton from fruit which is a capsule is used for making pillows and quilts (g, h) Deciduous tree in winter
Important Phytochemicals and Medicinal Value
The roots are a rich source of terpenoids like cadinane sesquiterpenoids, which include bombamalones, lactone, bombaxquinone B, and lacinilene. They also contain β-D-glucoside of β-setosterol, free β-setosterol, hentriacontane, hentriacontanol, kaempferol, and quercetin along with traces of an essential oil. Phytochemical screening revealed that fresh petals of the flowers are a source of anthocyanidine glycoside (A and B), which are characterized as pelargonidin-5β-D (El-Hagrassi et al. 2011). The seeds of plants also contain important metabolites like n-hexacosanol, palmitic acid, octadecyl palmitate, tannic acid, 1-gallayl-β-glucose, ethyl gallate, and a mixture of α-, β-, and γ- tocopherol.
The leaves, fruits, and heartwood are reported to possess antidiabetic activity due to the presence of C-flavonol glucoside shamiminol (Saravanamuttu and Sudarsanam 2012). At a dose of 600 mg/kg/bw, bark extract caused significant hypoglycemic and hypolipidemic effects in streptozotocin-induced diabetic rats, which might be due to the presence of triterpenoid compounds in the extract (Bhavsar and Talele 2013). The flowers are known to possess cardioprotective activities (Patel et al. 2011).
Traditional Uses
The plant is used as a source of fuel, furniture, and carving wood. Cotton from seeds is used for pillows quilts and matchsticks (Sheikh 1993). The stem bark is considered as diuretic, inflammation, and astringent. The flowers are also astringent. The seeds are used traditionally in the treatment of chickenpox and smallpox (Jain 1996; Williamsons 2002).
## 2.5 Barringtonia acutangula (L.) Gaertn.
* Vernacular names: Freshwater mangrove, Mango-Pine
* Family: Lecythidaceae
Ecological Distribution and Morphological Characteristics
It is an evergreen tree which grows up to a height of 15 m and is native to South Asian and northern Australia. The leaves are petiolate, oval, thick, and smooth, and inflorescence is a drooping raceme which bears white flowers. The fruits are oval (Fig. 2.4a–c).
Fig. 2.4
(a–c) (a) Barringtonia acutangula (Indian oak) is a medicinally important tree with antidiabetic activities of leaves, roots and fruit. Aqueous extract of fruit at dose of 400 mg/kg/bw possesses significant hypoglycemic activities in streptozotocin-induced hyperglycemic rats and its activity is comparable to standard drug glibenclamide. Important phytochemicals are barringtosides, quercetin, syringic acid and vanillic acid (b) Petiolate leaves (c) Drooping raceme
Important Phytochemicals and Medicinal Value
Methyl ether extract of dried seeds revealed the presence of three monodesmosidic glucuronide saponins of barringtogenol C, i.e., barringtosides A, B, and C (Pal et al. 1994). Phytochemical analysis showed the presence of flavonols 3′, 4′-diOMe quercetin, gossypetin, 3′-OMe gossypetin, and quinones in leaves, whereas the bark showed 8-oxygenated flavonols, gossypetin, 3′-methyl ether, and quinone (Daniel and Robin 2011).
Aqueous extract of the fruit at a dose of 400 mg/kg/bw possesses significant hypoglycemic activities in streptozotocin-induced hyperglycemic rats, and its activity is comparable to the standard drug glibenclamide (Khatib and Patil 2011). In another study, ethanolic extract of leaves significantly reduced (40–50%) blood sugar level in alloxan-induced diabetes in Wistar albino rats after treatment for 21 days at dose of 250 and 500 mg/kg/bw per day (Palanivel et al. 2013; Gregory et al. 2014). Aqueous ethanolic extract of roots is reported to cause significant improvement in glucose tolerance (Babre et al. 2010).
Petroleum ether extract of stem bark showed antimicrobial activities against B. subtilis and A. niger (Rahman et al. 2005). Flavonoids like myricetin and gossypetin can modify low-density lipoproteins and increase their uptake by macrophages. Many medicinal properties of the tree, like hepatoprotective, antiviral, antimicrobial, analgesic, anticarcinogenic, antidiabetic, and antimalarial activities, are due to compounds like quercetin, syringic acid, and vanillic acid (Miller 1996; Duke 1997).
Traditional Uses
Aqueous extract of the bark is hypoglycemic and effective against pneumonia, diarrhea, and malaria, and is used as a contraceptive in China. The leaf juice is also used to treat diarrhea. The fruit is anthelmintic and an expectorant (Daniel and Robin 2011). The young leaves are edible and popular in Vietnamese cuisine. It is also recommended that consumption of tea made from any of the plant parts is helpful in the management of diabetes (Gregory et al. 2014).
## 2.6 Casuarina equisitifolia L.
* Vernacular names: Australian pine tree, She-Oak
* Family: Casuarinaceae
Ecological Distribution and Morphological Characteristics
It is native to Myanmar, Vietnam and Southeast Asia to Australia. The plant is grown as an ornamental evergreen tree with drooping branches approximately 10–50 m high. Scale-like leaves are arranged in whorls of six to eight. Catkin flowers are monoecious(Fig. 2.5).
Fig. 2.5
Bark of Casuarina equisitifolia (Australian pine tree) is antidiabetic and antihelmintic
Important Phytochemicals and Medicinal Value
Important compounds isolated from different parts are alicyclic acids (shikimic and quinic acids), lupeol, kaempferol, lupenone, quercetin, sitosterol, taraxerol (Rastogi and Mehrotra 1998), ellagic acid, trifolin, catechin, epicatechin, casuarine, gallicin, nictoflorin, and rutin.
Phytosterols from the leaves possess antibacterial, hypoglycemic, antimicrobial (Gumgumjee and Hajar 2012), antidiarrheal (Kumar 2011), and cytotoxic activities. Ethanolic extract of the bark reduced blood sugar, total cholesterol, LDL cholesterol, and VLDL cholesterol in streptozotocin-induced diabetic rats, which indicates its hypoglycemic and antihyperlipidemic effect (Sriram 2011).
Traditional Uses
The plant is used as fuel and to control soil erosion. The bark is used traditionally as an astringent (Mhaskar et al. 2000) and for its antidiabetic (Prajapati et al. 2003) and antihelmintic activities (Aher et al. 2006, 2008). An infusion of the leaves is used to treat throat infections, menorrhagia, cough, asthma, and diabetes (Sriram 2011).
## 2.7 Conocarpus lancifolius Eng.
* Vernacular name: Damas tree
* Family: Combretaceae
Ecological Distribution and Morphological Characteristics
It is a fast-growing tree that produces a large amount of biomass in summer which can be shaped into hedges. The plant is native to coastal and riverine regions of East Africa (Baroon and Razzaque 2012) but is also cultivated in South Asia. It was introduced formerly into Kuwait, where it has been exposed to many environmental stresses including salinity, oil pollution, high temperature, and extreme climatic conditions.
The leaves are glossy, with trichomes, and lens shaped (Redha et al. 2011). Flowering takes place in early spring. The flowers are pentamerous and floral heads are globose and creamy white (Fig. 2.6a–f).
Fig. 2.6
(a–f) Methanolic extract of Conocarpus lancifolius (damas tree) is known to cause antidiabetic activity in alloxan-induced diabetic rabbits. Ethanolic extract of bark reduced blood sugar level in streptozotocin-induced diabetic rats (a) Young plant (b) Mature tree (c) Young flower (d, e) Developing flowers (f) Fruits
Important Phytochemicals and Medicinal Value
Analysis of the leaf extracts indicated the presence of phenolics, terpenoids, alkaloids, and fatty acids. The plant possesses prominent antidiabetic and cytotoxic activity against the MRC-5 cancer cell line and antiprotozoal and antibacterial activities (Al-Taweel et al. 2016). Total methanol-soluble extract is reported to have antibacterial and low antifungal activities (Saad et al. 2014).
Methanolic extract of the plant showed antidiabetic activity in alloxan-induced diabetic rabbits at a dose of 200 mg/kg/bw due to the presence of saponins (Saadullah et al. 2014).
Traditional Uses
The plant is used as a source of fuel and fodder (Sheikh 1993). Leaf extracts of C. lancifolius showed allelopathic effects in corn and bean seeds and plants.
## 2.8 Eriobotrya japonica Lindl.
* Vernacular names: Japanese plum, Loquat
* Family: Rosaceae
Ecological Distribution and Morphological Characteristics
E. japonica is an evergreen shrub or tree which is native to China. Its cultivation is known to be more than 2000 years old since the Han dynasty (100 BC) (Chen et al. 2008). It is widely planted as an edible fruit tree in China, Japan, Pakistan, and the Mediterranean region. Leaves are alternate, tough, leathery with serrate margins, and arranged in the form of whorls (Fig. 2.7a–d). Flowering starts in autumn.
Fig. 2.7
(a–d) Ethanolic extract of seeds Eriobotrya japonica (loquat) is reported to improve the glucose tolerance in mice and ethanolic extract of fruit reduced blood glucose level at dose of 200 mg/kg/bw in alloxan-induced diabetic rats (a) Leaves bearing young fruits (b) Flowers (c, d) Young fruits
Important Phytochemicals and Medicinal Value
Important metabolites include many alcoholic groups like camphene, camphor, cymene, farnesol, hexenol, geraniol, linalool, trans-linalool oxide, nerolidol, nerol, myrcene, and pinene (Zheng et al. 1998). Many phenolic compounds are found in the fruit, which include 4-caffeoylquinic acid, 5-caffeoylquinic acid, hydroxybenzoic acid, 5-p-feruloylquinic acid, protocatechuic acid, epicatechin, o-coumaric acid, ferulic acid, and p-coumaric acid (Lim 2012). Amygdalin is found in the leaf and kernel in significant amounts (Zhuang 2002).
Oleanolic and ursolic acids are extracted from flower extract (Cheng et al. 2001). Other isolated sesquiterpenes include sesquiterpene, sohumbertiol as aglycones, and branched oligosaccharidic chains made up of -L-rhamnopyranosyl and -D-glucopyranosyl units (Chen et al. 2008).
Antidiabetic potential of the plant is attributed to the presence of compounds like flavonoids, tannins, corosolic acid, 3-epicorosolic acid methyl ester, 2-α hydroxy-3-oxo urs 12-en-28-oic acid, tormentic acid methyl ester, and ursolic acid isolated from leaf extract (Saravanamuttu and Sudarsanam 2012). Ethanolic extract of the seeds is reported not only to suppress the rise of blood glucose for 4 months but also to improve glucose tolerance in mice (Tanaka et al. 2008). Further, 50% ethanolic extract of the fruit reduced blood glucose level at a dose of 200 mg/kg/bw and improved lipid profile in alloxan-induced diabetic rats (Shafi and Tabasum 2013).
Antidiabetic activity of the leaves is reported due to the presence of sesquiterpene glycosides which have hypoglycemic effects (Chen et al. 2008) and due to their potential to inhibit 11ß-HSD1 and 11ß-HSD2 (Christel et al. 2009). Leaf extract is also used as an oral hypoglycemic agent for the treatment of diabetes in Southeast Asia, China, Korea, and Japan (Ceylan-Isik et al. 2008). The presence of cinchonain Ib isolated from the plant is reported to increase insulin secretion in rats (Qa'dan et al. 2009).
Further, the plant is known to possess antioxidant (Hamada et al. 2004; Huang et al. 2006), cytotoxic (Ito et al. 2000), hepatoprotective (Nishioka et al. 2002), anti-inflammatory (Ju et al. 2003), and antiaging activities (Muramoto et al. 2011).
Traditional Uses
Almost all parts of the plant are used in traditional Chinese medicine. The plant is used to relieve cough and to treat skin problems in Japanese folk medicine (Namba 1994). The dried leaves are used to cure diarrhea and depression. Edible fruit is rich in sugar, high in vitamin C and low in fat (Lim 2012). Fragrant flowers contain essential oils which are used in cosmetics and for treating cough and common cold.
## 2.9 Euphorbia pulcherrima Willd.
* Vernacular names: Spurge, Poinsettia
* Family: Euphorbiaceae
Ecological Distribution and Morphological Characteristics
Euphorbia is a large genus of the family Euphorbiaceae and consists of over 2000 species. Poinsettia is a shrub or small tree native to Mexico and South America. The leaves are dentate, and the bracts are red and give an appearance of petals of flowers due to photoperiodism. The flowers are known as cyathia (Fig. 2.8a–c).
Fig. 2.8
(a–c) Euphorbia pulcherrima, poinsettia is an important ornamental and medicinal tree due to anticancer, antidiabetic and anti-eczemic properties (a) Young tree (b) Pink bracts bearing cyathia (c) Cyathia
Important Phytochemicals and Medicinal Value
Compounds of economic value comprise alkaloids, steroids, terpenoids, and saponins. Methanolic extract and ethyl acetate fraction showed antibacterial effect against E. coli, S. aureus, and S. typhi (Sharif et al. 2015).
It is an important medicinal plant due to anticancer, antidiabetic, antieczema, anti-inflammatory, and antitumor properties (Luo and Wang 2006; Valente et al. 2003; Eberle et al. 1999; Ferreira et al. 2006).
Traditional Uses
The bracts are used for ornamental purpose and for obtaining dyes. Decoction of flowers is used as a galactagogue. The plant is used traditionally to treat gastrointestinal problems in African ethnopharmacopeia (Yakubu and Mukhtar 2011), for malaria, asthma, and eczema.
## 2.10 Jasminum sambac L.
* Vernacular name: Arabian jasmine
* Family Oleaceae
Ecological Distribution and Morphological Characteristics
Jasmine is a large genus and comprises nearly 200 species. J. sambac grows as a small shrub and is native to Bhutan, Myanmar, Pakistan, India, and Sri Lanka. It is a popular ornamental plant and is cultivated for its fragrant flowers in many tropical and subtropical parts of the world. The leaves are broadly ovate or elliptic and opposite. The flowers are fragrant, white, and solitary (Fig. 2.9).
Fig. 2.9
Aqueous extract of leaves of Jasminum sambac (Arabic Jasmine) caused reduction in blood glucose at dose of 300 mg/kg/bw in alloxan-induced diabetic rat. Ethanolic extract of flowers also showed antidiabetic activity in alloxan and streptozotocin-induced rats
Important Phytochemicals and Medicinal Value
Important phytochemicals in the leaves include alkaloids, saponins, flavonoids, terpenoids, and iridoid glycosides like sambacin, jasminine, sambacoside A, sambacolingoside, and flavonoids like quercetin, isoquercetin, rutin, kaempferol, luteolin, and secoiridoids (Yu et al. 1995; Shen et al. 2000). The flowers contain 3-hexenol, 2-vinylpyridine, myrcene, linalool, geranyl linalool, alpha terpenol, benzyl alcohol, nerolidol, phytol, isophytol, cis-jasmone, 8,9-dihydrojasminine, and 9-deoxyjasminigenin (Inagaki et al. 1995).
The plant is used as an emmenagogue and analgesic as well as antidiabetic (Jensena et al. 2002). The leaves are used to control blood sugar level. Aqueous extract of leaves showed reduction in blood glucose at a dose of 300 mg/kg/bw in alloxan-induced diabetic rats for 21 days (Upaganlawar et al. 2009). The flowers also possess antidiabetic activity as blood glucose level was significantly reduced in alloxan- and streptozotocin-induced rats treated with ethanolic extract (Rambabu and Patnaik 2014). Many species of jasmine are also being used in research on its anticarcinogenic properties (Nadkarni and Basu 1996; Khare 2004).
Traditional Uses
The flowers are used in making jasmine tea and are commonly used in Buddhist ceremonies (Roberts 2000). They are considered to be antidepressant and a relaxing herb by aromatherapists. Decoction of dried flowers is used in conjunctivitis, dermatitis, opthalmopathy, leprosy, and dementia, as well as for menstrual flow (Rath et al. 2008; Joy and Raja 2008; Talib and Mahasneh 2010). The aroma of jasmine flower is antidepressant, and essential oils are widely used in aromatherapy and in making perfumes. Jasmine oil is used in vapor therapy and can be used to relieve stress. Flowers are also used to make garlands.
## 2.11 Kigelia pinnata Jacq. (Syn: Bignonia africana)
* Vernacular name: Sausage tree
* Family: Bignoniaceae
Ecological Distribution and Morphological Characteristics
K. pinnata is popularly called the sausage tree due to its huge fruit which is almost 4 kg in weight. It is native to East Africa.
The leaves are ovate to oblong in shape, alternate, and pinnately compound. It is a spreading tree which bears long, pendulous racemes of bell-shaped, mottled dark flowers which appear in spring or summer. The fruits are long and woody, sausage-like in appearance with long cord-like stalks (Fig. 2.10a–e).
Fig. 2.10
(a–e) Kigelia pinnata (sausage tree) has many medicinal properties due to the presence of many phytochemicals like iridoids, flavonoids, and naphthoquinones. Methanolic extract of flowers are reported to have significant antidiabetic and hypolipidemic activity in streptozotocin-induced diabetic Wistar rats at dose of 500 mg/kg/bw for 21 days (a) Tree (b) Bark (c) Leaves (d) Branch bearing leaves (e) Pods
Important Phytochemicals and Medicinal Value
Phytochemical screening studies revealed the presence of many flavonoids, iridoids, glycosides, steroids, naphthoquinones, pinnatal, and isopinnatal (Saini et al. 2009). Some of these phytoconstituents may be responsible for anthelmintic activity. Phytochemicals extracted from the roots include kigelinone, isopinnatal, dehydro-a-lapachone, lapachol, coumaric acid, and ferulic acid. The fruits contain many antibacterial and antifungal compounds, such as kigelinone and caffeic acid (Binutu et al. 1996).
The plant has many medicinal properties due to the presence of numerous iridoids, flavonoids, and naphthoquinones (Gormann et al. 2004; Asekun et al. 2011). Methanolic extract of flowers is reported to have significant antidiabetic and hypolipidemic activity in streptozotocin-induced diabetic Wistar rats at a dose of 500 mg/kg/bw for 21 days (Kumar et al. 2012). Extract from the leaves is also reported to have antidiabetic activity, which is compared with acarbose standard (Dhriti et al. 2014). Oral administration of leaf extract showed significant reduction in blood glucose level at a dose of 200 mg/kg/bw in alloxan-induced diabetic rats (Raju and Hemamilini 2012).
The fruit and stem bark are known to possess antibacterial activity (Grace et al. 2002). Root extract is known to have antimalarial activity due to the presence of laphachol (Saini et al. 2009). The plant is also known to have many antioxidant (Olalye and Rocha 2008) and antidiabetic activities (Nyarko, et al. 2005). Furthermore, ethanolic stem bark extract of the plant acts as a stimulant on the CNS(central nervous system) and is being explored for therapeutic advantage to treat sedation and dizziness (Owolabi et al. 2008).
Traditional Uses
The fruit is reported to be purgative, is known to induce abortion, and is also used in criminal poisoning (Quattrocchi 2012). The plant is used traditionally for treatment of malaria, asthma, cancer, gynecological disorders, renal ailments, epilepsy, and rheumatism, and as a detoxifier. The fruit is used as therapy for ulcers in Africa. It is also pickled in jam and used as an appetizer.
## 2.12 Lagerstroemia indica rosea
* Vernacular name: Crepe myrtle
* Family: Lythraceae
Ecological Distribution and Morphological Characteristics
Approximately 50 species of Lagerstroemia are known in the family Lythraceae. L. indica is native to China and has a long history of cultivation of 1800 years (Zhang 1991). It is a deciduous tree and raised as an ornamental plant in many parts of Southeast Asia including the Philippines, Vietnam, Malaysia, and southern China. The leaves are small and oval shaped. The flowers are formed in panicles and are pinkish purple in color (Fig. 2.11a–f).
Fig. 2.11
(a–f) Lajerstroemia indica (crape myrtle) is known to have antidiabetic activity. Main antidiabetic compound is gallotanin penta-o-galloyl-glucopyranose (PGG) which is isolated from leaves (a) Tree (b) Tree in autumn (c) Leaves in autumn appear red (d) Pink panicle (e) Flowers (f) Young fruits (g) Ripen fruits
Important Phytochemicals and Medicinal Value
Phytochemical screening showed the presence of alkaloids, cardiac glycosides, anthraquinones, saponins, essential oils, tannins, and flavonoids (Wani et al. 2012; Ghannadi et al. 2012), which makes the plant important for commercial purposes. Many microsatellite markers are being developed from crepe myrtle to analyze genetic diversity within Lagerstroemia cultivars and related species (Liu et al. 2013).
Hypoglycemic effects are due to crosolic acid, ellagitannins, and galactotannins. Crosolic acid is reported to reduce blood glucose level within 60 min and possesses antihyperlipidemic and antioxidant activities (Miura et al. 2012). Tannins like gallotanin penta-o-galloyl-glucopyranose (PGG) isolated from leaves are reported to have antidiabetic activity (Saravanamuttu and Sudarsanam 2012). Ellagitannins like lagerstroemin are also considered to be effective against diabetes (Klein et al. 2007). A significant decrease (16.6%) in blood glucose level is observed in individuals with fasting glucose level greater than 110 mg/dL (Miura et al. 2012).
Traditional Uses
The leaves are traditionally consumed in the Philippines for treatment of diabetes and for kidney diseases. The insulin-like hypoglycemic effect of the leaves was published as early as 1940 (Garcia 1940). It is also being used in health-promoting tea products in Japan, the Philippines, South Korea, and the United States.
## 2.13 Hibiscus rosa sinensis L.
* Vernacular names: Chinese rose, Shoe flower
* Family: Malvaceae
Ecological Distribution and Morphological Characteristics
It is an evergreen shrub which is native to East Asia. The leaves are glossy and solitary, the margins are serrate, and the flowers are complete, solitary, and red in color. The stamens are fused in the form of a cylinder, which surrounds the style (Fig.2.12a–c). Many cultivars of the plant in yellow, orange, and pink shades are being used in floriculture.
Fig. 2.12
(a–c) Ethanolic extract of flowers of Hibiscus rosa sinensis (shoe flower) reduced blood glucose level at dose of 250 mg/kg/bw and at 500 mg/kg/bw in alloxan-induced diabetic rats and also reduced glucose level in streptozotocin-induced rats (a, b) Bush bearing red flowers (c) Flower
Important Phytochemicals and Medicinal Value
The leaves and stems comprise important phytochemicals like β-sitosterol, stigmasterol, and taraxeryl acetate. The flowers are reported to have many glucosides and flavonoids like cyanidin diglucoside, thiamine, riboflavin, niacin, and ascorbic acid. In addition, other glucosides extracted from yellow flowers include cyanidin- 3,5-diglucoside, 3,7-diglucoside, cyanidin-3-sophoroside-5- glucoside, and quercetin-3-diglucoside.
Antidiabetic activity of Hibiscus is considered to be due to secondary metabolites like alkaloids, flavonoids, saponins, glycosides, and polyphenols as many alkaloids can regenerate β-cells of the pancreas, and polyphenols can decrease blood glucose level.
The antidiabetic activity of the plant in hyperglycemic rats is also reported (Sachdewa et al. 2001; Sachdeva and Khemani 2003a, b). Ethanolic extract of the flowers reduced blood glucose level at a dose of 250 mg/kg/bw and at 500 mg/kg/bw in alloxan-induced diabetic rats (Venkatesh et al. 2008) and also reduced glucose level in streptozotocin-induced rats (Sachdewa and Khemani 2003a, b). Aqueous extract of the flowers exhibited significant hypoglycemic and hypolipidemic activities at a dose of 500 mg/kg/day in streptozotocin-induced diabetic rats (Bhaskar and Vidhya 2012). Oral administration of root extract at a dose of 500 mg/kg/bw also caused significant decrease in blood glucose and plasma lipids after 15 days in alloxan-induced diabetic rats, indicating the role of the plant in treating diabetic dyslipidemia and related complications (Kumar et al. 2013). Ethanolic extract of the leaves also reduced blood glucose level and improved lipid profile in alloxan-induced diabetic dyslipidemia in rats after four weeks (Mamun et al. 2013). Insulin-secreting activity of leaf extract is reported in diabetic Wistar rats (Vimala et al. 2008).
The plant is also reported to have anticancer potential in a study conducted on mice exposed to ultraviolet radiation (Sharma et al. 2004). Current research is being targeted to explore its antiaging potential and use in the treatment of scopolamine-induced amnesia, which suggests its further role against cognitive disorders (Nade et al. 2011).
Traditional Uses
It is used in traditional medicine to induce abortion, to ease menstrual cramps, to facilitate childbirth, and to relieve fever, headache, and inflammation (Arullappan et al. 2009). It is considered a laxative and is used for treatment of stomach ulcer, as an aphrodisiac and emmenagogue, and as a reliable oral contraceptive. The petals are used as a remedy for treating chronic constipation. It is also used as a tonic for hair health and in psychiatric ailments. In order to induce periods, crushed petals are consumed early in the morning before the expected date of the period.
## 2.14 Morus alba L.
* Vernacular name: White mulberry
* Family: Moraceae
Ecological Distribution and Morphological Characteristics
M. alba is a medium-sized deciduous tree that can reach up to 3–10 m in height. It is native to China and Pakistan.
Long leaves are petiolate, cordate, and acuminate near the tips. The bark is gray and thick, with many irregular longitudinal cracks. The plant produces ovate buds in winter which are reddish brown and bears bud scales which are covered with hairs resembling those on the twig surface. The flowers are unisexual catkins on the same or different plants (Fig.2.13). Green unisexual catkins appear with leaves in April to May and bloom axillarily.
Fig. 2.13
Morus alba (white mulberry) is antidiabetic and anticancer tree. Leaf extract showed antidiabetic effects at a dose of 600 mg/kg/bw in streptozotocin-induced Wistar rats
Important Phytochemicals and Medicinal Value
The plant is a very good source of ascorbic acid and carotene. It contains many natural antioxidants like vitamin B1, folic acid, folinic acid, isoquercetin, quercetin, tannins, flavonoids, and saponins. The leaves are rich in lupeol, sterols, bioflavonoids (rutin, moracetin, quercetin-3-triglucoside, and isoquercitrin), coumarins, volatile oil, alkaloids, amino acids, and organic acids.
The leaves contain rutin, quercetin, and apigenin as bioactive constituents (Doi et al. 2001). One of the major constituents is 1-deoxynojirimycin (Chu et al. 2006). Many flavones were isolated from the root bark as active principles. Other important biochemical compounds isolated from the plant include moranoline, albafuran, albanol, morusin, kuwanol, and calystegin. Mulberroside F isolated from mulberry leaves is used as a skin-whitening agent (Sang et al. 2002).
Extract of the plant is reported to reduce blood glucose level through regeneration of β-cells (Jamshid and Prakash 2013). Methanolic and aqueous extract of leaves showed significant antidiabetic activity, i.e., 18.88% and 9.1% respectively in streptozotocin-induced diabetic rats at a dose of 200 mg/kg/bw after 15 days (Chaurasia et al. 2011). Leaf extract of mulberry is also reported to have antidiabetic effects at a dose of 600 mg/kg/bw in streptozotocin-induced Wistar rats (Mohammadi and Naik 2008). Antidiabetic activity of mulberry leaves is due to the presence of compounds like trigonelline bases, moran A (Burman 1985), and moranoline (Yoshikuni 1988).
The fruits have radical-scavenging activity, which indicates their role as important antioxidants (Chon et al. 2009; Shahid et al. 2012). The leaves have adaptogenic and anxiolytic activities (Yadav et al. 2008), which indicates its possible role as an antistress agent (Vandana et al. 2009) and in the management of psychiatric disorders (Adhikrao and Vandana 2008).
Traditional Uses
It is a source of silkworm food, fodder, and plywood furniture, and it is one of the ideal trees for reforestation projects. Major use of the plant is as an antidiabetic, immunomodulatory, antimicrobial, antioxidant, and anticancer agent (Chon et al. 2009). Leaves are used as diuretic, expectorant, and antidiabetic in traditional Chinese medicine (Chen et al. 1995).
## 2.15 Murraya koenigii L.
* Vernacular name Curry tree
* Family: Rutaceae
Ecological Distribution and Morphological Characteristics
M. koenigii is native to India, Sri Lanka, and other South Asian countries.
The plant grows as a deciduous shrub or tree having a short trunk with a dense shady crown. The leaves are bipinnately compound, with 11–25 leaflets which are arranged alternately on rachis. The flowers are small, aromatic, white, bisexual, and funnel shaped, borne on terminal cyme, comprising 60–90 flowers. The fruit is ovoid, subglobose, and purplish black in color upon maturity (Figs. 2.14a–e and 2.15).
Fig. 2.14
(a–e) Leaves and roots of Murraya koenigii (curry tree) are known to control blood cholesterol and possess antidiabetic and memory-enhancing activities due to mahanimbine and koenigine alkaloids. Antidiabetic activity of leaf extract is well reported in alloxan-induced and streptozotocin-induced diabetic rats (a) Tree (b) Leaflets on rachis (c) Terminal raceme of white flowers (d) Young globose fruits (e) Ripening of fruits
Fig. 2.15
(a–b) Leaves of Opuntia (prickly pear) are used to treat type 2 diabetes and hyperlipidemias
Important Phytochemicals and Medicinal Value
Phytochemicals such as alkaloids, sterols, tannins, volatile oils, saponins, anthroquinone glycosides, and flavanoids are reported from different parts of the plant (Handral et al. 2010). The leaves are aromatic and contain carotene, nicotinic acid, and vitamin C. The leaves contain high amounts of oxalic acid, crystalline glycosides, carbazole alkaloids, koenigin, and resin. The fresh leaves contain volatile oil which is rich in vitamin A, calcium, girinimbin, iso-mahanimbin, koenine, koenigine, koenidine, and koenimbine.
Other compounds isolated from leaves include mahanimbicine, bicyclomahanimbicine, phebalosin, coumarine, murrayastine, murrayaline, and pypayafolinecarbazole alkaloids. The bark comprises carbazole alkaloids as murrayacine, murrayazolidine, murrayazoline, mahanimbine, girinimbine, koenioline, and xynthyletin.
The roots and leaves are antihelmintic, analgesic, and effective against leucoderma and blood disorders. They are also known to control blood cholesterol, are antidiabetic (Saravanamuttu and Sudarsanam 2012; Quattrocchi 2012), and have memory-enhancing activities (Xie et al. 2006; Tembhurne and Sakarkar 2010). The leaves have antioxidant activity due to mahanimbine and koenigine alkaloids (Rao et al. 2006).
There has been significant research conducted on the antidiabetic potential of Murrya leaves which reported that feeding different doses of leaves to alloxan-induced diabetic rats can control mild to moderate diabetes (Yadav et al. 2002; Sarji et al. 2016). Oral administration of chloroform extract of Murraya leaves in alloxan-induced diabetic albino rats caused significant reduction in blood glucose level at doses of 250 mg/kg/bw and 500 mg/kg/bw possibly by increasing insulin secretion and enhancement of glycogenesis, decreasing oxidative stress, and preserving pancreatic cell integrity (Vijayanand 2015). Oral administration of aqueous extract of leaves showed 75% reduction in urine sugar in normal and streptozotocin-induced diabetic rats (Kesari et al. 2007). Further, oral administration of aqueous extract (600 mg/kg/bw) and methanolic extract (200 mg/kg/bw) of Murraya leaves in alloxan-induced diabetic rats caused reduction in blood glucose level and plasma insulin level (Vinuthan et al. 2004). Aqueous extract of roots also showed reduction in blood glucose level of almost 57.6% at a dose of 400 mg/kg/bw in alloxan-induced diabetic rats (Singh et al. 2012).
Traditional Uses
The leaves are the most important part of the plant due to their medicinal value. They are used as a herb and ingredient in Asian cuisine to promote appetite and digestion. Tree branches are used for strengthening of teeth and gums. Recently, it has been reported that formulation of a cream comprising essential oil of the leaf is found to have sun protection factor (Patil et al. 2010).
## 2.16 Opuntia ficus-indica L.
* Vernacular names: Cactus pear, Prickly pear
* Family Cactaceae
Ecological Distribution and Morphological Characteristics
Cacti are unique ornamental plants and are also useful in health and medicine. The cactus family comprises more than 350 families, and the plants are commonly known as desert plants, spiny plants, or succulents due to their ability to store water. Their leaves are reduced to form thorns or modified spines to reduce the rate of evaporation. The highly reduced stem of cacti is known as areoles, which form tubular flowers. These adaptations help cacti to survive in extreme weather. The joints of prickly pear swell when water is in abundance; however, they shrink in periods of drought, releasing excess water.
Important Phytochemicals and Medicinal Value
Cactus plants have antitumor, antiulcer, and antirheumatic properties due to many phytochemicals in their leaves and flowers. Flavonoids are widely present in pear cactus. The pads (stem joints) of prickly pear are rich in vitamins, minerals, amino acids, potassium, magnesium, calcium, and iron. When the pads of pear cactus are sliced, they release sticky mucilage, which contains soluble dietary fibers and many polysaccharides. Prickly pear cacti are a rich source of vitamin C, carotenoids, thiamin, riboflavin, and niacin (Dominguez-Lopez 1995; Tous and Ferguson 1996).
Traditional Uses
The fruit of prickly pear is pleasant in taste, with small black seeds. Tea is also made commercially from cactus flowers and sold in tea bags. Prickly pear is one of those exceptional plants which are vegetables, fruits, and flowers all in one and work both as food and medicine. Prickly pear is being used to treat type 2 diabetes and hyperlipidemias. It is under investigation for antiviral diseases and as a potential cancer prevention agent (Knishinsky 2004). The fruit of opuntia is reported to have antidiabetic activity (Saravanamuttu and Sudarsanam 2012).
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_3
# 3. Trees with Anticancer Activities
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
Many trees are known to possess important anticancer compounds which include flavonoids and other phenolic compounds, glucosinolates, phytosterols, phytoestrogens, tannins, and protease inhibitors which interrupt with the mitotic activities of abnormal cells. More than 500 compounds belonging to 25 different classes are identified as anticarcinogens. This chapter describes anticancer properties of plants like Bauhinia variegata, Butea monosperma, Callistemon citrinus, Carica papaya, Chukrasia velutina, Cycas revoluta, Dillenia indica, Jasminum officinale, Plumeria rubra, Plumeria obtusa, Magnolia grandiflora, Schleichera oleosa, Sapium sebiferum, and Thuja occidentalis.
## 3.1 Introduction
Cancer is an abnormal and uncontrolled cell division which results in the formation of tumors. It may be caused due to aberration in somatic cells, genetic disorders which may include atherosclerosis, cardiovascular diseases, exposure to carcinogens (cancer-causing agents), or due to some degenerative disorders. Environmental pollution is also responsible for causing cancer, particularly skin cancer.
Currently, many plants are being evaluated for their bioactive compounds, which have tumor-inhibiting properties. These compounds which have anticarcinogenic potential are mostly flavonoids and other phenolic compounds, glucosinolates, phytosterols, phytoestrogens, tannins, and protease inhibitors which are produced in different parts of plants. So far, more than 500 compounds belonging to 25 different classes are identified as anticarcinogens (Fig. 3.1a–d).
Fig. 3.1
(a) Celtis spp. exhibit cytotoxic activities against many human cell lines due to their flavonoids which act as strong antioxidants. (b) Delonix regia (royal poinciana, flame of forest) is a medicinally important tree with anticancer activities against hepatocellular carcinoma due to flavonoid contents in leaves. (c) Mimusops elengi (bullet wood tree) is a plant of the future with potential to be used against gynecologic cancer and possible anti-HIV activities due to mimusopic acid (a triterpene isolated from plant seeds). (d) Anticancer activity of the fruits of Citrus spp. is associated with β-carotenoids and retinoic acids which are anticancer against breast, lung, and mouth cancer
An anticarcinogen prevents cancer by either inactivating carcinogen or its mutagenic reaction with DNA. Major steps involved in anticarcinogenesis involve enzymatic inactivation, prevention of formation of active species usually in liver, scavenging of carcinogens, and inhibition of reactive oxygen species which can cause mutation. One of the major challenges in curing cancer involves tissue nonspecificity of anticancer compounds which can also damage normal tissues; however, research is being conducted to isolate tissue-specific natural anticancer compounds from plants. Various anticarcinogens from plants are being used clinically which include betulinic acids, camptothecin, combretastatin, etoposides from epipodophyllotoxin, flavopiridol, irinotecan, silvestrol, taxols, topotecan, vinblastine, and vincristine (Fig. 3.2a–c). Phenolic compounds like ellagic acid are reported to inhibit tumor cells proliferation and break DNA binding to carcinogens, and are found naturally in grapes, nuts, berries, green tea, and pomegranates.
Fig. 3.2
(a) Erythrina suberosa (coral tree). Many Erythrina spp. possess anticancer phytochemicals like ferulates, phenolates, stigmasterol, sitosterols, and campesterol which are effective against leukemia. (b) Antileukemiatic potential of leaves, bark, and fruit of Dillenia indica (elephant apple) is due to flavonoids and terpenoids. (c) Lapachol isolated from Tectona grandis (teak tree) is a compound with antitumor activities
## 3.2 Important Trees with Anticancer Activities
Many plants with anticancer potential are ornamental, ecologically important, and possess other medicinal properties as well, but in this section they are mainly characterized on the basis of their anticancer properties. Anticancer role and medicinal value of these trees along with their ecological and morphological characteristics are discussed in the next section.
## 3.3 Bauhinia variegata L.
* Vernacular name: Orchid tree
* Family: Leguminosae (Fabaceae)
* Subfamily: Caesalpinioideae
Ecological Distribution and Morphological Characteristics
Bauhinia is an important genus of family Leguminosae and it has more than 250 species. It is native to Southeast Asia and found in the sub-Himalayan tracts. It grows as a medium-sized deciduous tree in tropical and subtropical climate. Its leaves are petiolate and bilobed away from the base. Flowers are pinkish-red in color. Fruit is a pod with many seeds (Fig. 3.3a–f).
Fig. 3.3
(a–f) Bauhinia variegata (orchid tree) possesses many medicinal activities like anticancer, antimicrobial, anti-inflammatory, nephroprotective, and hepatoprotective due to the presence of many phenanthraquinones, flavonoids, tannins, saponins, steroids, and cardiac glycosides. (a) Mature tree (b) Bilobed leaf (c) Edible buds (d) Orchid-like flower (e) Pod emerging out of the flower (f) Pods
Important Phytochemicals and Medicinal Value
The plant is a rich source of terpenoids, phenanthraquinone flavonoids, tannins, saponins, steroids, and cardiac glycosides as revealed through phytochemical screening. The stem of B. variegata comprises many sitosterols, lupeol, naringenin, and rhamnosides. The bark contains compounds like quercitrocides, rutoside, myricetol, and kaempferol glycosides (Zhao et al. 2005). Many essential amino acids like lysine, threonine, valine, methionine, isoleucine, leucine, and phenylalanine are found in the leaf extracts. Leaves also possess many alkaloids, phenolics, saponins, β-sitosterol, kaempferol-3-glucoside, rutin, quercetin, apigenin-7-O-glucoside, amides, vitamin C, fibers, calcium, and phosphorus (Dhale 2011). Anti-inflammatory activity of the plant is attributed to many flavonoids, kaempferols, and cytokines present in the leaves. The seeds of these plants are also a rich source of myristic, palmitic, stearic, oleic, and linoleic acids.
B. variegata is reported to have a wide range of medicinal activities which include anticancer, antimicrobial, anti-inflammatory, nephroprotective, hepatoprotective, antiulcer, immunomodulating, molluscicidal, and wound-healing effects (Yadava and Reddy 2002).
The stem possesses antibacterial and anticancer properties (Lim 2014; Sharma and Kumar 2012). In vitro explants are reported to inhibit 60% of anticancer activity due to the presence of compounds like alkaloids, steroids, triterpenoids, and flavonoids (Kanak and Anita 2012). Antitumor activity of ethanolic extract is reported against Dalton's ascitic lymphoma (DAL) in Swiss albino mice and in N-nitrosodiethylamine-induced liver tumor in rats and human cancer cell lines (Rajkapoor et al. 2003a, b). Its flowers are also known to have chemopreventive potential on 7,12-dimethylbenz[a]anthracene (DMBA)-induced skin carcinogenesis in Swiss albino mice.
Leaves are used traditionally to cure jaundice, liver, and thyroid problems. The bark is effective against stomach disorders due to its anti-inflammatory and antidiabetic properties (Saravanamuttu and Sudarsanam 2012).
Traditional Uses
Dried flowers are edible and can be used to make energy drinks. It is considered as a sacred tree to Buddha and the red flowers of the plant are compared with a woman's beauty due to their pinkish-red color. Its bark is also used in treating leprosy.
## 3.4 Callistemon citrinus L.
* Vernacular name: Crimson Bottlebrush
* Family: Myrtaceae
Ecological Distribution and Morphological Characteristics
Genus Callistemon has over 30 species of great medicinal importance. It is native to Australia but also grown as an ornamental and medicinal shrub in many tropical and subtropical countries. It is commonly called bottlebrush due to the cylindrical shape of the flowers. Its leaves are narrow and bark is white papery. Red and yellow stamens are the most conspicuous parts of the flower. Flower color may vary from yellow, green, orange to red (Fig. 3.4a–f). Flowering starts in October and lasts till December.
Fig. 3.4
(a, b) Callistemon spp. (bottlebrush) are medicinally important and possess many antioxidants which are antimicrobial and anticancer. (c) Leaves (d, e) Developing flower (f) Withered flower
Important Phytochemicals and Medicinal Value
Phytochemical studies showed the presence of many flavonoids, steroids, saponins, and terpenoids (Shinde et al. 2012). Essential oils of leaves and flowers are known to have anticancer potential against human lung carcinoma, colon, and cervical cancer cells (Kumar et al. 2015). Ethanolic extract of stem possesses free radical scavenging activity (Kim et al. 2009). Its bark is also reported to have cytotoxic effects against A549 cells (Mahapatra 2013).
The extracts from C. citrinus exhibited antimicrobial activity against Bacillus subtilis, Bacillus pumilus, and Escherichia coli (Krishna et al. 2012). Leaves and flowers are reported to have anthelmintic activity due to the presence of 1,8-cineole and alpha-terpineol. Ethanolic and methanolic extracts of leaves possess antimicrobial and hepatoprotective properties (Seyydnejad et al. 2010; Firoz et al. 2011). Leaf oils also possess antinociceptive and anti-inflammatory activities. The plant is also being considered to be environment friendly due to the presence of compound 1,8-cineole with the potential to replace ozone depleting industrial solvents (Babu and Singh 2009; Kumar et al. 2015).
Traditional Uses
C. citrinus is used as weed control and also serves as a bioindicator for environmental management (Burchett et al. 2002). The plant is known to possess many antistaphylococcal, antithrombotic, and antioxidant activities (Williams 2013).
## 3.5 Carica papaya L.
* Vernacular names: Papaya, Papaw
* Family: Caricaceae
Ecological Distribution and Morphological Characteristics
Carica papaya is commonly known as papaya. It is native to southern Mexico and Costa Rica but grown widely in many tropical and subtropical countries. It has been introduced in many countries and widely grown due to its edible fruit. Four species of papaya are an important source of breeding for developing resistance against viruses, i.e., C. cauliflora Jacq, C. pubescens Lenne & K. Koch, C. quercifolia Benth. & Hook.f. ex Hieron, and C. papaya. Red and yellow papaya are commonly grown in Australia; however, Brazil and India are major papaya producing countries. Genetically engineered cultivars of papaya are viral-resistant cultivars incorporated with viral DNA and have been introduced by the University of Hawaii.
Papaya is an evergreen tree which grows up to a height of 10 m. Leaves are large, palmately lobed, and spirally arranged at the top of the trunk. Flowers are white, fragrant, and both monoecious and dioecious. Female flowers are large and solitary. Male flowers only pollinate but never produce fruits (Fig. 3.5a–e). Fruit is an elongated globose berry, fleshy from inside having a large cavity which bears black seeds surrounded by transparent aril. Hermaphrodite flowers can self-pollinate and be used in commercial papaya orchards.
Fig. 3.5
(a–e) Fruit of Carica papaya (papaya) inhibits the growth of tumor cells in different types of cancer cell lines. Important phytochemicals which contribute to antimicrobial, antidengue, antidiabetic, and antimalarial activities include carpaine, pseudocarpaine, dehydrocarpaine, with choline, carposide, and vitamins C and E. (a) Papaya tree (b) Palmately lobed leaf (c) Male flower (d) Female flower (e) Developing fruit
Important Phytochemicals and Medicinal Value
Leaves and unripe fruit contain a white milky fluid which contains many phytochemicals of medicinal value, most noticeable is papain protein. Leaves comprise many alkaloids including carpaine, pseudocarpaine, dehydrocarpaine I and II, with choline, carposide, and vitamins C and E. Fruits are edible and nutritious due to proteins, fibers, fats, vitamin C, thiamin, riboflavin, niacin, carotene, calcium, phosphorus, iron, and citric and malic acids. Both young and ripe papaya are sources of minerals and nutrients which include sodium, potassium, magnesium, phosphorus, calcium, iron, copper, zinc, manganese, selenium, and vitamins C, B6, B12, A, K, E, thiamine, riboflavin, niacin, folate, and pantothenic acid with carotenes, cryptoxanthin, lutein, and zeaxanthin (Gunde and Amnerkar 2016). Due to the presence of vitamins A and C, papaya fruit is a good source for maintaining eye health and can also prevent blindness in young children. Volatile compounds like linalool, benzyl isothiocyanate, alkaloids like carpaine, and glucosides are also present in the fruit. Seeds are a source of fatty acids, fibers, proteins, papaya oil, carpaine, benzyl isothiocyanate, β-sitosterol, and caricin. Plant latex comprises many proteolytic enzymes, papain, chemopapain, and chymopapain. Red-fleshed papaya contains five β carotene, β-cryptoxanthin, β carotene 5,6-epoxide, lycopene, and zeta-carotene. However, yellow-fleshed papaya is a source of β carotene, β-cryptoxanthin, and zeta-carotene (Chandrika et al. 2003). Papaya possesses strong antioxidant activity due to the presence of carotene which is more than apples, guava, and plantains (Zhou et al. 2011; Addai et al. 2013).
Papaya is reported to have anticancer activity against cervical, breast, lung, and pancreatic cancer (Otsuki et al. 2010). Papaya fruit is known to alter the growth of tumor cells in different types of cancer cell lines (Nguyen et al. 2013). Papain, lycopene, and carotene are important phytochemicals from papaya which are responsible for its anticancer value along with other antioxidants. Possible mechanism involved is that the papain breaks down fibrin cancer cell wall and protein in amino acid form (Gunde and Amnerkar 2016). Lycopene is highly reactive toward oxygen and free radicals, and isothiocyanates are effective against breast, lung, colon, pancreatic, and prostate cancer, and leukemia, which can interfere with the activity of cancer development (Fauziya and Krishnamurthy 2013). Benzyl isothiocyanates are isolated from fruit extract-induced cytotoxic effect in proliferating human colon cells (Miyoshi et al. 2007). Further benzyl isothiocyanates from seed homogenate of papaya are known to be highly effective against superoxide generation (Nakamura et al. 2007) and in inducing apoptosis in acute promyelocytic leukemia cell line HL-60. Aqueous extract of papaya flesh caused significant inhibition against breast cancer cell line MCF 7 (Garcia-Solis et al. 2009). Petroleum ether extracts of aerial parts of papaya are also reported to show significant activity against breast cancer (Rashid and Fouche 2013).
Papaya seed possesses antimicrobial activities against B. subtilis, E. cloacae, E. coli, S. typhi, S. aureus, P. aeruginosa and T. vaginalis. Papaya latex exhibits antifungal activities against C. albicans and anti-amoebic activity against E. histolytica. Papaya rind is known to have antimalarial activities (combretastatins). Recently, antidengue activity of papaya is also reported (Ahmad et al. 2011; Saran et al. 2016). Antihyperglycemic and hypolipidemic activities of papaya leaves are also reported in alloxan-induced diabetic rats (Maniyar and Bhixavatimath 2012).
Traditional Uses
The young green papaya fruit is used as a vegetable and to make meat tender. It is also used in salad, juices, and desserts. Papain helps in digestion of food and is also prescribed for dyspeptic patients (Krishna et al. 2008). Papain is also used for brewing, wine making, and in tanning and textile industries. Air-dried papaya seeds can be used for treating intestinal parasites if consumed with honey. It is also used for making jellies, pickles, candies, marmalades, chutneys, sauce, burfi, and leather (Saran et al. 2016). Papaya leaf tea is antiseptic and useful for chronic indigestion, weight loss, and for maintaining cardiovascular health (Mantok 2005). Papaya-based skin care products and scrubs are available in markets. Papain is widely used in cosmetics and pharmaceutical industries due to its ability to degrade collagen. In Thailand, papain is imported from other countries for use in cosmetic products. Chymopapain and papaya proteinase are also important enzymes from papaya which are widely used in hair and skin care products.
## 3.6 Cycas revoluta Thunb.
* Vernacular name: King Sago Palm
* Family: Cycadaceae
Ecological Distribution and Morphological Characteristics
It is an evergreen cycad which is native to China and Japan.
Cycas revoluta forms a crown of shiny, dark green leaves on a thick trunk. The trunk is very low, to subterranean in young plants but increases in height as it grows older. The plant is very slow-growing and requires about 50–100 years to achieve normal height. Trunks can branch multiple times, thus producing multiple heads of leaves (Fig. 3.6).
Fig. 3.6
Leaves of Cycas revoluta (king sago palm) are known to have anticancer potential
Important Phytochemicals and Medicinal Value
Seeds of C. revoluta are utilized as an antirheumatic and as an expectorant. They are also known to produce certain neurotoxic metabolites (Chang et al. 1995; Cren-Olive et al. 2002; Nelson et al. 2007).
Leaves of the plant are reported to have anticancer potential as metabolites extracted from the seeds are used to inhibit the growth of malignant tumors (Fabricant and Farnsworth 2001). Peptides isolated from the plant are repressors of cell proliferation of human epidermoid cancer and colon carcinoma through insertion of cell cycles arrest at G0–G1 phase of Hep2 cells (Mandal et al. 2012).
Traditional Uses
Toxicity of seeds is reported due to the presence of some neurotoxins. The plant is also being used in the treatment of rheumatism.
## 3.7 Dillenia indica L.
* Vernacular name: Elephant Apple
* Family: Dilleniaceae
Ecological Distribution and Morphological Characteristics
Genus Dillenia comprises almost 60 species of medicinally important plants. D. indica is an evergreen shrub with long leaves which is native to Southeast Asia. Leaves are large, light green in color, having corrugated surface and impressed veins (Fig. 3.7a–b). Flowers are large and white in color.
Fig. 3.7
(a–b) Juices of leaves, bark, and fruits of Dillenia indica (elephant apple) are given orally for the treatment of cancer and diarrhea. (a) Tree (b) Large leaves with impressed venation
Important Phytochemicals and Medicinal Value
Phytochemical screening revealed the presence of many compounds like flavonoids, steroids, terpenoids, and phenolic compounds. Fruits and leaves extracts are reported to have antioxidant properties (Abdille et al. 2005). The alcoholic extracts of leaves are known to possess central nervous system (CNS) depressant activity (Bhakuni et al. 1969).
Anticancer, anti-HIV, antidiabetic, anti-inflammatory, antimalarial, and analgesic activities of the plant are due to the presence of compounds like betulin, betulinic acid, lupeol, and stigmasterol (Theo et al. 2009; Boparai et al. 2016). Methanolic extract of leaves is reported to have antidiabetic activity (Sood et al. 2005; Kumar et al. 2011).
Fruits of D. indica exhibited anticancer activities against different human cancer cell lines (Vedasiromoni et al. 2010). Betulinic acid is known to have inhibitory activity on the growth of tumor and can induce apoptosis and can also be tolerated by mice at a dose of 500 mg/kg without showing any side effects (Pisha et al. 1995; Fulda and Debatin 2000).
Traditional Uses
Different parts of the plant are used to relieve indigestion, asthma, influenza, dysentery, jaundice, weakness, and rheumatic pain. The bark is used in making charcoal. The bark and leaves of D. indica are astringent (Kritikar and Basu 2003). Traditionally, the plant is used in the treatment of diabetes (Sood et al. 2005; Kumar et al. 2011). Mixed juices of leaves, bark, and fruits are given orally for the treatment of cancer and diarrhea (Sharma et al. 2001). Fruit is used as a laxative and for abdominal pains. It is also used to give flavor to Assamese cuisine (Begum and Gogoi 2007). Fruits are rich in nutrients and processed to make commercial products such as beverages and squash (Kerrigan et al. 2011). Leaves are also used as contraceptive. A young leaf is mixed with one handful of rice and soaked in water overnight. In the morning, the rice and leaf are crushed and taken on empty stomach (Boparai et al. 2016).
## 3.8 Jacaranda mimosifolia D. Don
* Vernacular names: Blue Jacaranda, Brazilian Rosewood, Fern tree
* Family: Bignoniaceae
Ecological Distribution and Morphological Characteristics
Jacaranda mimosifolia is a tree with intense blue or violet clusters of flowers. It is native to south-central South America.
Leaves arrangement is opposite, bipinnate, and symmetrical like fern leaves. Monoecious flowers create a beautiful carpet of violet color on the ground when they fall. Fruit is a capsule, ovate in shape, and bears many compressed seeds (Fig. 3.8a–e).
Fig. 3.8
(a–e) Jacaranone, jacraninoside, polyphenols, and flavonoids are main anticancer compounds in many Jacaranda species. (a) J. mimosifolia (blue jacaranda, fern tree) (b) Fern-like leaves (c) Flowering branch (d) Flowers (e) Young fruits (green) and ripe fruits (brown)
Important Phytochemicals and Medicinal Value
Jacaranda is a genus of 49 medicinally important species. Many triterpenes, flavonoids, acetosides (Moharram and Marzouk 2007), quinones (Rana et al. 2013), phenylpropanoid derivatives (Salome et al. 2010), fatty acid, and anthocyanins are present in different parts of J. mimosifolia.
Compounds extracted from the root bark include isoquercitrin, isovitexin, apigenin, luteolin, scutellarein, apigenin 7-O-b-D-glucopyranoside methyl ester, luteolin 7-O-b-D-glucopyranoside methyl ester, E and Z acetosides, cistanoside, campneoside, and jacraninoside A (Moharram and Marzouk 2007). Many of these compounds are isolated from the stem bark using 1D and 2D Nuclear magnetic resonance (NMR), and Electrospray ionization (ESI) mass spectrometry (Sidjui et al. 2014).
Jacaranone is the main constituent isolated from Jacaranda spp. which possesses anticancer activities (Gachet and Schuhly 2009). Anticancer activity of J. mimosifolia is known due to the presence of compounds like jacaranone, phenylethanoid glucosides, polyphenols, and flavonoids in the leaves and flowers (Rana et al. 2013). Hydroethanolic extract of J. decurrens leaves possesses antioxidants and cytotoxic potential due to phenolic and flavonoid compounds (Casagrande et al. 2014).
Ethanolic extract of J. caucana is reported to have in vitro antitumor activity against P-388 lymphocytic leukemia system. J. mimosifolia is also known to possess antimicrobial activities against B. cereus, E. coli, and S. aureus (Nisar et al. 2014). The plant is also reported to have many antifungal activities (Sidjui et al. 2014).
Traditional Uses
It is planted as an avenue plant and due to its aromatic wood, it is used commercially. Bark decoction and tea are used to regulate fertility and lactation (Quattrocchi 2012). Traditionally, flowers are used to cure hepatitis. Leaves and bark are used to cure neuralgia. Dried leaves are used as an ointment to heal wounds. Infusion of bark is applied as a lotion to treat ulcers. In Ecuador, bark of the plant is used as a blood purifier (Acosta-Solis 1992; Gachet and Schuhly 2009).
## 3.9 Jasminum officinale L.
* Vernacular name: White Jasmine
* Family: Oleaceae
Ecological Distribution and Morphological Characteristics
Over 300 species of Jasmine are reported in many tropical and subtropical regions. Jasminum officinale is also the national flower of Pakistan. It reaches up to a height of about 8–10 feet (Fig. 3.9a–b). Leaves are either evergreen or deciduous, ovate, and rounded at the base.
Fig. 3.9
(a–b) Jasminum officinale (white jasmine) possesses important anticarcinogenic, antimicrobial, and antidepressant activities. (a) Tree (b) White flowers
Important Phytochemicals and Medicinal Value
Ethanolic extract of J. officinale indicated the presence of many alkaloids and flavonoids, i.e., leucoanthocyanins and anthocyanins (Hongratanaworakit 2010).
Antimicrobial activity of flowers is also reported against human pathogens (Hussain et al. 2013). Oleuropein isolated from flower extract is suggested to be a potential inhibitor of hepatitis B virus (Zhao et al. 2009). Aqueous extracts of leaves showed cytotoxic and genotoxic potential (Ghurde et al. 2012).
Anticancer activity is reported to be due to the presence of compounds like jasminum, jasminigenin, flavonoids, and saponins. Methanolic extract of flowers caused inhibition in the growth of cancer cells in Swiss albino rats (Kalaivani et al. 2012).
Traditional Uses
Jasmine flower is an important ingredient of almost all Ayurvedic medicines. It is used for the treatment of jaundice and other venereal diseases (Priya and Patric 2008). The flower buds help in the treatment of ulcers, skin diseases, and eye disorders. Jasmine tea is thought to have anticancer potential. Flowers are also used for making perfumes which are effective in aromatherapy and anxiety-related sexual problems in men and women (Roberts 2000).
## 3.10 Magnolia grandiflora L.
* Vernacular names: Bull bay, Magnolia
* Family: Magnoliaceae
Ecological Distribution and Morphological Characteristics
Magnolia grandiflora is an evergreen tree of medicinal value and native to the United States. More than 50 cultivars of the plant are grown for commercial purposes.
The leaves are alternate, petiolate, thick, dark green, glabrous, tomentose, underneath with yellow-brown pubescence. Their shape is elliptical and they are pinnately veined. The mature leaves have acute apex whereas young leaves are obtuse (Fig. 3.10a–g). Solitary flowers appear on stout tomentose pedicel. They are creamy white in color, fleshy, aromatic, bisexual, and hypogynous. Gynoecium are numerous and apocarpous. Fruits are aggregate of follicles, cone shaped, hard, and woody. Large, shiny seeds are bright red in color.
Fig. 3.10
(a–c) Magnolia spp. are medicinally important due to compounds like magnolol and honokiol; they have antitumor, analgesic, and sleep-enhancing properties. (d) Leaves and young cones of M. grandiflora (bull bay). (e) Flower has antioxidant activity and can decrease the melanin content in B16F10 melanoma cells and therefore is used in skin care products. (f) Extract of seed cones can cause apoptosis against B-cell lymphocytic leukemia. (g) Mature cone
Important Phytochemicals and Medicinal Value
Phytochemicals like alkaloids, flavonoids, gums and mucilage, phenolics, phlobatannins, saponins, steroids, tannins, and terpenoids are extracted from fruits and seeds. Many sesquiterpenes lactones isolated from the plant possess anti-inflammatory and analgesic activities (Feltenstein et al. 2004). Essential oils of flowers comprise geraniol, germacrene, and pinene, which are isolated through headspace solid-phase microextraction (Lim 2014).
It is reported that extract of seed cones of the plant induced apoptosis against B cell lymphocytic leukemia (B-Cell). Recently, stem, leaves, and flowers of the plant are reported to have anticancer, antidepressant, antioxidant, and anti-inflammatory activities (Marin and Mansilla 2010; Lee et al. 2011). Additionally, the flower extract has antioxidant activity as it is known to decrease the melanin content in B16F10 melanoma cells, which indicates the potential use of the plant in skin care products (Huang et al. 2012).
Compounds like magnolol and honokiol (Rao and Davis 1982) have muscle-relaxing and antitumor (Ikeda and Nagase 2002) properties. The plant is also reported to protect the skin from photoaging by inhibiting NF-kappaB transcription (Lim 2014). Sleep enhancement activity is observed due to magnolol (Ma et al. 2009).
Traditional Uses
Timber is used commercially for making furniture. The plant is used as a traditional medicine for curing diseases like diarrhea, rheumatic arthritis, high blood pressure, epilepsy, fever, and for problems related to fertility (Schuhly et al. 2001). Leaves and flower extracts are used for cardiovascular diseases (Mellado et al. 1980).
## 3.11 Plumeria obtusa L.
* Vernacular name: Frangipani
* Family Apocynaceae
Ecological Distribution and Morphological Characteristics
Plumeria plants are commonly known as frangipani and well known for their fragrant and beautiful flowers. Plumeria obtusa is an evergreen deciduous tree, native to Bahamas and cultivated in tropical climates, due to its ornamental and medicinal value.
Its leaves are spiral, alternate, obovate, and obtuse at both ends. It produces flowers of different colors which are aromatic. Flowers have five petals, which are arranged in the form of a short funnel at the base; however, they widen up toward the top (Fig. 3.11a–f). Cylindrical pods are produced which are arranged in pairs after the flowering season.
Fig. 3.11
(a–f) Plumeria spp., commonly known as frangipani, possess many medicinal activities like anticarcinogenic, antimicrobial, and anti-inflammatory. Main anticancer compounds isolated from the plants include iridoids like fulvoplumierin, allamcin, and allamandin. (a–d) P. rubra, (a, b) Tree (c, d) Developing flower (e, f) Pink flowered Plumeria sp.
Important Phytochemicals and Medicinal Value
Phytochemical screening showed the presence of alkaloids, glycosides, saponins, tannins, carbonyls, flavonoids, phlobatannins, and steroids. Essential oils of flowers have antimicrobial and antibacterial properties and are rich in benzyl salicylate, farnesol, methyl stearate, nerolidol, and linalool (Norista et al. 2006a; Lim 2013). It is also reported to have anticarcinogenic activity (Banu and Jayapakar 2011; Shinde et al. 2014).
Traditional Uses
Milky latex and decoction of the bark are known to possess purgative, emmenagogic, febrifuge, and diuretic properties. The plant is also used in the treatment of skin diseases and for bringing down fevers.
## 3.12 Plumeria rubra L.
* Vernacular names: Frangipani, Temple tree
* Family: Apocynaceae
Ecological Distribution and Morphological Characteristics
It is native to Mexico and South America.
All species of genus Plumeria are small trees with very thick branches and produce a milky juice when the leaves or branches are cut. The leaves are alternate, elliptic or ovate, and arranged spirally near the ends of the swollen branches. Its aromatic flowers are in various shades of red, pink, orange, and yellow (Fig. 3.11a–f).
Important Phytochemicals and Medicinal Value
The presence of flavone glycosides isolated from P. rubra shows its antioxidant potential. Flowers are a rich source of many flavonoids, phenols, and many antioxidants (Sirisha et al. 2013). Different chemical constituents extracted from bark are bitter like glycosides, plumieride, plumeric acid, β-sitosterol, lupeol, plumieride, amyrin and fulvoplumierin, plumericin, isoplumericin, 4-hydroxyacetophenone, plumieride, coumarylplumieride and protoplumericine (Ye et al., 2008; Ye et al., 2009). Essential oils of pink flowers include lauric acid, myristic acid, palmitic acid, nonadecane, linalool, docosane, and tricosane (Norista et al. 2006a). Yellow-white flowers of P. acuminata comprise benzyl salicylate, geraniol, camphor, cinnamyl cinnamate, and nerolidol (Norista et al. 2006b).
The plant is also reported to have anticancer activities against cell lines of murine lymphocytic leukemia and also against a number of human cancer cell types (breast, colon, fibrosarcoma, lung, and melanoma). Main anticancer compounds isolated from the bark of the plant include iridoids like fulvoplumierin, allamcin, and allamandin (Lim 2013). Rubrinol isolated from the leaves possesses antimicrobial activities against Bacillus anthracis, Pseudomonas aeruginosa, and Burkholderia pseudomallei (Akhtar et al. 1994).
Traditional Uses
Traditionally, it is used for the treatment of diarrhea, blennorrhea, and leprosy. Leaves of the plant are effective against inflammation and have antibacterial, anticancer, and antifungal properties (Baghel et al. 2010). Flowers of the plant are one of the main ingredients of "five flower herbal tea" (Kong et al. 2006).
## 3.13 Sapium sebiferum L. (syn: Triadica sebifera)
* Vernacular name: Tallow tree
* Family: Euphorbiaceae
Ecological Distribution and Morphological Characteristics
It is native to southern and central China and can grow up to 20 m in height (Renne et al. 2002). Euphorbiaceae family in the plant kingdom is a complex heterogeneous family consisting of about 322 genera and 8900 species in the world (Singh et al. 2011).
It is a monoecious and deciduous tree. Leaves are simple and heart-shaped, and the flowers are monoecious forming drooping branches (Fig. 3.12a–f).
Fig. 3.12
(a–f) Extracts of Sapium sebiferum (Chinese tallow) leaves possess strong antioxidant and anti-inflammatory activities. (a) Tree (b) Autumn tree (c) Heart-shaped leaves (d) Leaves in autumn (e) Flower (f) Fruit
Important Phytochemicals and Medicinal Value
Many glucosides isolated and detected through NMR include moretenone, moretenol, xanthoxylin, sitosterol β-D-glucoside, and 2-acetyl-3,5-dimethoxyphenyl-O-β-D-xylopyranosyl-(1–6)-β-D-glucopyranoside. Extracts of leaves possess strong antioxidant and anti-inflammatory activities (Fu et al. 2013).
Anticancer activity of stem bark of S. baccatum is reported against different cancer cell lines due to the presence of compounds like malaytaraxerate, taraxerol, taraxerone, docosyl isoferulate, and docosanoic acid 2′,3′-dihydroxypropyl ester (Al-Muqarrabun et al. 2014).
Traditional Uses
It is used as a source of fodder and fuel. Wax from the seed coat is used to make candles, soaps, crates, and boxes. The seed contains about 20% of a drying oil. The oil is used in making varnishes and native paints because of its quick-drying properties. It is also used in machine oils and as a crude lamp oil. The leaves are rich in tannin, a black dye that can be obtained by boiling them in alum water. The plant is also used as a soil binder along the sides of roads and canals. Fruits are sweet and edible and used to treat ulcers by folks in Malaysia (Eswani et al. 2010).
## 3.14 Schleichera oleosa Lour.
* Vernacular names: Ceylon Oak, Kusum
* Family: Sapindaceae
Ecological Distribution and Morphological Characteristics
It grows as a large deciduous tree which is found in the mountains of South Asia having pinnate leaves forming an expanded crown. Flowers are small, yellow in color, and form a compact cluster. Fruit is ovoid, oblongated bearing brown seeds (Fig. 3.13a–c).
Fig. 3.13
(a) Schleichera oleosa (Ceylon oak) is an important anticancer and antidiabetic tree which is also a source of biofuel. (b) Young leaves appear red in color. (c) Seed oil comprises oleic, stearic, gadoleic, and arachidonic acids
Important Phytochemicals and Medicinal Value
Seed oil comprises oleic, stearic, gadoleic, and arachidonic acids. Extracts of leaves in aqueous and methanol revealed the presence of many secondary metabolites. Agar disk diffusion method indicated the presence of many compounds having antioxidant and antimicrobial activities.
The plant is known to possess antimicrobial, antioxidant, and anticancer activities (Bhatia et al. 2013). Extracts from tree bark are known to possess strong antioxidants which have anticancer activities (Bhatia et al. 2013); however, research is being done to further explore anticancer potential of the plant (Meshram et al. 2015).
Traditional Uses
All parts of the plant are used in Indian traditional healing. It is also being used for the production of biodiesel. Wood is used for building materials, farming, forestry, and hunting. Many dyes, stains, inks, tattoos, and mordants are made from the floral products. Due to the presence of low tannins, the plant is also used as a source of food for livestock. Seeds are a source of kusum oil which is used for curing skin problems, rheumatism, hair dressing, and to promote hair growth.
## 3.15 Thuja occidentalis L.
* Vernacular name: White Cedar
* Family: Cupressaceae
Ecological Distribution and Morphological Characteristics
Thuja grows as an evergreen, small coniferous tree in wet forests, comprising scale-like alternate leaves. It is native to the United States and Canada.
Plant is monoecious and produces cones of yellow green in color. Male cones are reddish brown while female cones are yellow green and ellipsoid in shape (Fig. 3.14a–d).
Fig. 3.14
(a–d) Thujone is an alkaloid extracted from ethanolic extract of Thuja occidentalis (white cedar) which is known to possess anticancer potential. Active compounds of essential oils include carnphor, fenchor, isothujone, and thujone. (a) Tree (b) Leaves (c, d) Young and mature cones
Important Phytochemicals and Medicinal Value
The plant is rich in phenolic compounds which increase blood glutathione level (Dubey and Batra 2009a) and many antioxidants (Dubey and Batra 2009b). The plant has antibacterial, antioxidants, anticancer, antidiabetic, and hepatoprotective properties (Meenu et al. 2011).
Thujone, an alkaloid, is extracted from ethanolic extract of the plant which is reported to possess anticancer potential. Active compounds of essential oils of Thuja include carnphor, fenchor, isothujone, and thujone (Asili et al. 2007). The plant is reported to have activity against breast cancer cell lines by inducing apoptosis.
Traditional Uses
The plant is known to reduce effects of chemotherapy. Its ointment is very popular for skin ailments like eczema, muscular pains, and rheumatism. Essential oil is used in steam bath. It is also used in small doses for treatment of flu, headache, and cough.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_4
# 4. Trees with Antimicrobial Activities
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
This chapter describes the antibacterial, antifungal, and antiplasmodial activities of important trees. Antimicrobial properties of many trees are associated with alkaloids, tannins, terpenoids, polyamines, isothiocyanates, thiosulfinates, glucosides, and polyacetylenes. Many antimicrobial compounds are being extracted from plants to be used for therapeutic purposes due to problems related to increase in antibiotics resistance. They include Ailanthus altissima, Bouganvillea spectabilis, Cedrus deodera, Madhuca longifolia, Melia azerdarach, Opuntia ficus-indica, Podocarpus macrophyllus, Polyalthia longifolia, Swietenia macrophylla, Toona ciliata, Tecoma stans, and Terminalia mantaly.
## 4.1 Introduction
Many trees are known to possess antibacterial, antifungal, antiplasmodial, antiprotozoal, and antiviral properties. Antimicrobial properties of plants have been investigated worldwide especially in Latin America against S. aureus, B. subtilis, P. vulgaris, P. aeruginosa, and E. coli through diffusion methods. Further research is being conducted to explore antimicrobial potential of medicinal plants and to develop plant-based drugs which do not have any side effects like antibiotics.
Antimicrobial properties of many of trees are due to the presence of active compounds including alkaloids, flavonoids, glycosides, and terpenoids group. Some plants belonging to family Apocynaceae, Longaniaceae, and Rubiaceae comprise alkaloids with antimicrobial properties in all parts of plant (Fig. 4.1a–h). Essential oils of many plants are also known to have inhibitory properties against many microbes and can kill food-borne pathogens. Leaves of trees like Laurus nobilis (bay leaf), Murraya koengii (curry leaf), and Citrus spp. are used in Asian cooking due to their fragrance and germicidal properties. Aromatic oil of Cinnamomum zeylanicum (cinnamon tree), Illicium verum (star anise), and Syzygium aromaticum (clove) are also antimicrobial and commonly used in cooking due to their aroma and flavor.
Fig. 4.1
(a–h) (a) Stem bark of Bursera serrata exhibits antimicrobial activities due to the presence of compounds like β-amyrin and β-sitostenone and a coumarin, scopoletin (b, c) Camphor oil from Cinnamomum camphora (camphor tree) is well known for having antibacterial and antifungal activities (d, e) Fruits of Duranta repens (golden dewdrop) revealed antimalarial activity against Plasmodium berghei and methanolic extract possesses antiviral activities (f) Euphorbia cotinifolia, red spurge is known to possess a wide range of phytochemicals with antibacterial activities (g) Leaves (h) Flower buds
Members of Lamiaceae like Azadirachta indica (neem tree) contains essential oils which are toxic for many insects and neem-based biopesticides are being used which are biodegradable and environmentally friendly.
## 4.2 An Account of Woody Plants with Antimicrobial Activities
Although, many trees are reported to have antimicrobial activities, however, the following section will explain antimicrobial activities of relatively more important trees.
## 4.3 Ailanthus altissima (Mill.) Swingle
* Vernacular name: Tree of heavens
* Family: Simaroubaceae
Ecological Distribution and Morphological Characteristics
It is grown as an ornamental tree in Europe and North America. It originated in China, but it is now introduced in many countries of world including Japan, Australia, and Pakistan.
It grows as a large deciduous tree up to a height of over 50 feet having rounded head of branches. Leaves are pinnate, having 15–30 leaflets, long, ovate with almost entire margins (Fig. 4.2a–d). Flowers are small, yellowish green in terminal panicles, and fruits are orange to red in color. Male flowers are more conspicuous and emit an odor to attract insects.
Fig. 4.2
(a–d) (a) Ailanthus altissima commonly known as tree of heavens is an important tree with antimicrobial activities due to active compounds which include ailanthone, ailanthinone, chaparrine, and ailanthinol B. The quassinoid alkaloid from aqueous extract of bark and leaves is antiamoebic against Entamoeba histolytica (a) Tree (b) Leaves with flowers in terminal panicles (c, d) Young fruits
Important Phytochemicals and Medicinal Value
Active compounds isolated include ailanthone, ailanthinone, chaparrine, and ailanthinol B (quassinoid derivatives), among which main compound with herbicidal activity is ailanthone (Feo et al. 2013). Phytochemicals like ailanthone and galucarubinone are found to be effective against lymphocytic leukemia in mice (Bajaj 1991).
Ethanolic extract of stem bark revealed antibacterial activity against B. subtilis, S. aureus, P. aeruginosa, and antifungal activity against A. niger, A. fumigatus, and P. chrysogenum (Manikandan et al. 2015). The quassinoid alkaloids from aqueous extract of bark and leaves is used for the treatment of dysentery and is antiamoebic against Entamoeba histolytica. Antimicrobial activity of fruit is reported to be due to the presence of compounds like 5alpha-stigmastane-3,6-dione, 3β-hydroxystigmast-5-en-7-one, stigmast-5-ene-3β, 7alpha-diol, 6alpha-hydroxystigmast-4-en-3-one, 5alpha-stigmastane-3β, 6β-diol, stigmast-4-ene-3β, 6alpha-diol, stigmast-5-ene-3β, 7alpha, 20xi-triol (Zhao et al. 2005).
Plant is known to have antioxidant, antiviral, antimalarial, antifungal, insecticidal, antiproliferative, and antiasthmatic activities (De Feo et al. 2005; Rahman et al. 2009; Lu and He 2010; Luis et al. 2012). A triterpenoid, AECHL1 ase isolated from root bark, possesses anticancer activity (Lavhale et al. 2009). Plant is known to have potential to control birth rate and possesses antifertile activity and can also be used as an environmentally safe pesticide (Lavhale and Mishra 2007).
Traditional Uses
In traditional Chinese medicine, different parts of plants are used as astringent. In Asia, extract from fruit and bark is used as antimicrobial, anthelmintic, and amoebicide. Bark is harvested in spring and then dried. Leaves are slightly toxic and boiled in water to treat skin problems like acnes, boils, and itches. Plant is a folk remedy for asthma, cancer, diarrhea, malaria, sores, breast tumors, epilepsy, and fever.
## 4.4 Bougainvillea spectabilis Willd.
* Vernacular name: Bougainvillea
* Family: Nyctaginaceae
Ecological Distribution and Morphological Characteristics
Genus Bougainvillea has almost 14 species which grow as ornamental shrubs. It is native to South America, Brazil, Peru, and Argentina. Horticulturally important species include B. spectabilis, B. glabra, and B. peruvina. Leaves are simple and ovate. Flowers are small usually white, surrounded by bracts of pink, white, and red color (Fig. 4.3a–d).
Fig. 4.3
(a–d) Bougainvillea spectablis (bougainvillea) is a medicinally important shrub with antimicrobial and antidiabetic activities due to compounds like D-pinitol. Leaves possess antibacterial activities against B. spectabilis (a, b) Bouganvillea spp. (c) White bracts surrounding white flowers (d) White flowers surrounded by red bracts
Important Phytochemicals and Medicinal Value
Phytochemical screening of different plant parts indicated presence of many metabolites like alkaloid, flavonoids, glycosides, phlobotannins, saponins, steroids, tannins, and terpenoids in water, acetone, chloroform, ethanol, and methanol.
Plant is known to have broad-spectrum antibacterial activity (Farzana et al. 2013) due to the presence of myristic acid and steroidal compounds. Ethanolic, methanolic, chloroform, and ethylacetate extracts showed inhibitory activities against S. aureus, B. subtilis, S. faecalis, M. luteus, E. coli, P. aeruginosa, S. typhi, K. pneumoniae, P. vulgaris, and S. marcescens (Umamaheswari et al. 2008). Leaves showed highest biological activity as phytochemical analysis of B. spectabilis leaves showed presence of chemical components such as saponins, tannins, terpenoids, and phlebotannin (Sofowora 1993). Leaves also revealed antiviral, antibacterial, and insecticidal properties. Bougainvillea leaf extracts also possess antioxidant activity (Venkatachalam et al. 2012). Flower extracts also showed antimicrobial activities against B. subtilis (Swamy et al. 2012).
The leaves of B. spectabilis are also reported to have antidiabetic activity due to the presence of D-pinitol (Saravanamuttu and Sudarsanam 2012). Aqueous extract of leaves showed antifertile activity in Swiss albino mice (Mishra et al. 1998).
Traditional Uses
Traditionally plant is used to cure diarrhea, stomach disorders, cough, sore throat, and hepatitis. Leaves and flowers are used in Mexican traditional system to treat cold and bronchitis (Aguilar et al. 1994).
## 4.5 Cedrus deodera Roxb. (Syn Pinus deodara Roxb. ex D. Don)
* Vernacular names: Deodar cedar, Himalayan cedar
* Family: Pinaceae
Ecological Distribution and Morphological Characteristics
Plant is distributed in Mediterranean region and in Western Himalayas. It is the national tree of Pakistan. It is an evergreen tall coniferous tree with needle-like leaves (Fig. 4.4).
Fig. 4.4
Cedrus spp. are known to possess antiseptic, antibacterial, and antifungal properties
Important Phytochemicals and Medicinal Value
Himachalol is an active constituent of wood with antispasmodic activity. Its spasmolytic activity is similar to papaverine (Kar et al. 1975). Alcoholic extract of stem has anticancer and antispasmodic activities (Singh et al. 2007; Jain et al. 2014). Main antispasmodic compounds include allohimachol, centdarol, and himadarol. Juvabional compounds with juvenile hormonal activity are delta-7-dehydrodomatuic acid, delta-10-dehydroepitodomatuic acid, and 7-hydroxytodomatuic acid (Duke et al. 2002).
Plant is also known to have antiviral, antifertile, anti-inflammatory, astringent, and carminative properties (Duke et al. 2002). Essential oil of pine needles is known to possess antiseptic, antibacterial, and antifungal properties (Parihar 2003). Antibacterial activity is reported against food-borne bacteria due to the presence of compounds like terpineol, linalool, limonene, anethol, caryophyllene, and eugenol (Zeng et al. 2012; Tarranum et al. 2014) and can be used as an antimicrobial agent in the food industry. Ethanolic extract revealed antimicrobial activities against S. aureus, S. faecalis, B. cereus, K. pneumoniae, and E. coli (Devmurari 2010).
Traditional Uses
Wood of this plant is used in Ayurvedic medicine to treat inflammation and rheumatoid arthritis. Plant is bitter, slightly pungent, and used in inflammation, dyspepsia, insomnia, cough, fever, blood, and skin diseases. Oil is analgesic and is used for joint injuries and skin problems.
## 4.6 Chukrasia velutina Roxb. (Syn: Chukrasia tabularis A. Juss)
* Vernacular name: Chickrassy, Jamaica cedar
* Family: Meliaceae
Ecological and Morphological Description
It is a valuable and dominant canopy tree in many Asian countries. It is cultivated in China, India, Pakistan, Sri Lanka, Thailand, Malaysia, and Vietnam due to its role in agroforestry. It is a deciduous tree which grows up to a height of 40 m. Leaves are pinnate and bear alternate leaflets. Unisexual flowers are born as terminal panicles. Flowers are aromatic, unisexual, and appear from April to July. Fruit is capsule (Fig. 4.5a, b).
Fig. 4.5
(a, b) Antimicrobial activity of Chukrasia tabularis (Jamaica cedar) is due to chukrasins, busseins, limonoids, tannic acids, sitosterols, cedrelons, and tabularin (a) Tree (b) Leaves
Important Phytochemicals and Medicinal Value
Important compounds isolated are tannic acids, sitosterols, cedrelons, and tabularin. Limonoids are also common with modified triterpenes (Roy and Saraf 2006). Wood comprises diverse chukrasins and busseins. Different phragmalin limonoids are extracted from root bark which include cedrelone, tabulalins through droplet countercurrent chromatography (DCCC) and reverse-phase High-performance liquid chromatography (HPLC) (Nakatani et al. 2004).
Antibacterial and antifungal activity of petroleum ether, benzene, chloroform, ethyl acetate, and methanolic extract is reported against B. subtilis, S. aureus, S. aeruginosa, E. coli, P. vulgaris, K. pneumoniae, A. fumigatus, A. niger, and F. oxysporum (Nagalakshmi et al. 2001). Crude extract of leaves exhibits mild antibacterial activities while bark extract is reported to have central analgesic properties at a dose of 400 mg/kg and peripheral activity at 200 mg/kg and 400 mg/kg (Aktar et al. 2015).
Traditional Uses
Bark, leaves, and fruits are reported to have biopesticidal activities due to the presence of some limonoids, terpenoids, and phenolic compounds (Kaur and Arora 2009) which makes it resistant to many pathogens. Bark is used traditionally as antipyretic, astringent, antidiarrheal, and anti-influenza drug (Nakatani et al. 2004).
## 4.7 Madhuca longifolia L.
* Vernacular name: Mahwa
* Family: Sapotaceae
Ecological Distribution and Morphological Characteristics
M. longifolia is a large evergreen tree distributed in India, Sri Lanka, and Nepal and is grown in many South Asian countries. Madhuca is commonly known as mahwa or butternut tree and grows up to a height of 17 m with a large top (Ramadan et al. 2006). The bark is yellowish gray to dark brown red in color and milky inside (Fig. 4.6a–d).
Fig. 4.6
(a–d) Medicinal activities of Madhuca longifolia (mahwa) are due to the presence of many important compounds like terpenoids, anthraquinone glycosides, cardiac glycosides, saponins, and tannins (a) Tree (b) Leaves (c) Branch bearing flowers (d) Close view of flowers
Important Phytochemicals and Medicinal Value
Phytochemical studies of stem bark with ethanol, water, and chloroform extract revealed the presence of many important compounds like terpenoids, proteins, mucilage, anthraquinone glycosides, cardiac glycosides, saponins, and tannins (Gopalkrishnan and Shraddha 2012).
Important constituents of leaves are β-carotene and xanthophylls; erthrodiol, palmitic acid, myricetin, rhamnoside, quercetin, galactoside; oleanolic acid, β-sitosterol, stigmasterol, β-sitosterol-β-D glucoside, n-hexacosanol, n-octacosanol, sitosterol, and quercetin. Important constituents from trunk bark include lupeolacetate, α-amyrin acetate, α-spinasterol, erythrodiol monocaprylate, betulinic acid, and oleanolic acid caprylates (Khare 2007).
Antimicrobial properties of alcoholic extract of leaves and flowers are reported against S. aureus, B. subtilis, E. coli, P. aeruginosa, A. oryzae, and A. nigar at dose level ranging from 50 to 250 μg/ml (Kalaivani and Jegadeesan 2013; Pandey et al. 2013). Even dose of 50 μg/ml showed significant inhibitory activity against B. subtilis which is comparable to standard antibiotic (ciprofloxacin); however, for rest of bacteria, dose of 100 μg/ml proved to be effective.
Antifungal activity of alcoholic extract of leaves is comparable with standard antifungal drug (clotrimazole). Alcoholic extract of flowers also revealed significant inhibitory activity against the above-mentioned bacteria and fungi. Methanolic extracts of inner bark is also reported to have significant antibacterial activity against S. aureus (24-mm zone of inhibition in methanolic extract), B. subtilis (20-mm zone of inhibition in methanolic extract), E. coli (15 mm), and S. epidermidis (10-mm zone of inhibition in methanolic extract) (Nimbekar et al. 2012). Antibacterial and antifungal activities are reported to be due to the presence of compounds like 2-furan methanol, 4H pyran 4-one, 2,3-dihydro 3,5-dihydroxy-6-methyl, thiophene, 2-furancarboxyaldehyde-5-(hydroxymethyl), and 1,4-tetra decanediol (Kalaivani and Jegadeesan 2013).
Traditional Uses
Madhuca oil is used for making laundry soaps, detergents, and also for cooking purposes. It is also considered to be a part of cultural heritage by many Indian tribes and a symbol to conserve wild forest. It is also a source of food, fodder, fuel, and used to control erosion. Flowers are edible, source of vitamins A and C, and used in many traditional sweets (Patel and Naik 2008). Distilled extract of flower is used to make vinegar. The bark is used in rheumatism, to relieve itching, swelling, fractures, and snake-bite poisoning (Priyanka et al. 2011). Leaves are applied as a poultice to relieve eczema.
## 4.8 Melia azerdarach L.
* Vernacular names: White cedar, Chinaberry
* Family: Meliaceae
Ecological Distribution and Morphological Characteristics
M. azedarach is a medium-sized deciduous tree. It is native to tropical Asia and grown due to its medicinal and economic importance. The plant is characterized by the presence of a spreading, dense, and dark-green crown. The leaves are alternate and leaflets are short stalked. Flowers are white with purple stripes and aromatic. Fruits or berries are yellow, round, smooth, and fleshy (Fig. 4.7a–e). Dried fruits are hard with 4–5 seeds.
Fig. 4.7
(a–e) Antimicrobial properties of Melia azedarach (white cedar) are well known due to phytochemicals like azadirachitin, nimbin, and nimbidin. Seed oil possesses antidiabetic activity and bark has antifertility activity (a) Tree (b) Leaves (c) Flowers (d) Branch bearing fruits (e) Fruits
Important Phytochemicals and Medicinal Value
Main compounds isolated from M. azerdarach are mainly triterpenes with the most effective being the limonoids abundant in its oil. At least nine limonoids are effective in inhibiting insect growth, especially some of the most deadly varieties found in human health and agriculture worldwide.
M. azedarach is well known to possess antimicrobial activities against infectious diseases. A wide range of microorganisms, i.e., B. cereus, S. aureus, E. coli, P. aeruginosa, S. flexneri, A. niger, A. flavus, F. oxisporum, and R. stolonifer showed inhibitory activities against methanolic, ethanolic, petroleum ether, and aqueous extracts; however, organic solvents proved to be more powerful than aqueous solvents (Sen and Batra 2012). Antimicrobial activity of ethanolic extract of leaves is reported against E. coli, E. faecalis, B. subtilis as well as antifungal activity against A. alternate, F. solani, F. oxysporum, and B. cinerea (Akacha et al. 2016). Methanolic extract of bark is effective against S. enterica ser typhi and S. mitis (Khatoon et al. 2014).
Seed extracts of plants have potential to be used as antibiotics against gram-positive and gram-negative disease-causing infectious bacteria (Khan et al. 2011). Medicinal activity of seeds is due to the presence of compounds like β-sitosterol, vanillin, benzoic acid, vanillic acid, daucosterol, α-D-glucopyranose, limonoid glycoside, epoxymeliacin, scopoletin, coumaramin, melianol, meliacin, meliacarpin, meliartenin vanillin, hydroxyl-3-methoxcinnamaldehyde, and pinoresinol (Carpinella et al. 2005; Chong et al. 2009). Azadirachitin is the most important limonoid of plant. Nimbin and nimbidin have antiviral and antifungal properties useful to humans and animals. Gedunin, a minor limonoid, is effective in treating malaria through infusion of leaves (Mugnai 2009).
Plant is also known to possess anthelmintic, anticancer, astringent, antimalarial, analgesic, and anti-inflammatory activities (Warrier et al. 1995; Ntalli et al. 2010; Vishnukanta 2010).
Traditional Uses
Plant is used as a source of furniture, timber, agricultural implements, and sports equipment. Extracts from different parts of plant are used in Ayurvedic remedies for common colds, headaches, stomach disorders, inflammation, diabetes, various forms of poisoning and malaria. Extracts from leaf, seed, and bark possess a wide spectrum of antibacterial action (Mahafuzul et al. 2007).
Essential oil extracted from leaves is used as an antiseptic to cure diseases like eczema, cancer (Ahana 2005), leprosy (Rai et al. 2011), chickenpox (Chakrabarty et al. 2012), tuberculosis (Ramya et al. 2009), and to inhibit acne development (Jain and Basal 2003). Seed oil is used as an antidiabetic (Vijayanandand and Wesely 2011), and bark has antifertility activity (Abdel-Shafy and Zayed 2002) and blood purifying properties (Hirt and Pia 2001).
Young branches of plants are effective against dental problems (Ismail et al. 2010). Stem bark is also reported to have metabolites with anticancer and antiviral properties and is useful against cold, fever, thirst, and skin diseases.
## 4.9 Podocarpus macrophyllus Thunb
* Vernacular name: Star-pine
* Family: Podocarpaceae
Ecological Distribution and Morphological Characteristics
Podocarpus is a large genus of over 100 species which are distributed in Asia, Africa, Australia, Central and South America. P. macrophyllous is a medium-sized evergreen tree native to China and Japan. Leaves are long and star shaped (Fig. 4.8a, b). Cones are solitary and comprise cone scales having one or more seeds.
Fig. 4.8
(a, b) Podocarpus macrophyllus (star-pine) possesses antitumor, antimicrobial, anti-inflammatory, antioxidants, and larvicidal activities (a) Tree (b) Leaves
Important Phytochemicals and Medicinal Value
Phytochemical elucidated various terpenoids and dilactones. Taxol and totarol are isolated from Podocarpus and commercially produced as a potent antibacterial and antioxidant agents. Flavonoids and podolactones play an important role in mediating cytotoxic activities against cancer (Park et al. 2004; Kuo et al. 2008).
In vitro and in vivo studies showed antitumor, antimicrobial, anti-inflammatory, antioxidant, and larvicidal activities (Abdillahi et al. 2010). Some novel compounds are isolated from methanolic bark extract and their structures are elucidated through spectroscopic method which include diterpenes, inumakiols A, B, C, H, totarol type diterpenes, and abietane. Some of them are known to possess antibacterial activities against pathogenic microorganism (Sato et al. 2008). Strong inhibitory activity is reported against B. subtilis, S. aureus, E. coli, K. pneumoniae, and C. albicans (Abdillahi et al. 2008). Totarals isolated from stem bark possess antibacterial activities against β-lactam-resistant strains of bacteria (Moorhead and Bigwood 2003).
Traditional Uses
P. macrophyllus is used in the treatment of fevers, asthma, coughs, and cholera. It is also used as a source of timber and wax. Fruits of Podocarpus are used for jams and preservatives. Decoction of fruit is also used as a tonic for heart and kidney problems. However, decoction from leaves is used for treatment of arthritis and rheumatism (Johnson 1999). Cones are edible parts of plant. In South Africa, bark and sap of Podocarpus spp. are used by Zulu and woodmen to treat chest problems and as a herbal remedy for gallsickness (Hutchings et al. 1996).
## 4.10 Polyalthia longifolia Sonn.
* Vernacular names: Indian fir tree, false ashoka
* Family: Annonaceae
Ecological Distribution and Morphological Characteristics
It is a tall and evergreen tree which is native to Southeast Asia. It is also known to reduce the level of noise pollution. Plant has a conical crown which has many short branches. Leaves are long, tapering, narrow, and drooping with wavy margins. Leaves are narrow, lanceolate, quite glabrous, faintly gland dotted with wavy margins (Fig. 4.9a, b). Flowers are star shaped and yellowish green, and form very short umbles. Flowering lasts only for few weeks.
Fig. 4.9
(a, b) Flavonoids and tannins in Polyalthia longifolia (false ashoka) are responsible for its antibacterial activity (a) Tree (b) Leaves
Important Phytochemicals and Medicinal Value
Phytochemical study of P. longifolia leaves indicated the presence of tannins, phenolic acids, glycosides, and steroids. Important phytochemicals include asclerodanediterpenes, polyalthialdoic acid, kolavenic acid, liriodenine, noroliveroline, oliveroline-β-N-oxide, and azafluorenealkaloids (Ghosh et al. 2008; Katkar et al. 2010).
Extracts and compounds isolated from plants have revealed promising activity against bacteria and fungi. Presence of flavonoids and tannins in P. longifolia are responsible for its antibacterial activity (Phadnis et al. 1988; Kokate et al. 2006; Paradhan et al. 2007; Jothy et al. 2013). Bark extract possesses alkaloids with antibacterial and antifungal activities (Hasan et al. 1988). Acetone, methanolic, and 1,4-dioxan extracts of leaves showed antimicrobial activities against more than 91 clinically important strains of microorganisms (Chanda and Nair 2010). Solvent extracts of leaves showed inhibitory activity against human pathogenic yeasts, C. albicans, C. neoformans, T. beigelli, and A. candidus due to the presence of compounds like clerodanediterpenes (Nair and Chanda 2006; Sashidhara et al. 2009). Alcoholic extract of leaves proved to be inhibitory against F. solani (Ramteke and Kamble 2011) while aqueous extract showed inhibition against aflatoxin producing A. parasiticus (Rajani et al. 2012).
Unripe extract of pericarp is reported to have strong inhibitory activities against bacteria and fungi due to the presence of compounds like flavonoids, tannins, steroids, and glycosides (Manasa et al. 2014).
Traditional Uses
The wood is tough and flexible. In South India, it is used for making drums. In China, it is used for matches. Tree is planted for its dense shade and elegant appearance. The fruits in times of scarcity are eaten by humans and all times by birds or monkeys. Leaves are used for ornamental decoration in festivals. The tree can be cut into various shapes and maintained in required sizes. Recently, plant is reported to absorb Cr, and its potential as an hyperaccumulator is being explored to be used in phytoremediation. Bark is used as a febrifuge in village areas of India (Kritikar and Basu 1993). Leaves are used to treat fever, uterus ailments, mouth ulcer, and heart problems. Stem bark is used for diabetes and hypertension in West Bengal.
## 4.11 Toona ciliata M. Roem (Syn: Cedrela toona)
* Vernacular names: Toon tree, red cedar
* Family: Meliaceae
Ecological Distribution and Morphological Characteristics
It grows as a medium-sized deciduous tree in southern Asia and Australia. Leaves are compound, lanceolate, acute having the entire margin, and crowded at the end of branchlets. Flowers are small, white, in drooping panicles (Fig. 4.10a–d). Capsules are oblong and become dark-brown on maturity. Bark is fragrant; therefore, plant is named as "cedrela" from Latin "cedrus," the cedar.
Fig. 4.10
(a–d) Ethyl acetate extract of Toona ciliata (red cedar) is reported to have antibacterial activity against S. typhi and S. epidermidis while methanolic extract is significant against K. pneumoniae, S. typhi, S. aureus, and S. epidermidis (a) Tree (b) Leaves (c) Flowers (d) Fruits
Important Phytochemicals and Medicinal Value
Seeds are source of 12-deacetoxytoonacilin and 6α-acetoxy-14α,15α-epoxyazadirone. Petroleum extract of plant revealed the presence of C-methyl coumarins, 12-α-Hydroxystigmast-4-en-3-one, and steroids (Chowdhury et al. 2002). Leaves are known to comprise norlimonoids and limonoids from leaves and stem (Liao et al. 2007; Chen et al. 2009) whereas bark is a source of protolimonoids and norlimonoids (Wang et al. 2011). Bark of tree comprises sesquiterpenes, cycloartanes, stigmasterol, campestrol, sitostenona, limonoids, apotirucallane, tirucallanes, and catechin (Das et al. 1999). Flowers contain nyctanthin and quercetin.
Ethyl acetate extract of plant showed antibacterial activity against S. typhi and S. epidermidis while methanolic extract is significant against K. pneumoniae, S. typhi, S. aureus, and S. epidermidis (Kavitha and Satish 2013). Plant is also known to have gastroprotective activity (Malairjan et al. 2006).
Traditional Uses
Plant is used traditionally for curing illness and dye preparation. Bark is used as astringent, expectorant, and antipyretic. Flowers are used as emmenagogue.
## 4.12 Tecoma stans L Juss. Ex Kunth Syn (Bigonia Stans)
* Vernacular names: Yellow elder, yellow trumpet bush
* Family: Bignoniaceae
Ecological Distribution and Morphological Characteristics
T. stans is native to America and the national flower of the US Virgin Islands and Bahamas. It is also found in many tropical and subtropical countries.
It grows as a small tree up to 5–9 m tall. Leaves are lanceolate, opposite, imparipinnate having 2–5 pairs of leaflets with a large single leaflet. Flowers are yellow, appear in clusters, trumpet shaped having round lobes. Fruits are narrow elongated pods, having winged seeds, green when young but turn brown upon maturity (Fig. 4.11a–d).
Fig. 4.11
(a–d) Ethanolic extract of Tecoma stans (yellow elder) flowers possesses antimicrobial activities against B. subtilis, E. coli, S. aureus, P. mirabilis, K. pneumoniae, and antifungal activities against P. spp. (a) Tree (b) Flower (c, d) Young and mature pods
Important Phytochemicals and Medicinal Value
Phytochemical screening showed presence of tannins, flavonoids, alkaloids, cardiac glycosides, and saponins (Rajamurugan et al. 2013) which increase wound healing potential of plant.
Ethanolic extract of flowers possesses antimicrobial activities against B. subtilis, E. coli, S. aureus, P. mirabilis, K. pneumoniae and antifungal activities against Penicillium spp. Tecomanine, tecostanine and chlorogenic acid (CGA) which is a phenylpropanoid all possess antidiabetic effects (Hammouda and Khalafallah 1971; Costantino et al. 2003). Leaves contain glycosides, iridoids, indolic compounds, and phenylethanoid, 2-(3,4-dihydroxyphenyl)ethyl-2-O-[6-deoxy-alpha-L-mannopyranosyl-4-(3, 4 dihydroxyphenyl)-2-propenoate]-β-D-glucopyranoside. Fruits and flowers contain monoterpene alkaloid, 5-hydroxy-skytanthine hydrochloride, along with many other compounds (Marzouk et al. 2006). Aqueous extract of leaves controls diabetes through stimulation of glucose uptake in insulin-sensitive and insulin-insensitive human adipocytes without any antiadipogenic effects. Plant also exhibits antifungal activities against all species of Aspergillus and Alternaria.
Fruits and flowers possess phenylethanoid, 2-(3,4-dihydroxyphenyl)ethyl-2-O-[6-deoxy-alpha-Lmannopyranosyl-4-(3,4-dihydroxyphenyl)-2-propenoate]-β-D-glucopyranoside, 5-hydroxy-skytanthine hydrochloride, along with 11 known compounds; 4-O-E-caffeoyl-alpha-Lrhamnopyranosyl-(1′,3-alpha) β-D-glucopyranose, E/Z-acetoside, isoacetoside, rutin, luteolin 7-Oβ-D-neohespridoside, luteolin 7-O-β-D-glucopyranoside, and sucrose were isolated from the fruits, while luteolin 7-O-β-D-glucuronopyranoside, diosmetin 7-O-β-D-glucuronopyranoside, diosmetin 7-O-β-D-glucopyranoside, diosmetin 7-O-β-D-glucuronopyranoside methyl ester, and acetoside as revealed through phytochemical screening. Fruit extract also possesses antioxidant activities.
Traditional Uses
Leaves are used in Mexico and Central America for treatment of diabetes and for problems related to urinary disorders (Shapiro and Gong 2002). Roots are used as diuretic and vermifuge (Khare 2007). Leaves, bark, and root are used as muscle relaxant and heart tonic. Leaves are used traditionally in Mexico to control diabetes. Flowers and leaves are used for treatment of cancer. Plant is used as herbal medicine for treating diabetes and digestive problems, and used as a vermifuge (Roman-Romas et al. 1991).
## 4.13 Terminalia mantaly L.
* Vernacular names: Terminalia, umbrella tree
* Family: Combretaceae
Ecological Distribution and Morphological Characteristics
The generic name comes from the Latin terminalis (ending), and refers to the habit of the crowded leaves present at the ends of the shoots. It grows up to 10–20 m, with an erect stem having conspicuously layered branches. Leaves are bright green when young and born in terminal rosettes of 4–9 unequal leaves on short, and thickened stem (Fig. 4.12a–c). Flowers are small, greenish, produced in erect spikes up to 5 cm long. Fruits are oval, with seeds of about 1.5 cm long.
Fig. 4.12
(a–c) Terminalia mantaly (umbrella tree) possesses antibacterial activity against strains of P. aeruginosa (a) Tree (b) Leaves (c) Fruit
Important Phytochemicals and Medicinal Value
Plant is used in the treatment of many fungal infections (Baba-Moussa et al. 1999). In fact, in traditional medicine it is used to treat arterial hypertension and gastroenteritis (Ouattra et al. 2007; Neofytos et al. 2009).
T. mantaly is also reported to have antibacterial activity against strains of P. aeruginosa (Ackah et al. 2014).
Traditional Uses
It is used as a forage and fuel as well as a source of timber. Bark and wood are used for dyeing due to the presence of tannins.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_5
# 5. Woody Plants with Possible Anti-HIV Activity
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
Many natural compounds produced within plants like alkaloids, flavonoids, and terpenoids are reported to have possible anti-HIV potential which can target reverse transcriptase or protease and inhibit replication. Many flavonoids like swertifrancheside, glycyrrhizin, and chrysin are known to be effective against HIV infection. Plants like Acer okamotoanum, Aegle marmelos, Argemone mexicana, Andrographis paniculata, Asparagus racemosus, Azadirachta indica, Bulbine alooides, Canna indica, Coleus forskohlii, Garcinia speciosa, Geum japonicum, Humulus lupulus, Magnolia spp., Musa acuminata, Ocimum sanctum, Polyalthia suberosa, Rubia cordifolia, Syzygium cumini, Terminalia chebula, and Tinospora cordifolia revealed promising anti-HIV activities, and their potential against AIDS is being explored. Antiviral activities of different trees are described, which include Artocarpus integrifolia, Aegle marmelos, Caesalpinia pulcherrima, Gleditsia triacanthos, Euphorbia royleana, Jatropha curcas, Heterophragma adenophyllum, Mimusops elengi, Platanus orientalis, Syzygium cumini, and Tamarix aphylla.
## 5.1 Introduction
Over 35 million people worldwide are suffering from AIDS, 40% of whom are between the ages 15 and 24. The US Federal Drug Administration has approved 25 drugs for the treatment of AIDS, but they are not derived from natural products. The World Health Organization (WHO) suggests a need to evaluate ethnomedicines in order to control AIDS.
Many natural compounds produced within plants like alkaloids, flavonoids, and terpenoids are reported to have possible anti-HIV potential which can target reverse transcriptase or protease and inhibit the replication of virus. Therefore, research is being conducted to isolate novel compounds from plants with possible anti-HIV activity, but till date although some research reports isolation of anti-HIV compounds from some medicinal plants, no drug is made for HIV treatments which is derived from natural products.
## 5.2 Anti-HIV Compounds
Many natural products are reported to have anti-HIV activities. Different compounds are isolated from many plants, but forming and introducing drugs from these plant-based products is still a challenge in medical science. Many trees possess compounds which possess anti-HIV activity. The active principle against HIV isolated from Rhus succedanea includes biflavonoids, while michellamines are isolated from Ancistrocladus korupensis, suksdorfin from Lomatium suksdorfii, caffeic acid from Arnebia euchroma, oleanolic acid from Xanthoceras sorbifolia, and nigranoic acid from Schisandra sphaerandra. Solamargine isolated from Solanum khasianum is also reported to be useful against HIV infection. Flavonoids like swertifrancheside, glycyrrhizin, and chrysin are also known to be effective against HIV infection (Fig. 5.1a–f).
Fig. 5.1
(a–f) (a–c) Flavonoids of Couroupita guianensis, commonly known as cannonball tree (due to its large fruits), are reported to inhibit HIV-infected cell cultures. (a) Tree bearing flowers; (b) close view of flower; (c) fruit of cannonball tree; (d) extract of Terminalia bellerica (bahera) fruit rind contains termilignan, thannilignan, 7-hydroxy-3′,4′-(methylenedioxy)flavan, and anolignan B, which contribute to its anti-HIV-1, antimalarial, and antifungal activities; (e) leaves of T. bellerica; (f) Phoenix dactylifera is an important palm tree with antiviral and possible anti-HIV activities
## 5.3 Trees with Possible Anti-HIV Potential
This section explains anti-HIV activities of important trees with their phytochemicals.
## 5.4 Artocarpus integrifolia L. (Syn: Artocarpus heterophyllus Lam.)
* Vernacular names: Jackfruit, kathal
* Family: Moraceae
Ecological Distribution and Morphological Characteristics
A. heterophyllus is native to South Asia, including Nepal, Bangladesh, Pakistan, and Sri Lanka. It is the national fruit of Bangladesh and Indonesia. It is cultivated for its commercial value in South Asia, Southeast Asia, and northern Australia. It grows as an evergreen, perennial, medium-sized tree having many branches. Mature tree has tubular roots, leaves are spirally arranged, petiolate, having elliptic to ovate blades. It is a monoecious plant, having male and female spikes on the same plant. Male flowers are small and green in color, while female flowers are large and rounded. Compound or multiple fruit is large, ellipsoid to globose, almost 4–5 kg, having an yellowish brown external rind with hexagonal tubercles (Fig. 5.2a, b).
Fig. 5.2
(a, b) Anti-HIV activity of Artocarpus integrifolia (jackfruit) is due to the presence of lectins. Jacalin can block HIV-1 infection of CD4+ lymphoblastoid cells and also exhibits antiproliferative activity on human cancer cell line. (a) Tree, (b) fruit
Important Phytochemicals and Medicinal Value
The fruits and seeds are rich in many micronutrients. Many flavones are isolated from the wood, comprising morin, dihydromorins, cynomacurins, artocarpin, isotocarpin, and artocarpetin. The bark contains compounds like cyclomorusin, cycloartocarpin, brosimone, and β-sitosterol, while the fruit possesses artocarpesin, Norartocarpetin, and oxyresveratrol.
A. heterophyllus possesses antimicrobial, antidiabetic, antioxidant, and immunomodulatory properties. Prenylated flavonoids isolated from the plant have strong inhibitory effects on lipid peroxidation and are potent antioxidants. The methanolic extract of the bark has anti-inflammatory effects on carrageenan-induced models in albino rats. The plant is also known to have skin-whitening activity due to the flavonoid artocarpanone isolated from the wood. It has inhibitory effects on mushroom tyrosinase activity and melanin production in B16 melanoma cells (Arung et al. 2006). The methanolic extract of different parts of the plant is known to have antibacterial activities (Khan et al. 2003).
Anti-HIV activity of the plant is reported due to the presence of lectins like jacalin, which interacts with the lymphocyte cell surface molecule CD4, which is a receptor of HIV-1 (Corbeau et al. 1994; Silprasit et al. 2011; Akkouh et al. 2015). Jacalin is also known to block HIV-1 infection of CD4+ lymphoblastoid cells. Further, jackfruit lectin has inhibitory activity in vitro toward herpes simplex virus type 2 (HSV-2), varicella zoster virus (VZV), and cytomegalovirus (CMW) (Wetprasit et al. 2000). Jacalin can also exert antiproliferative activity on human cancer cell line.
Traditional Uses
The perianths, seeds, and immature and ripe fruits are eaten. The seeds, also known as jack nuts, are nutritive and eaten raw, roasted, or boiled (Lim 2012). Different parts of the plant are used to treat fever, boils, wounds, skin diseases, ophthalmic diseases, and snakebites (Prakash et al. 2009). The root is antiasthmatic, and the decoction of the root is used in diarrhea and fever. The wood is sedative, and its pith is an abortifacient. Roasted seeds are known to have aphrodisiac properties.
## 5.5 Aegle marmelos L.
* Vernacular name: Bael
* Family: Rutaceae
Ecological Distribution and Morphological Characteristics
A. marmelos is a perennial gum-producing tree. It is native to Southeast Asia and found in the sub-Himalayan tract. It is used in folk medicines and considered to be a sacred tree in India. The leaves are alternate, compound, and trifoliate. The flowers are perfect, greenish white, aromatic, and appear in April–May. The fruit is edible and produced a year after flowering (Fig. 5.3a–d).
Fig. 5.3
(a–d) Fruit of Aegle marmelos (bael) possesses antioxidants like butylated hydroxy anisole, butylated hydroxy toluene, tertiary butylated hydroxy quinone, and gallic acid esters, which make it an important antibacterial and antiviral plant. (a) Tree, (b) leaves, (c) spiny branches, (d) fruits
Important Phytochemicals and Medicinal Value
The plant is reported to produce antioxidants like butylated hydroxy anisole (BHA), butylated hydroxy toluene (BHT), tertiary butylated hydroxy quinone, and gallic acid esters, which are being evaluated for their anticancer properties.
The methanolic extract of the fruits revealed the presence of furanocoumarins like imperatorin and xanthotoxol, which can inhibit the replication of HIV, suggesting their role as possible anti-HIV compounds (Sabde et al. 2011). Imperatorin showed 60% of inhibition at a dose of 8 μg/ml through the inhibition of cyclin D1 expression and arresting the cells at the G1 phase of cell cycle (Sancho et al. 2004).
Several pharmacological activities reported include anti-inflammatory, antipyretic, analgesic (Arul et al. 2005), antioxidant (Upadhya et al. 2004), and antidiabetic (Sabu and Kuttan 2004) activities. The fruit also possesses a wide range of therapeutic effects like free radical scavenging, antioxidant, antibacterial, and antiviral activities (Nugroho et al. 2011).
Traditional Uses
It is a source of timber and fuel. The fruit pulp is used to prepare sweets like murabba, puddings, and juice. The extract of the fruit pulp is found to be a good natural antioxidant (Vilioglu et al. 1998; Rajani et al. 2011). The leaves are consumed as dietary supplement due to the presence of an alkaloid, aegeline.
## 5.6 Caesalpinia pulcherrima L.
* Vernacular names: Poinciana, peacock flower
* Family: Leguminosae (Fabaceae)
Ecological Distribution and Morphological Characteristics
It is an evergreen shrub which grows up to 3 m tall. C. pulcherrima is mainly cultivated for its ornamental value in tropical gardens.
Leaves are bipinnate, 20–40 cm long, having three to ten pairs of pinnae, each with six to ten pairs of leaflets which are 15–25 mm long and 10–15 mm broad with oblong to ovate shape. The flowers are bright red in color, and flowering takes place in late summer (Fig. 5.4).
Fig. 5.4
Leaves of Caesalpinia pulcherrima (poinciana) possess antiviral, antipyretic, and aphrodisiac activities
Important Phytochemicals and Medicinal Value
The phytochemical screening of the plant showed the presence of many compounds like alkaloids, glycosides, phenolics, tannins, and flavonoids. The leaves are used for antipyretic, antimicrobial (Sudhakar et al. 2006), antibacterial (Pawar et al. 2007), antioxidant (Pawar et al. 2009), and antitubercular activities (Rao et al. 2005). Leaf extracts revealed gastric antiulcer activity, while the flowers have analgesic and anti-inflammatory activities due to presence of compounds like lupeol, myricetin, and quercetin (Duke 1992; Patel et al. 2010).
The plant is also reported to have antiherpes virus (HSV-1, HSV-2) activity due to the presence of flavonoids like quercetins (Chiang et al. 2003).
Traditional Uses
C. pulcherrima is used for various purposes in herbal medicine. It is used as an emmenagogue, stimulant, and abortifacient. The plant is used in bronchitis, asthma, and malarial fever. The different parts of the plant are used for the treatment of many disorders like pyrexia, menoxenia, and wheezing (Chiu and Chang 1992). The seeds become poisonous upon maturity.
## 5.7 Gleditsia triacanthos Linn.
* Vernacular names: Honey locust, thorny locust
* Family: Fabaceae
Ecological Distribution and Morphological Characteristics
It is native to North America and Asia but widely grown in many countries for its economic, medicinal, and ornamental values. Almost 14 species of the genus Gleditsia are reported. It is adapted to semiarid and subtropical climates.
It is a large deciduous tree with a spreading crown. The stem is thorny which may form a cluster (Fig. 5.5a–c). The leaves are bipinnately compound. The flowers are aromatic and appear in clusters from the base of leaf axil. Flowering takes place in May–June. The fruit is a legume up to 50 cm in length. The fruit pulp is edible. The seed ripens in September–October.
Fig. 5.5
(a–c) Medicinal importance of Gleditsia triacanthos (honey locust) is due to alkaloids, flavonoids, sterols, terpenoids, and phenolic compounds, which possess anti-HIV, antifungal, and antihyperlipidemic activities. (a) Tree, (b) leaves, (c) thorny bark
Important Phytochemicals and Medicinal Value
G. triacanthos is a rich source of alkaloids, flavonoids (Li et al. 2005; Zhang et al. 2016), sterols, terpenoids, and phenolic compounds which possess antitumor, antiallergic, antipyretic, anti-inflammatory, anti-HIV, antibacterial, antifungal, and antihyperlipidemic activities (Zhou et al. 2007; Ragab et al. 2010; El-sayed et al. 2013). Over 60 compounds of pharmacological importance are isolated from different species (Zhang et al. 2016). Eight glycosides, including six flavonoid glycosides, are isolated from the plant, i.e., vicenin-II, lucenin-I, isoorientin, orientin, vitexin, isovitexin, and aglycones like luteolin and apigenin (Mohamed et al. 2013). The leaves are a source of alkaloid tricanthine as revealed through Gas chromatography mass spectrometry (GC-MS). Liquid chromatography diode array detector method is used for the identification and characterization of flavonoids from the plant. This method is useful for the isolation of flavonoids from Gleditsia spp.
The presence of triterpene saponin C, triacanthosides, and lupane acid in the fruits of Gleditsia spp. is reported to have significant anti-HIV activity (Konoshima et al. 1995; Li et al. 2007). Analgesic property of the plant is reported in the methanolic extract (Saleh et al. 2015). Total ethanol hexane extract has potential cytotoxic and anticancer activities against larynx, breast, cervix, liver, and colon cancer cell lines, which justifies the use of the plant by folklore (Mohammed et al. 2014).
Traditional Uses
The plant is known as honey locust due to its sweet-tasting pulp, which, however, is not as sweet as honey. It was used by native Americans to make beer. The plant is a source of fodder, timber, and furniture. The seeds are a source of vitamins (A, B, and K), proteins, palmitate, oleic, stearic, and linoleic acids (Mariod and Matthaus 2008). The seed oil is a rich source of phosphatidylcholine, phosphatidylinositol, and phosphatidic acid. Young seeds taste like green peas.
Gleditsia spp. are used traditionally in China for treating measles, indigestion, whooping cough, smallpox, constipation, and diarrhea (Zhang et al. 2016). An infusion is made from the bark and used to treat dyspepsia. It is also used to treat cough, measles, and chickenpox. Tea is also made from the pods to treat symptoms of cold.
## 5.8 Euphorbia royleana Boiss
* Vernacular name: Sullu spurge
* Family: Euphorbiaceae
Ecological Distribution and Morphological Characteristics
The genus Euphorbia comprises more than 2000 species, many of which are medicinally important plants. E. royleana is commonly found in Asian countries, including Pakistan, India, and Nepal.
It is a succulent, prickly, and latex-producing cactus-like shrub which grows up to a height of 5 m. Stem is fleshy with winged ridges and spines on their margins. The leaves are fleshy, spoon shaped, and the flowers are yellowish green. The fruit is three lobed (Fig. 5.6a–c). The plant sheds its leaves in extreme cold or hot seasons.
Fig. 5.6
(a–c) Monocyclic terpenes from Euphorbia spp. are known to possess anti-HIV and anticancer activities. (a) E. royleana (sullu spurge) is known to contain diterpenoids with cytotoxic and antiviral activities, and antipyretic and analgesic importance; (b) leaves; (c) cyathia
Important Phytochemicals and Medicinal Value
Many monocyclic terpenes from Euphorbia spp. are known to possess antibacterial, anti-inflammatory, and anticancer activities. Many phytochemicals in the latex of Euphorbia species are responsible for anti-HIV activities (Gyuris et al. 2009; Upadhyay et al. 2010).
E. royleana is known to contain diterpenoids having cytotoxic (Fatope et al. 1996; Shi et al. 2005) and antiviral activities (Zheng et al. 1998). The latex contains metabolites of medicinal value, including ingenol, and ingol terpenoid derivative, epitaraxerol, ellagic acid, euphol, taraxerol, sitosterol, m-hydroxy benzoic acid, 7-hydroxy-3,4-benzcoumarin, 7-methoxy-3,4-benzcoumarin, and 2′,7-dihydroxy-3,4-benzcoumarin (Rastogi and Meharotra 1993; Tiwari et al. 2008).
Anti-inflammatory, piscicidal, molluscicidal (Abdel-Hamid 2003), insecticidal (Tiwari et al. 2004), antiacetylcholinesterase, antiarthritic, and immunosuppressive activities (Bani et al. 2005) of the stem extract are reported. Antioxidant and antitumor activities are reported from the hexane extract due to phenolic and flavonoid contents (Ashraf et al. 2015). Analgesic and antipyretic activities are reported in rabbits and rats (Bani et al. 1998).
Traditional Uses
The plant is used as a folk medicine in India and Turkey to cure skin disease, wounds, warts, migraines, and intestinal parasites. The latex is purgative in small doses; however, in higher doses, it is acrid. It is used traditionally for the treatment of paralysis, ear pain, and gastrointestinal disorders. Plant latex is used traditionally as cathartic and anthelmintic, for curing skin and eye infections, as well as for snakebites (Basak et al. 2009).
## 5.9 Jatropha curcas L.
* Vernacular names: Barbados nut, purging nut
* Family: Euphorbiaceae
Ecological Distribution and Morphological Characteristics
J. curcas is native to Mexico and America. It is a drought-resistant and perennial plant.
The leaves are alternate, petiolate, stipulate, five lobed and exhibit spiral phyllotaxy. The flowers are generally dioecious, and the fruit is an ellipsoid capsule (Fig. 5.7a, b).
Fig. 5.7
(a, b) Anti-HIV activity of Jatropha curcas (Barbados nuts) is due to the presence of compounds like 12-deoxyphorbol-13-phenylacetate, which inhibits HIV entry into target cells. (a) Tree, (b) leaves with fruit
Important Phytochemicals and Medicinal Value
Phytochemical screening of stem bark extracts revealed the presence of alkaloids, flavonoids, saponins, steroids, and tannins. Plant latex contains alkaloids like jatrophine, jatropham, and curcain, which have anticancer properties (Thomas et al. 2008). Steroidal compounds present in stem extracts are important due to their relationship with various anabolic hormones, including sex hormones (Okwu 2001). Flavonoids extracted from stem bark extracts possess antimicrobial, anti-inflammatory, anti-angiogenic, analgesic, antiallergic, and antioxidant properties (Hodek et al. 2002).
Recently, the aqueous and methanolic extracts of leaves showed anti-HIV activity, indicating Jatropha to be a good candidate for anti-HIV therapy due its antireverse transcriptase activity (Chinsembu and Hedimbi 2010; Dahake et al. 2013; Kaur and Kharb 2011). The aqueous extract of the branches is also reported to inhibit HIV-induced cytopathic effects with low toxicity (Igbinosa et al. 2009). The leaves are used in Tanzania to treat conditions related with AIDS such as skin rash and oral candidiasis (Park et al. 2009). The stem bark extract has also shown to inhibit HIV-induced cytopathic effects (Matsuse et al. 1999). Anti-HIV activity is due to the presence of compounds like 12-deoxyphorbol-13-phenylacetate, which inhibits HIV entry into target cells (Makkar and Becker 2009; Wender et al. 2008).
Traditional Uses
leaves are used as a remedy for jaundice. They contain apigenin, vitexin, and isovitexin, which enable them to be used against malaria and rheumatic and muscular pains (Thomas et al. 2008). The oil has a strong purgative action and is used for the treatment of eczema. It is also used to relieve pain and for making soaps. The seeds are poisonous, cathartic, and antimalarial, and the seed oil is massaged to treat rheumatism. The roots are also used to induce abortion (Quattrocchi 2012).
The plant is being used commercially for the production of biofuel (Thomas et al. 2008). Traditionally, it is used to cure diseases like cancer, paralysis, and dropsy. The paste of the root is used to treat inflammation in areas of Rajasthan.
## 5.10 Heterophragma adenophyllum Seem.
* Vernacular name: Katsagon
* Family: Bignoniaceae
Ecological Distribution and Morphological Characteristics
Heterophragma is a small genus of trees distributed in Southeast Asia and Africa.
The leaves are large and give beautiful shade in autumn. The margins of the leaves are wavy. The flowers are large, brownish yellow and appear in November. The fruits are large twisted pods which give an appearance of hanging snakes (Fig. 5.8a–f).
Fig. 5.8
(a–f) Leaf extracts of Heterophragma adenophyllum (katsagon) possess anti-HIV activities due to the presence of naphthoquinones. (a) Tree, (b) autumn tree, (c) tree in autumn, (d) leaves, (e) flowers, (f) pods
Important Phytochemicals and Medicinal Value
Phytochemical screening has revealed the presence of important metabolites like β-amyrin, β-sitosterol, tecomaquinone-I, tectol, lapachones like α-lapachone, β-lapachone, and dehydro-α-lapachone. In addition, many saponins, alkaloids, glycosides, and tannins are also reported from leaf extracts (Akhtar et al. 2012), which have antimicrobial activities. Dilapachone and adenophyllone are isolated from the heartwood, and their structures are elucidated through high-resolution mass, UV, 1D-, and 2D-NMR spectroscopies.
Antiviral and anti-HIV activities of the plant are reported to be due to the presence of novel compounds like naphthoquinones (Jassbi et al. 2004). Lapachones are considered to be a potent inhibitor of transcriptase activity of myeloblastosis virus and Rauscher murine leukemia virus.
Traditional Uses
The wood is used as a source of timber, fuel, and furniture (Sheikh 1993). The roots are used in various skin diseases. A paste of the leaves is used for the treatment of snakebites (Kalita et al. 2014).
## 5.11 Mimusops elengi L.
* Vernacular names: Bullet wood, Spanish cherry
* Family: Sapotaceae
Ecological Distribution and Morphological Characteristics
It is a medium-sized evergreen tree. It is native to Southeast Asia.
The leaves are arranged spirally, and are oval shaped, petiolate with caducous stipules, and acuminate at apex. The flowers are bisexual or functionally unisexual, aromatic, axillary, solitary, and yellow to white in color. The fruit is ovoid, fleshy, edible, and orange in color (Fig. 5.9a–c).
Fig. 5.9
(a–c) Anti-HIV activity of the fruits and seeds of Mimusops elengi (bullet wood) is due to compounds like mimusopsides A and B, mimusopins and mimusopsins. (a) Tree, (b) young fruits, (c) mature fruits
Important Phytochemicals and Medicinal Value
Commercially important compounds extracted from its bark are taraxerone, taraxerol, betulinic acid and spinasterol, sodium salt of betulinic acid and ursolic acid, and fatty acid esters of α-spinasterol. A new farnane-type pentacyclic triterpene, mimusopfarnanol, is isolated along with the known triterpenoids (Akhtar et al. 2009). A lupene-type triterpene, lupeol is also reported from the bark of tree (Jahann et al. 2001). The other major constituents are α-cadinol, tau-muurolol, hexadecanoic acid, diisobutyl phthalate, and octadecadienoic acid. New gallic acid esters are characterized as phenyl propyl gallate (Ruikar et al. 2009). The fruits and seeds are rich sources of triterpenoid saponins like mimusopsides A and B, mimusopins, and mimusopsins (Lim 2013).
The plant is also reported to have possible anti-HIV potential due to the presence of a triterpene, i.e., mimusopic acid, in the seeds (Sahu et al. 2001). The leaf extract is known to possess anti-HIV integrase activity and prevents conditions causing AIDS by inhibiting the activity of enzymes integrase, reverse transcriptase, and protease, which can allow the HIV virus to enter a host cell (Suedee et al. 2014).
Leaf-and-bark extract is reported to have antimicrobial activity against a wide range of bacteria and fungi (Ali et al. 2008; Satish et al. 2008; Shahwar and Raza 2009; Prabhat et al. 2010; Lalitha et al. 2011).
Traditional Uses
The bark is valuable for timber and is a source of fuel. It is used for cooling and as a cardiotonic, alexipharmic, anthelmintic, and astringent. It also cures diseases of the gums and teeth. The flowers are used to cure diseases of the blood and headaches, and is also effective against asthma. The fruit is used to make preservatives and pickles. The seeds yield oil, which is used for cooking purposes (Lim 2013). The powder of dried flowers is used as a brain tonic. Extract from root and fruits is aphrodisiac, diuretic, and astringent. The flowers are also used for making garlands, for stuffing pillows, and for making perfumes.
## 5.12 Platanus orientalis L.
* Vernacular names: Oriental plane, chinar
* Family: Platanaceae
Ecological Distribution and Morphological Characteristics
P. orientalis, also known as oriental plane or chinar, is a deciduous tree which is native to Southwest Asia. It is a deciduous tree with palmately lobed leaves which are alternately arranged. The monoecious flowers form dense spherical heads and are covered with hairs (Fig. 5.10a–c).
Fig. 5.10
(a–c) Antiviral activity of Platanus orientalis (oriental plane) is due to compounds like kaempferol, coumaroylglucopyranoside, tiliroside, kaempferol-3-rhamnopyranoside (afzelin), quercetin, nicotiflorin, rutin, and coumaric acid. (a) Tree, (b) leaves, (c) fruit
Important Phytochemicals and Medicinal Value
Phytochemical screening revealed the presence of many metabolites, like coumaryl rhamnopyranoside, platanoside, kaempferol, coumaroylglucopyranoside, tiliroside, kaempferol-3-rhamnopyranoside (afzelin), quercetin, nicotiflorin, rutin, and coumaric acid (Mitrokosta et al. 1993; Dimas et al. 2000; Natakani et al. 2000).
Isolation of betulinic acid and betulonic acid from the bark extract indicates the anti-HIV and anticancer activities of the plant (Bastos et al. 2007).
Traditional Uses
Chinar is an important tree in many cultures. The wood is used as a source of fuel and for making furniture. The leaves were used in folk medicine in ophthalmia, and the bark was boiled in vinegar and given for toothache. Plant leaves are used in Iranian folk and traditional medicines, for the treatment of dermatological, rheumatic, and inflammatory diseases.
## 5.13 Syzygium cumini L.
* Vernacular name: Jambul
* Family: Myrtaceae
Ecological Distribution and Morphological Characteristics
Syzygium is a large genus which comprises of over 1000 species. S. cumini is a tall tree that can live for more than 100 years. It is native to many regions of Southeast Asia, including Bangladesh, Burma, Nepal, Indonesia, and Pakistan. In India, it is planted near temples and considered to be sacred.
The leaves are simple and oval shaped, slightly pointed near the apices. They look red when young, but turn green upon maturity. The flowers are greenish white in color and fragrant. The fruits look like berries and are edible (Fig. 5.11a–f).
Fig. 5.11
(a–f) Fruit and seeds of Syzygium cumini (jambul) are rich in antioxidants, which have anticancer and anti-HIV properties. (a) Tree, (b) leaf, (c) flower, (d) young fruit, (e) developing fruit, (f) ripe fruits
Important Phytochemicals and Medicinal Value
The plant is rich in compounds containing flavonoids, anthocyanins, glucoside, ellagic acid, isoquercetin, kaempferol, and myricetin (Vaishnava and Gupta 1990; Timbola et al. 2002). The seeds contain alkaloid, jambosine, and glycoside jambolin or antimellin, which inhibit the diastatic conversion of starch into sugar. The seeds have been reported to be rich in flavonoids, which account for the scavenging of free radicals and protective effect on antioxidant enzymes (Ravi et al. 2004; Bajapi et al. 2005). The roots are rich in flavonoid glycosides and isorhamnetin (Veishnava et al. 1992).
Antidiabetic activity of the fruits, bark, and seed bark is reported in many experiments (Ravi et al. 2004; Schossler et al. 2004; Villasenor and Lamadrid 2006). The fruit and seeds are rich in antioxidants (Ravi et al. 2004) that have anticarcinogenic and anti-HIV properties (Kusumoto et al. 1995).
Traditional Uses
The wood is used as a source of fuel and for making furniture. It is an important tree for reforestation projects due to the high-quality wood. The peel powder is applied as a colorant in food and also in pharmaceuticals. Jaman vinegar is carminative and used as a diuretic (Noomrio and Dahot 1996).
## 5.14 Tamarix aphylla L.
* Vernacular name: Athel pine
* Family: Tamaricaceae
Ecological Distribution and Morphological Characteristics
The tree is native to Central Asia and North Africa. It is also planted in the sand dune area of Thal Desert.
T. aphylla has a rounded crown of many stout branches with drooping twigs. It reaches a height of 10–18 m and may attain a diameter of 0.8 m. The leaves are minute, gray-green, having long scales which overlap closely on the twigs (Fig. 5.12a, b). Flowers are small, pink in color, and arranged in spike-like racemes. The fruit is a small capsule with three valves. The plant flowers from spring through the summer.
Fig. 5.12
(a, b) Tamarix (salt cedar) spp. possess antiviral, antibacterial, and anticancer properties. (a) T. aphylla, (b) leaves
Important Phytochemicals and Medicinal Value
The leaves are rich in alkaloids, flavonoids, saponins, and polyphenols. In addition, gums, glycosides, anthraquinones, terpenoids, and tannins are also reported from different parts.
Tamarix spp. possess antibacterial, antiviral, antiallergic, antioxidant, antidiabetic, and anticancer properties as reported in different experiments (Djurdjevic et al. 2006; Mohammedi and Atik 2011; Drabu et al. 2012).
Traditional Uses
It is a valuable tree in arid areas, which is used in carpentry, agricultural implements, and shelterbelts.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_6
# 6. Trees with Hepatoprotective and Cardioprotective Activities
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
Many natural products like polyphenols and flavonoids from woody plants are known to maintain liver and heart health. These products have antihepatitis, anticirrhosis, and antiatherosclerosis with lipid-lowering activities. Trees included are Alstonia scholaris, Anogeissus acuminata, Crataeva religiosa, Carissa carandas, Cupressus sempervirens, Diospyros spp., Nerium oleander, Terminalia arjuna, and Thevetia peruviana.
## 6.1 Introduction
Phenolic compounds, flavonoids, and many anthocyanins which are present in fruits and vegetables possess cardioprotective and hepatoprotective activities. Effects of six anthocyanidins (cyanidin chloride, delphinidin chloride, malvidin chloride, pelargonidin chloride, peonidin chloride, and petunidin chloride) and seven anthocyanins (cyanidin 3-O-β-galactopyranoside chloride, cyanidin 3-O-β-glucopyranoside chloride, delphinidin 3-O-β-glucopyranoside chloride, malvidin 3-O-β-glucopyranoside chloride, pelargonidin 3-O-β-glucopyranoside chloride, peonidin 3-O-β-glucopyranoside chloride, and petunidin 3-O-β-glucopyranoside chloride) are reported to improve cell viability against DOX-induced cardiotoxicity by scavenging reactive oxygen species. Apigenins present in edible fruits like oranges, cherries, apples, and grapefruits, and avicularins from crabapple are also reported to have cardioprotective properties. Carotenoids present in edible fruits also provide protection against cardiovascular diseases by improving antioxidant defense and by maintaining the myocardial membrane. Many plants are known to possess cardioactive properties, which are discussed in Sect. 6.3 (Fig. 6.1a–c).
Fig. 6.1
(a–c) Petroleum ether extract of the stem bark of Millettia pinnata (Indian beech) is revealed to be effective against cardiomyopathy in diabetic rats due to presence of phytochemicals like kanugin, gamatay, glabrin, glabrosaponin, kaempferol, kankone, kanugin, karangin, neoglabrin, pinnatin, pongamol, pongapin, quercitin, saponin, β-sitosterol, and tannin. (a) Tree (b) Flower (c) Pods
Many phytochemicals from plants, such as alliin, coumarins, phenolic compounds, silymarin, β-sitosterol, betalain, neoandrographolide, phyllanthin, andrographolide, curcumin, picroside, hypophyllanthin, kutkoside, shallomin, quercetin, kaempferol, stigmasterol, lupeol, berberine, and glycyrrhizin, have been demonstrated to have potent hepatoprotective properties. Woody plants like Acacia caetchu, Aegle marmelos, Alstonia scholaris, Cassia fistula, Diospyros spp., Kigelia africana, Hamelia patens, Mallotus japonicus, Phyllanthus emblica, Prunus armeniaca, Spondias pinnata, Vitex trifolia, and Zanthoxylum armatum are known to possess hepatoprotective properties (Fig. 6.2a, b). The extract from many herbs is used as liver tonic which include many members of sunflower family and legume family like Calendula spp., Silybum marianum, and Glycyrrhiza glabra. Many edible fruits like Citrus paradisi, Vaccinium spp., and Vitis vinifera also possess hepatoprotective properties.
Fig. 6.2
(a, b) Hepatoprotective activity of Hamelia patens (firebush) is reported due to the presence of compounds like terpenoids, coumarins, sterols, and anthocyanins. (a) Bush (b) Flowers
## 6.2 An Account of Some Trees with Hepatoprotective and Cardioactive Activities
This section deals with hepatoprotective and cardioprotective activities of some trees.
## 6.3 Alstonia scholaris L. R. Br (Syn: Echites scholaris)
* Vernacular name: Blackboard Tree
* Family: Apocynaceae
Ecological Distribution and Morphological Characteristics
A. scholaris is a large deciduous tree native to India, Pakistan, Malaysia, and Australia.
Its leaves are petiolate, glossy, obovate, and occur in a whorl of 3–10 and rounded at apices. The bark is grayish-white in color having conspicuous lenticels and secretes milky sap. Flowers are small, greenish-white in color and arranged in an umbellate manner. They bloom in October and are aromatic at night time due to their nocturnal pollinators. Fruit is pendulous, dehiscent, and woody, bearing many oblong seeds of brown color (Fig. 6.3a–g).
Fig. 6.3
(a–g) Flowers of Alstonia scholaris (devil tree) contain many phenols, tannins, cardioactive glycosides picrinine, nareline and strictamine. Plant is known to possess hepatoprotective and cardioprotective activities. (a) Tree (b) Trunk (c, d) Old branches (e) Spiral leaves (f) Flower (g) Fruit
Important Phytochemicals and Medicinal Value
The main constituents of the plant include iridoids, coumarins, leucoanthocyanins, phenolics, saponins, and tannins. Many alkaloids of economic value are extracted from the stem bark, for example, echitamine, tubotaiwine. Rhazmanine has recently been isolated from the leaves (Khyade and Vaikos 2009). However, some alkaloids are also extracted from flowers, and they include compounds like picrinine, nareline, and strictamine. In addition, flowers also contain many phenols, tannins, cardioactive glycosides, and essential oils like linalool.
It is one of the most important medicinal trees and is reported to have a wide spectrum of biological activity (Meena et al. 2011). Alstonia is reported to possess analgesic, antioxidant, antimicrobial, immunostimulant, anticancer, and hepatoprotective effects (Khan et al. 2003; Shang et al. 2010; Sinnathambi et al. 2010, 2011). The bitter milky exudate is used for treating ulcers and wounds (Arulmozhi et al. 2007). The methanolic extract of the root bark is reported to have some cytotoxic activity against human lung cancer (Keawpradub et al. 1997).
Cardioprotective activity of the ethanolic extract of the plant is reported at doses of 200 and 400 mg/kg in isoproterenol-induced myocardial infarction in rats (Pullaiah et al. 2013) possibly by reducing the number of free radicals and reducing oxidative stress due to the presence of compounds like flavonoids, phenolics, and phytosterols. Hepatoprotective activity of the fruit of A. scholaris is reported in rats induced with CCl4 hepatotoxicity (Shankar et al. 2012). Hepatoprotective activity of the methanolic extract of the bark is also observed at 200 mg/kg in Swiss albino rats with CCl4-induced damaged liver (Husain et al. 2012).
Traditional Uses
A. scholaris is an important source of pulai timber. The wood is used for light indoor construction purposes and for pulp and paper production (Pratap et al. 2013). The wood of the plant has been used for school blackboards; therefore, the plant is named "scholaris." The bark is used for making fibers and paper. The latex is used for making high-quality chewing gum.
Wood charcoal is also used as gun powder. The bark of the plant is used in the treatment of malarial fever, abdominal disorders, and skin diseases.
## 6.4 Anogeissus acuminata (Roxb. ex DC.)
* Vernacular name: Axlewood
* Family: Combretaceae
Ecological Distribution and Morphological Characteristics
A. latifolia grows as a medium- to large-sized canopy tree widely distributed in Nepal, Sri Lanka, Pakistan, and India. Its leaves are opposite or alternate; young leaves are silvery and covered with long silky hairs (Fig. 6.4a, b).
Fig. 6.4
(a, b) Anogeissus spp. possess hepatoprotective, hypoglycemic, and antioxidant activities. Hydroalcoholic extract of A. latifolia (axlewood tree) is known to have hepatoprotective activities due to the presence of polyphenols and flavonoids. (a) A. acuminata (b) Leaves
Important Phytochemicals and Medicinal Value
The plant is rich in tannins and flavonoids (Navale and Paranjape 2016) and can inhibit activity of protein tyrosine phosphatase (PTP), which is a negative regulator of insulin. Phytochemistry of various parts of plants is therefore being evaluated in order to use plant for the treatment of diabetes due to its inhibitory role on PTP. The plant contains triterpenoids like 3-β-hydroxy-28-acetyltaraxaren and β-sitosterol (Rahman et al. 2007). The bark contains leucoanthocyanidin, ellagic acid, and two glycosides of ellagic and flavellagic acids, while the leaves contain gallotannins.
The hydroalcoholic extract of A. latifolia is known to have hepatoprotective activities due to the presence of polyphenols and flavonoids like quercetin, rutin, and gallic acid (Pradeep et al. 2009). Quercetin and rutin are potential therapeutic agents due to their effects on reducing oxidative DNA damage, lipid peroxidation, and quenching free radicals.
A. acuminata can be useful for the development of therapeutic drugs for long-term diabetes due to its hypoglycemic action and its antioxidant activity (Navale and Paranjape 2013; Govindarajan et al. 2005). The gum is also known to have hypolipidemic activities (Parvathi et al. 2009).
Traditional Uses
Many parts of the plant are used traditionally to treat diabetes and inflammatory conditions. The bark is used for the treatment of sores, boils, itching, cough, snake and scorpion bites, and stomach diseases (Singh et al. 2010). The bark decoction is used for the treatment of dysentery, and infusion is used for mouthwash, skin burns, and bruises. The bark juice is used on wounds, and crushed seeds are given to cattle to improve milk quality (Quattrocchi 2012).
## 6.5 Crataeva religiosa Forst f.
* Vernacular names: Sacred Garlic Pear, Temple Tree
* Family: Capparaceae
Ecological Distribution and Morphological Characteristics
C. religiosa grows as a branched deciduous tree. Its leaves are trifoliate, glabrous, and ovate. The flowers are whitish and milky-white in color in terminal dense corymb. The fruit is berry, or may become oblong with woody rind (Fig. 6.5a, b). It is distributed in Pakistan, India, Sri Lanka, Malaysia, and China.
Fig. 6.5
(a, b) Crataeva religiosa (sacred garlic pear) possesses cardioprotective, hepatoprotective, nephroprotective, rubifacient, contraceptive, and antipyretic activities due to lupeol, which is a hepatoprotective agent, and its effect is comparable with silymarin, which is a standard hepatoprotective drug. (a) Tree (b) Leaves
Important Phytochemicals and Medicinal Value
The bark contains compounds like lupeol, lupeol acetate, varunaol, spinasterol acetate, taraxasterol, 3-epilupeol, triterpene, diosgenin, cadabicine, and cadabcine acetate. The leaves showed the presence of compounds like dodecanoic anhydride, methyl pentacosanoate, Kaemferol-3-O-α-D-glucoside, querscitin-3-O-α-D-glucoside, isovitexin, proanthocyanidins, myricetin, phenolic acids, p-hydroxyl benzoic acid, vanilic acid, ferulic acid, and sinapic acid.
The plant is known to have antifungal activities against Candida spp. and Aspergillus spp. Lupeol and lupeol acetate from the stem bark possess anti-inflammatory, antinociceptive, and antipyretic activities.
Lupeol is also a hepatoprotective agent and its effect is comparable with silymarin, which is a standard hepatoprotective drug. The effect of lupeol is studied in rats with aflatoxin-induced peroxidative hepatic damage. Aflatoxin treatment caused hepatic damage by increasing the levels of lactate dehydrogenase, alkaline phosphatase, and alanine and aspartate aminotransferase with an increase in lipid peroxide level; however, oral treatment of lupeol normalized the mechanism of hepatic system (Jaikanth et al. 2014).
Traditional Uses
The plant is used as a diuretic, cardioprotective, hepatoprotective, nephroprotective, rubifacient, contraceptive, and antipyretic. The bark is used in the treatment of urinary disorders and kidney stones (Patil and Gaikwad 2011). Varunal is a decoction which is prepared with Eclipts, Picrorrhiza, Achillea, Cichorium, Solanum, Arjuna, and Cassia seeds for treating hepatitis, edema, and arthritis (Warrier 1997). The bark and leaves are applied in the form of poultice in rheumatism. Fruit is used as a spice due to its garlic-like taste (Seidmann 2005). In Tamil Nadu, leaves and the bark are used to treat jaundice, eczema, and rabies.
## 6.6 Carissa carandas L.
* Vernacular name: Bengal Currant
* Family: Apocynaceae
Ecological Distribution and Morphological Characteristics
C. carandas is native to Africa, Australia, and Asian countries. It is drought resistant and can survive in tropical and subtropical climates. It is widely cultivated due to its edible fruit with high medicinal value. Over 100 species of genus Carissa are reported.
It grows as a small tree which can attain a height of 10 m and bears thorns. The stem is rich in white milky latex. The leaves are small, thick, simple, oblong, conical, and lanceolate. White flowers possess five petals, are fragrant, and may be solitary or arranged in umbel or corymb. Flowering takes place in early spring and fruit maturation in May–June (Fig. 6.6a–h). Fruit is a round to oval berry produced in groups of 3–10 fruits which bear many seeds. It is green when young, but turns red upon maturity. They are poisonous when they are young and secrete latex.
Fig. 6.6
(a–h) Leaf extracts of Carissa spp. like C. carandas (Bengal currant) possess cardioprotective effects, while root extracts are reported to have hepatoprotective effects. Anticancer activity of the plant is reported against human breast carcinoma cell line due to the presence of orientin, isoquercetin, myricetin, and apigenin. (a) Tree (b) Flower (c–f) Developing fruit (g) C. grandiflora (h) Flower
Important Phytochemicals and Medicinal Value
The chlorophyll and aqueous extract of the fruit of another important species of Carissa, i.e., C. opaca, revealed the presence of phenolic and flavonoid contents with antioxidant activities. Anticancer activity of the plant is reported against human breast carcinoma cell line (Nisa et al. 2013) due to the presence of orientin, isoquercetin, myricetin, and apigenin, revealed through High-performance liquid chromatography (HPLC) (Sahreen et al. 2013). Phytochemical screening showed the presence of compounds like oleanolic acid, ursolic acid, stigmasterol, and beta-sitosterol, terpenes, tannins, and coumarins (Ya'u et al. 2008). Many phytochemicals contribute to gut-stimulatory activity of the plant through activation of muscarinic and histaminergic receptors (Mehmood et al. 2014).
Cardioprotective effects of the leaf extract of the plant are also reported against CCl4 (Sahreen et al. 2014). Hepatoprotective effects of the root extract of C. carandas are also reported (Hedge and Joshi 2009). Unripe fruit possesses antidiabetic activities (Itankar et al. 2011). The dried fruits of the plant are also known to have anti-inflammatory activity as revealed through Gas chromatography-mass spectrometry (GC-MS) (Anupama et al. 2014).
The plant is also known to possess antifungal activity against A. niger, A. flavus, and F. solani. Antibacterial activity is reported against B. subtilis, E. aerogenes, E. coli, K. pneumoniae, M. luteus, P. aeruginosa, S. typhi, and S. aureus (Sahreen et al. 2013).
Traditional Uses
The fruits, leaves, bark, and root of the plant are traditionally used in ethnomedicine systems, including Ayurveda, Unani, and Homeopathy, to cure diseases like anorexia, fever, mouth ulcer, sore throat, syphilitic pain, burning sensations, scabies, and epilepsy (Begum et al. 2013). Traditionally, the plant can also be used to cure cancer by applying a dressing of the plant extract to cancerous wounds. A decoction is also prepared by taking equal amounts of different parts of the plant and making a thin paste, by crushing them, and boiling it until half the amount of water is left, and then applied during the early stages of cancer. Oil is extracted from plant parts after boiling in mustard oil and then applied to wounds (Maheshwari et al. 2012). This paste is considered to be more effective than neem. The leaf decoction is used to treat fever, diarrhea, and earache. The stem is known to possess antipyretic activity and tendon-strengthening property. It is popular in the indigenous system of medicine and used in gut motility disorders like diarrhea and constipation (Mehmood et al. 2014). The plant wood is used to manufacture household utensils and spoons.
Berries are edible and used for cooking. They possess antimicrobial and antifungal activities and can be used to remove intestinal worms. They are used traditionally for treating asthma, hepatitis, and microbial infections. The plant is a good source of vitamin C and iron. It is a good appetizer and used in making chutneys, jams, and jellies due to its high pectin content. In Rajasthan, it is also cooked with green chilies and consumed with bread (Maheshwari et al. 2012). A refreshing red drink is prepared from the ripe fruit.
## 6.7 Cupresses sempervirens L.
* Vernacular name: Cypress
* Family: Cupressaceae
Ecological Distribution and Morphological Characteristics
C. sempervirens is a medium-sized conifer which is native to the Mediterranean region. The plant can survive up to a period of 500 years (A Guide to Medicinal Plants in North Africa 2005). Leaves are evergreen, acicular in young stages, and reduced to form scales. The female cones are globular, shiny, peltate, opposed crosswise on a short axis. The plant produces seed cones that are ovoid and narrowly winged (Fig. 6.7a, b).
Fig. 6.7
(a, b) Cupresses sempervirens (cypress) possesses cardioprotective and antidiabetic activities. Important phytochemicals include cosmosiin, cupressuflavone, amentoflavone, rutin, quercitrin, quercetin, and myricitrin. (a) Tree (b) Fruit
Important Phytochemicals and Medicinal Value
Three phenolic compounds, cosmosiin, caffeic acid, and p-coumaric acid, are isolated from the leaves of C. sempervirens. In addition to this, flavones like cupressuflavone, amentoflavone, rutin, quercitrin, quercetin, and myricitrin are also isolated from leaves (Ibrahim et al. 2007). Branchlets release an essential oil which contains biflavonoids, monoterpenic carbides, and sesquiterpenoids (Emami 2013).
The methanolic extract of the leaves revealed hepatoprotective activity in CCl4-treated rats with a significant decrease in glutamate oxaloacetate transaminase, glutamate pyruvate transaminase, cholesterol, and triglycerides levels and an increase in the total protein level (Ibrahim et al. 2007). Hepatoprotective activity of leaves is also reported in rats against paracetamol-induced liver injury (Ali et al. 2012).
Fruits of C. sempervirens var. horizontalis possess antioxidant and antiglycation properties. They are reported to have antidiabetic activity and are useful in treating cardiovascular complications. The essential oil is known to have antimicrobial activity against B. cereus, E. faceless, S. marcescens, S. aureus, and Gram-negative bacteria A. hydrophila, E. coli, K. pneumoniae, P. vulgaris, P. aeruginosa, and S. indica. However, the methanol extract strongly inhibited the growth of many bacteria (Ismail et al. 2013). Taxodione isolated from cones is reported to have anticancer activity (Tumen et al. 2012). The fruit extract is also known to possess antiherpes virus activity (Amouroux et al. 1998).
Traditional Uses
The cones and leaves are used internally as an astringent. The decoction of the cones and leaves is used in a sitz bath three times a day for 1 week for the treatment of hemorrhoids. The essential oil is used as an antiseptic and antispasmodic for cough.
## 6.8 Diospyros Spp.
* Vernacular names: Ebony, Persimmon Tree
* Family: Ebenaceae
Ecological Distribution and Morphological Characteristics
Over 20 species of Diospyros are reported which comprise mostly of deciduous trees and are used traditionally for several medicinal purposes. The most popular of them are D. cordifolia, D. kaki, D. melanoxylon, D. peregrina, and D. sylvatica (Fig. 6.8a–i).
Fig. 6.8
(a–i) Many species of Diospyros are known for their antimicrobial, anticancer, hepatoprotective, and cardioactive properties. (a) D. cordifolia (persimmon tree) with ripe fruits (b) Leaves (c, d) D. peregrina (e) Leaves (f) Flowers (g, h) Developing fruits (i) Seeds
Important Phytochemicals and Medicinal Value
The main constituents of medicinal importance comprise pentacyclic triterpenes and naphthoquinones (Kantamreddi and Wright 2008). Other compounds like α-amyrin, β-amyrin, β-sitosterol, hentriacontanol, lupeol, nentriacontane, taraxerol, and ursolic acid are isolated from the leaves of many Diospyros spp. Methylanthroquinone, plumbagin, diosindigo, diospyrin, isodiospyrin, and microphyllon are isolated from the root (Ganapaty et al. 2004), while lupeol, betulin, and betulinic acid are reported to be important constituents of the bark of many Diospyros spp. The fruits are rich sources of compounds like carotenoids, flavonoids, terpenoids, and tannins.
D. melanoxylon bark is reported to have hepatoprotective, wound-healing, analgesic, anti-inflammatory, and antitumor (Jain and DePhillips 1991) properties(Rath et al. 2009).
Traditional Uses
Many species of Diospyros, like D. heudelotii, D. lycioides, and D. tricolor, are known for their antimicrobial properties (Watson and Preedy 2008). Leaves are used as fish poison. The leaf extract is also applied over bone fractures (Quattrocchi 2012). The plant is used in Indian folk medicine for the treatment of liver disorders, leprosy, and wounds.
### 6.8.1 Diospyros cordifolia Roxb.
* Vernacular names: Indian Ebony, Persimmon Tree
* Family: Ebenaceae
Ecological Distribution and Morphological Characteristics
D. cordifolia Roxb. is a deciduous tree and used traditionally in the Indian subcontinent for several medicinal purposes (Fig. 6.8e–i).
Important Phytochemicals and Medicinal Value
Compounds like α-amyrin, β-amyrin, β-sitosterol, hentriacontanol, lupeol, nentriacontane, taraxerol, and ursolic acid are isolated from the leaves of D. cordifolia. Many species of Diospyros, like D. heudelotii, D. lycioides, and D. tricolor, are known for their antimicrobial properties (Watson and Preedy 2008).
D. melanoxylon bark is reported to have hepatoprotective, wound-healing, analgesic, and anti-inflammatory activities (Rath et al. 2009). Significant hepatoprotective activity of D. cordifolia is reported from the stem bark extract in male Wistar rats due to presence of compounds like ursolic acid, lupeol, and betulin (Mankani et al. 2006). Hepatoprotective activity of the ethanolic extract of D. melanoxylon leaves is also reported in CCl4-induced hepatotoxicity in albino rats (Patel et al. 2015).
Traditional Uses
The leaves are used as fish poison. The leaf extract is also applied over bone fractures (Quattrocchi 2012). The plant is used in Indian folk medicine for the treatment of liver disorders, leprosy, wounds, whooping cough, eye infections, abdominal pain, ulcers, and fever.
## 6.9 Nerium oleander L.
* Vernacular names: Oleander, Rose-bay
* Family: Apocynaceae
Ecological Distribution and Morphological Characteristics
Nerium is an evergreen perennial shrub, native to Southwestern Asia and is cultivated for its ornamental and medicinal importance.
Its leaves are thick, narrow, lanceolate with entire margins (Fig. 6.9a–c). The flowers are aromatic, pink to white in color and are formed in clusters from March–June. The fruit is a capsule and contains two fused follicles.
Fig. 6.9
(a–c) Nerium oleander (rose-bay) possesses important cardioactive glycosides, neriin and an alkaloid, oleandrin. Oleandrin and its aglycone oleandrigenin are shown to have anticancer properties. (a) Bush (b) Flowers (c) Flowers of white oleander
Important Phytochemicals and Medicinal Value
All parts of the plant are poisonous, even the honey produced from the nectar is toxic. The most well-known effects of oleander are due to two cardioactive glycosides, neriin and, an alkaloid, oleandrin. Other glycosides like gentiobiosyl-oleandrin, gentiobiosyl-nerigoside, and gentiobiosyl-beaumontoside are also extracted from the leaves (Duke 1985).
Many cardenolides (heterosides of uzarigenine) and inactive cardenolides (heteroside of adynergenine, of digitalose), triterpenoids, a resin, tannins, glucose, a paraffin, ursolic acid, vitamin C, and an essential oil make important constituents of this plant. Oleandrin and its aglycone oleandrigenin are the active compounds, which are shown to have anticancer properties. Anvirzel has also revealed cytotoxicity in human tumor cell lines, with evidence of apoptosis as a principal mode of cell death (A Guide to Medicinal Plants in North Africa 2005).
Cardioprotective activity of the hydroalcoholic extract of the flowers is reported against isoproterenol-induced myocardial oxidative stress in rats (Gayathri et al. 2011). Hepatoprotective and cardioactive activities of the methanolic extract of the flowers are observed at doses of 100, 200, and 400 mg/kg in CCl4-induced liver injury in rats (Singhal and Gupta 2012).
Traditional Uses
Oleander extracts have anticancer properties. They are diuretic and lenitive on dermatosis and contusion. In addition, the plant's lymph is rich in minerals and α-tocopherol, which is an important antioxidant. However, Nerium is also a poisonous plant and an overdose could result in heart failure and abortion (Shaw and Pearn 1997). The leaves and seeds can cause nausea and mental confusion. The plant is also used indigenously as a heart tonic, diuretic, and for treatment of epilepsy and skin conditions.
## 6.10 Terminalia arjuna Roxb.
* Vernacular name: Arjun Tree
* Family: Combretaceae
Ecological Distribution and Morphological Characteristics
T. arjuna is native to Southeast Asian countries. The plant is commonly found growing along riverbanks as a shade and ornamental tree; it also has high medicinal importance.
It grows as a tall evergreen tree with a spreading crown and drooping branches, reaching 80-feet height. The leaves are oblong and conical shaped. The flowers are yellow and appear in March and June. The fruit is five-winged, green in early stages and becomes brown on maturity (Fig. 6.10a–i). Fruit formation starts in late summer.
Fig. 6.10
(a–i) Terminalia arjuna (arjun tree) possesses important cardioprotective properties comparable to fluvastatin against many cardiovascular disorders which are attributed to presence of triterpenoids and flavonoids. Cardioactive role of the plant is regulated through alternation of thyroid hormones. (a) Tree (b) Tree in autumn (c) Leaves bearing flowers (d) Young flowers (e) Mature flowers (f) Branch bearing young green fruits (g) Mature fruits (h) Winged fruits on ground (i) Winter tree with fruits
Important Phytochemicals and Medicinal Value
The plant possesses a wide range of pharmacological activities, including cardioprotective, antioxidant, antipyretic, and antidiabetic activities (Dwivedi 2007; Jain et al. 2009; Sandhu et al. 2010; Haq et al. 2012; Maulik and Talwar 2012; Mythili et al. 2012). Compounds extracted from the stem bark include triterpenoids, beta-sitosterol, tannins, flavonoids (arjunone, arjunolone, and luteolin), glycosides and minerals (calcium, iron, and zinc), saponins (arjunic acid, arjunolic acid, and arjungenin), proanthocyanidins, and cardenolides (Yadava and Rathore 2000; Wang et al. 2010; Dhingra et al. 2012).
It possesses cardioprotective properties against chronic stable angina, endothelial dysfunction, heart failure, and ischemic mitral regurgitation (Gauthaman et al. 2001; Dwivedi 2007; Maulik and Katiyar 2010). Cardioactive properties are attributed to the presence of triterpenoids and flavonoids. It is comparable to fluvastatin. Arjunolic acid extracted from the bark is reported to have inhibitory effects on enzymes like superoxide dismutase, catalase, ceruloplasmin, lipid peroxides, and myeloperoxidase (Shahriar et al. 2012; Kokkiripati et al. 2013).
The bark of the plant has protective effects on doxorubicin-induced DNA damage and cardiotoxicity (Singh et al. 2008). It is also reported that the cardioactive role of the plant is regulated through alternation of thyroid hormones (Parmar et al. 2006). Impaired endothelial function of the plant is also reported in chronic smokers (Bharani et al. 2004). The presence of flavonoids and proanthocyanidins provide free-radical antioxidant and vascular strengthening activities. Two new cardenolides cardioactive glycosides extracted from the seeds contribute to increase the force of cardiac contraction through rise of intercellular sodium and calcium (Dwivedi 2007). Green synthesis of gold nanoparticles is also reported from the aqueous fruit extract of T. arjuna (Gopinath et al. 2014).
Traditional Uses
The stem bark is used in Ayurvedic medicine for treating cardiovascular diseases known as hritroga. Cardioactive properties of the plant are reported in ancient Indian texts, including Charaka Samhita and Ashtanga Hridayam. The bark is also used for treating hypertension, hyperlipidemia, atherogenic effects, anginal pain, fractures, ulcers, leukorrhea, diabetes, dyslipidemia, snakebites, and scorpion stings (Tripathi et al. 2000; Manna et al. 2007; Dwivedi and Chopra 2014; Subramaniam et al. 2011). The bark decoction is also used with milk in folklore medicine.
## 6.11 Thevetia peruviana Pers. (Syn: Cascabela thevetia)
* Vernacular name: Yellow Oleander
* Family: Apocynaceae
Ecological Distribution and Morphological Characteristics
T. peruviana is cultivated as an ornamental tree or shrub. It is native to Mexico and Central America.
Its leaves are lens-shaped with a glossy appearance. Flowering takes place during summer and early fall (Fig. 6.11a–e).
Fig. 6.11
(a–e) Medicinal importance of Thevetia peruviana (yellow oleander) is due to the presence of many active constituents, i.e., α-amyrin acetate, lupeol acetate, α-amyrin, β-amyrin, lupeol, and vetigenin and cardioactive glycosides. (a) Tree (b) Branch (c) Leaves (d) Flowers (e) Fruit
Important Phytochemicals and Medicinal Value
All parts of the plant are poisonous due to the presence of cardiac glycosides. Phytochemical screening of T. peruviana indicated the presence of many active constituents, i.e., α-amyrin acetate, lupeol acetate, α-amyrin, β-amyrin, lupeol. Due to presence of cardenolides, i.e., cardioactive glycosides, it is toxic to many vertebrates. These toxins include Thevetin A and Thevetin B, cerebrosides, Peruvosides, Nerifolin, Thevetoxin, and Ruvosides. The main activities of the plant are antifungal, antimicrobial, antispermatogenic, anti-inflammatory, and anti-HIV activities (Kareru et al. 2010; Gupta et al. 2011; Hammuel et al. 2011).
Thevefolin, isolated from seeds, is reported to have anticancer and cardiotonic activities (Save et al. 2015). Beta-sitosterol prevents the oxidation of LDL cholesterol and reduces the risk of atherosclerosis (Berry et al. 2009).
Traditional Uses
T. peruviana is reported to have inhibitory effects on spermatogenesis in rats, indicating the possibility of developing a herbal male contraceptive. The bark is used as a febrifuge, and the seeds are used in the treatment of rheumatism.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_7
# 7. Antipyretic and Analgesic Activities of Some Economically Important Woody Plants
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
This chapter describes antipyretic and analgesic activities of trees like Brachychiton populneus, Ceiba speciosa, Eucalyptus citriodora, Murraya exotica, Pinus roxbrghii, Pterospermum acerifolium, Putranjiva roxburghii, Salix babylonica, Salix tetrasperma, Tectona grandis, and Zizyphus mauritiana.
## 7.1 Introduction
Fever is generally caused by microbial infection, physiological stress, excess physical activities, or increase in thyroid secretion which produces pyrogens, which in turn results in increase of body temperature above 99 °C. It can disturb the metabolism of the body, causing increase in blood pressure, pulse rate, respiration rate, etc. However, many traditional systems of medicine report that fever itself is not a disease but an indication of symptoms of some other disease. Infectious agents cause increase in formation of cytokines, which in turn enhance the formation of prostaglandin. These prostaglandins act on the hypothalamus and increase the body temperature. Many antipyretic drugs inhibit activity of prostaglandin synthase with high selectivity and exert many side effects by disturbing metabolic activities of glomeruli, cortex, hepatic cells, and heart muscles.
There is sufficient scientific evidence which reports the role of plants as antipyretics and their uses in synthetic drugs. Use of medicinal plants to relieve fever and pain has a long history. People in many parts of the world still rely on traditional plants to relieve pain and fever and avoid synthetic drugs due to their side effects. A well-known example is aspirin, which has been used in the form of willow leaves for 2400 years; the presence of "salicylates" in their bark sap is the main reason for their antipyretic and analgesic activities.
## 7.2 An Account of Important Trees
Many woody plants are known to possess antipyretic and analgesic activities (Fig. 7.1a–d). Their important phytochemicals with potential pain- and fever-relieving activities are tested on animal models such as rabbits and rats. This is carried out by inducing pyrexia on these animal models through the use of Brewer's yeast, through some vaccines, or through acetic acid-induced writhing and tail immersion tests. Alcoholic extract of trees like Crataeva magna is reported to show reduction in fever at a dose of 200 and 400 mg/kg by typhoid vaccine-induced pyrexia in rabbits (Chidambaram et al. 2011).
Fig. 7.1
(a, b) Fruit extracts of Grewia asiatica (phalsa) are reported recently to have antipyretic, analgesic, and anti-inflammatory activities in albino mice, which might be attributed to the presence of anthroquinones, tannins, and flavonoids; (a) Fruits; (b) Close view of fruits; (c, d) Phyllanthus emblica (India gooseberry) possesses analgesic and antipyretic activities due to the presence of compounds like hydrolysable tannins, Emblicanin A, Emblicanin B, punigluconin, pedunculagin, quercetin, gallic acid, tannins, flavonoids, pectin, and vitamin C; (c) Tree; (d) Leaves
Plants like M. azedarach are used in folk medicine for antipyretic purposes, and there are scientific data which support its use as an antipyretic at a dose of 500 mg/kg (Sultana et al. 2013). Hydro-methanol extract of plant leaves is reported to reduce fever significantly in rabbits. Phytochemicals like kaempferol, quercetin, stigmasterol, β-sitosterol, campesterol, phytol, beta-carotene, tocopherol, and squalene present in the leaves of M. azedarach contribute to its antipyretic properties. Leaf extract at a dose of 500 mg/kg has activity similar to that of the standard drug paracetamol. Antipyretic activities of stem extract of Mangifera indica, mango, are also reported in mice (Awe et al. 1998).
Piper nigrum, commonly known as black pepper, is also used traditionally to reduce fever. However, a few scientific studies also reveal its antipyretic properties, which may be due to piperamides, piperidine, and piperine. Alcoholic extract of plant reduced fever in Wistar albino rats at doses of 250 and 500 mg/kg (Nagateja et al. 2013).
In India, root juice is used traditionally to reduce fever. Antipyretic activity of fruit and leaf extracts of Prosopis cineraria is reported at a dose of 200–300 mg/kg (Mahmood et al. 2013) due to the presence of compounds like heptacosanoate, patulitrin, sitosterol, spicigerine, and prosogerine.
Antipyretic activities of a few more important trees are discussed below.
## 7.3 Brachychiton populneus (Schott & Endl.) R. Br
* Vernacular name: Kurrajong
* Family: Malvaceae
Ecological Distribution and Morphological Characteristics
B. populneus grows as a medium-sized tree and is native to Australia. Some species of genus Brachychiton are found in Australia, including B. rupestris, B.discolor, B. diversifolius, B. acerifolius, and B. populnefolium.
B. populneus is an evergreen tree. The leaves are palmately lobed, and flowers are small, white, and bell-shaped (Fig. 7.2a, b) and appear in May to July. Male and female flowers are found on the same plant. The lamina of young leaves is shiny, red, and thick. The tree is also grown due to the ornamental value of its leaves. The seeds are edible, and roasted seeds were consumed by native Aboriginal Australians. The fruit is a woody, boat-shaped pod. The roots have water-retentive properties and are used as an artificial source of water.
Fig. 7.2
(a, b) Brachyton populneus (kurrajong) is known to possess antipyretic activities and is useful for treating skin infection; (a) Flower buds; (b) Flowers
Important Phytochemicals and Medicinal Value
The plant is known to possess antipyretic activities and is useful for treating skin infections (Williams 2010). Flexible bark from the tree is used as a bandage. Eye drops are also made from young branches (Smith 1991).
Traditional Uses
The young roots, stem, seeds, and nuts are the edible parts. The roots taste like boiled turnip.
Gum extracted from some Brachychiton spp. is used for fixatives. When mixed with ocher, they are used as adhesive body paint. Some species have high levels of potassium, calcium, sodium, iron, copper, and zinc. It is a popular shade tree in parks.
## 7.4 Ceiba speciosa A. St.-Hill (Syn: Chorisia speciosa)
* Vernacular names: Floss- silk tree, Kopak, Buddha Tree
* Family: Bombacaceae
Ecological Distribution and Morphological Characteristics
It is a medium-sized, deciduous, beautiful tree. It is native to tropical and subtropical forests of South America.
The leaves are palmate, alternate, and petiolate, bearing serrate and elliptic leaflets. The trunk is bottle-shaped; the bark and branches are covered with cone-shaped prickles. These stout prickles disappear from the bark upon maturity. The young stem is green and photosynthetic. The flowers are white with yellow margins and appear in September. They look like orchid flowers and give the appearance of a pink haze in a tree canopy. Monarch butterflies are one of their pollinators and feed on floral nectars. The fruit is a pod which produces black seeds, covered with fibers known as cotton silk (Fig. 7.3a–o).
Fig. 7.3
(a–o) Ceiba spp. (a–i) C. speciosa (floss-silk tree, Buddha tree) is used in preparing a psychedelic brew, i.e., Ayahuasca, which is used for healing purposes; (a, b) Young and old tree; (c) Leaves; (d) Conical bark; (e) Young bark; (f) Old bark; (g) Flower; (h) Fruit; (i–n) Different stages of C. insignis; (i) Tree; (j) Tree in bloom; (k) Conical and spiny bark; (l) Tree trunk; (m) Flower; (n, o) Fruits
Important Phytochemicals and Medicinal Value
Phytochemical analysis showed the presence of flavonoids, naphthalenes, sesquiterpenes, steroids, and lignans. Antipyretic activity is reported at a dose of 400 mg/kg in Wistar albino rats with pyrexia induced through Brewer's yeast (Khan et al. 2015).
Traditional Uses
The fibers of fruits are soft and flexible and used in making canoes, ropes, and paper. Vegetable oil is prepared from the seeds and used for cooking as well as industrial purposes. The silk in the seed pods is also used to fill life-jackets and pillows. The plant is also used in preparing a psychedelic brew, i.e., Ayahuasca, which is used for healing purposes.
## 7.5 Eucalyptus citriodora Hook. (Syn: Corymbia citriodora)
* Vernacular name: Lemon-scented Eucalyptus
* Family: Myrtaceae
Ecological Distribution and Morphological Characteristics
The genus Eucalyptus is native to Australia and comprises over 800 species of economically and medicinally important woody plants. E. citriodora is a tall, evergreen, and important forest tree which is a source of oil, timber, fuel, and nectar in honey production.
It is a tall tree with a straight trunk. The bark is smooth, gray, and whitish. The leaves are small, entire, alternate, elongated, lens-shaped, aromatic, and acuminate at the apex. The flowers are umbel borne on short stalks, the stamens are threadlike, and the ovary is inferior. The fruit is an ovoid capsule. The seeds are ellipsoid and black (Fig. 7.4a–e).
Fig. 7.4
(a–e) Eucalyptus citriodora (lemon-scented Eucalyptus) is a powerful antiseptic and is used for treating cold, cough, flu, headache, and sore throat and comprises citronellol, geranyl acetate, limonene, eucalyptol, camphene, alpha-pinene, and linalyl acetate; (a) Tree; (b) Leaves; (c) Flower; (d) Fruits; (e) Red resin of E. camaldulensis (river red gum)
Important Phytochemicals and Medicinal Value
The leaves are aromatic and release essential oil comprising terpenes. The plant oil is used to synthesize citronellol and related products. It possesses analgesic, anti-inflammatory (Silva et al. 2003), acricidal (Clemente et al. 2010), and antimicrobial properties (Muanza et al. 1994; Luqman et al. 2008). The essential oil of the plant possesses anti-inflammatory and antifungal (Ramezani et al. 2002; Javed et al. 2012) activities. It can be used as a therapeutic alternative to deal with inflammatory diseases (Gbenou et al. 2013).
Essential oil of Eucalyptus spp. is a powerful antiseptic; it is used for treating cold, cough, flu, headache, and sore throat and in cosmetics (Mhaskar et al. 2000; Vaghasiya et al. 2008). It comprises citronellol (80%), geranyl acetate, limonene, terpinen-4-ol, 1, 8-cineole (eucalyptol 44.1%), camphene (29.7%), alpha-pinene (8.4%), and linalyl acetate (4.4%) (Dagne et al. 2000; Adam 2001; Husain and Ali 2013). The main activity of the oil is antifungal, analgesic, and anti-inflammatory (Silva et al. 2003). However, at higher doses eucalyptus oil is neurotoxic.
Analgesic activity of essential oil of E. citriodora is reported with central and peripheral effects in mice (Silva et al. 2003). At a dose of 50 mg/kg, strong antipyretic and analgesic activities are observed in Wistar rats due to the presence of the main compound citronellal (Gbenou et al. 2013).
Traditional Uses
Eucalyptus oil is applied externally as an antiseptic to heal wounds and skin infections. It is widely used as nasal drops, in syrups, and in lozenges (Jean 2008). It is used in pharmaceutical applications for treating benign bronchial disease, and to relieve nasal congestion in the common cold. It is also used for catarrh of the upper respiratory tract and for bronchitis.
## 7.6 Murraya exotica L. (Syn: Murraya paniculata)
* Vernacular names: Mock orange, Mock Lime, Satinwood
* Family: Rutaceae
Ecological Distribution and Morphological Characteristics
M. exotica is an evergreen shrub or occasionally a small tree that is native to Southeast Asia. It is usually 2–3 m in height and 8 m–13 cm wide. The flowers are white and fragrant. It is commonly cultivated in many tropical and subtropical regions due to its glossy green foliage and fragrant flowers and it is a popular hedge (Fig. 7.5a, b).
Fig. 7.5
(a, b) Infusion of leaves and flowers of Murraya exotica (orange jasmine) is aromatic, refrigerant, digestive, and beneficial in rheumatic fever, and burning of the skin due to the presence of flavonoids, coumarins, phytosterols, alkaloids, and volatile oil, which comprises spathulenol, α-pinene, caryophyllene oxide, α-caryophyllene, and bicyclogermacrene and γ-selinene; (a) Plant; (b) Flower; (c) Young fruits (in red)
Important Phytochemicals and Medicinal Value
Phytochemical studies on M. exotica revealed the presence of flavonoids, coumarins, phenolic compounds, phytosterols, alkaloids, and volatile oil (Wang et al. 2007; Aziz et al. 2010; Saeed et al. 2011; Gill et al. 2014; Shah et al. 2014). The chemical composition of essential oil has been widely studied (Negi et al. 2005). The main constituents of the oil are spathulenol, α-pinene, caryophyllene oxide, α-caryophyllene, bicyclogermacrene, and γ-selinene (Wei et al. 2010).
The leaves and bark are also reported to have antinociceptive (Sharker et al. 2009), anti-inflammatory (Rahman et al. 2010), antidiabetic, antifungal (Sundram et al. 2011), and antioxidant activities. The bark extract contains methanolic, petroleum ether, and ethyl acetate and when introduced in albino mice at doses of 200–400 mg/kg produced significant analgesic effects (Podder et al. 2011). Analgesic activity is also reported to be due to alkaloid isomurrayafoline in mice and reduced the number of writhes induced by acetic acid (Fazal-ur-Rehman et al. 2014). Oral administration of ethanolic extract of leaves at doses of 50, 100, and 200 mg/kg caused significant inhibition at rates of 28.84%, 54.93%, and 67.91% respectively in Swiss albino mice administered with acetic acid (Nerkhede et al. 2012).
Traditional Uses
In Bangladesh, oral administration of leaves extract is used in traditional medicine to relieve pain. An infusion of the leaves and flowers of M. exotica is used as a tonic and stomachic. It is said to be aromatic, refrigerant, digestive, and beneficial in rheumatic fever, and burning of the skin. The leaves are also used as food additives in many Indian and Malay dishes due to their strong fragrance (Ng et al. 2012). In Thailand, the leaves are crushed and mixed with alcohol to treat joint pain, bone pain, and snake bites (Rodanant et al. 2012).
## 7.7 Pinus roxbrghii Sarg.
* Vernacular names: Chir pine
* Family: Pinaceae
Ecological Distribution and Morphological Characteristics
Chir pine is native to Bhutan, Nepal, and Pakistan. It is a large monoecious tree, with whorled branches and dark red bark which is deeply fissured with leaves in clusters of three. The leaves are finely toothed, light, green, and persist on average for a year (Fig. 7.6a–f). Male flowers or cones are small, many about 1.5 cm long; female cones are large, solitary or 2–5 together, ovoid, and woody, and seeds are winged.
Fig. 7.6
(a–f) The medicinal importance of Pinus roxburghii (chir pine) is due to its essential oil, which comprises new xanthone, 1,5-dihydroxy- 3,6,7-trimethoxy-8-dimethylallyloxy-xanthone, and 1-hydroxy-3,6-dimethoxy--β-D- glucopyranoxanthone, which are isolated from the methanolic extract of the bark; (a) Tree; (b) Leaves; (c) Bark; (d, e) Cones; (f) Female cone on the ground
Important Phytochemicals and Medicinal Value
Two new xanthones, 1,5-dihydroxy- 3,6,7-trimethoxy-8-dimethylallyloxy-xanthone and 1-hydroxy-3,6-dimethoxy-β-D- glucopyranoxanthone, are isolated from the methanolic extract of the bark of P. roxburghii (Rawat et al. 2006). Needles form a colorless and volatile oil known as pine oil on steam distillation.
Pine oil is commercially important and is a mixture of limonene, phellandrene, borneol, longifolene, and cadinene. Oleoresin from chir is the main source of turpentine oil in the country. The hydrolyzed fraction of chir pine oleoresin comprises phenolic acids and a lignan. Ferulic acid, p-coumaric acid, and pinoresinol are also extracted (El- Shaer 2002).
Ethanolic extract of bark showed analgesic activity, which is evaluated by acetic acid-induced writhing and tail immersion tests in Swiss albino mice at doses of 100, 300, and 500 mg/kg. Diclofenac sodium and indomethacin were employed as standard analgesic drugs (Kaushik et al. 2012). Analgesic activity is reported due to the presence of flavonoids, which can inhibit prostaglandins.
Traditional Uses
It is used as a source of resin, fuel, and food. The wood is used in making furniture and matchsticks. It is a valuable tree for reforestation and afforestation of denuded areas in the foothills of Pakistan. Pine is an important source of an oleoresin which yields turpentine oil. Indian turpentine oil comprises less pinene and is commercially used as a solvent for paints and varnishes. Turpentine oil is also used in the perfumery industry and in the manufacture of synthetic pine oil, disinfectants, and insecticides. It is one of the most important basic raw materials for the synthesis of terpene chemicals and is used in a variety of industries. Pine oil from the wood is used in paints, varnishes, lacquers, and pharmaceuticals (Langenheim 2003). It is also used in the textiles industry and as a degreasing agent in leather manufacture.
Rosin is used principally in the paper, soap, cosmetics, paint varnish, rubber, and polish industries.It also finds applications in linoleum and roofing cements, fireworks, match compositions, explosives, insecticides, and disinfectants. Rosin on distillation yields spirit and rosin oil, which is used in varnishes (Wiyono et al. 2006). The plant is also used for curing eye and ear problems, hemorrhages, flatulence, liver diseases, bronchitis, and skin diseases.
## 7.8 Pterospermum acerifolium L.
* Vernacular names: Maple-Leafed Bayur Tree
* Family: Sterculiaceae
Ecological Distribution and Morphological Characteristics
P. acerifolium is widely distributed in the Indian sub-Himalayan tract, the outer Himalayan valley, Pakistan, and North America (Dixit et al. 2011).
The leaves are elongated and palmately compound, with stipules, and are alternate. The flowers are white in color and aromatic at night. The fruit is a capsule with hard texture and produces winged seeds (Fig. 7.7a–f).
Fig. 7.7
(a–f) The aqueous and alcoholic extracts of Pterospermum acerifolium (maple-leafed bayur tree) possess analgesic, antipyretic, and antihyperglycemic activities due to bioactive secondary metabolites like saponins, steroids, caumarins, and betacyanins; (a) Tree; (b, c) Young leaves; (d) Mature leaves; (e) Flower; (f) Fruit
Important Phytochemicals and Medicinal Value
P. acerifolium assessed for phytochemical analysis exhibited the presence of bioactive secondary metabolites like saponins, steroids, caumarins, and betacyanins. Flavonoids like kaempferol, keampferide, luteolin, and steroids and triterpenoids like sitosterol, taraxerol, friedelin, sugars, and fatty acids are present in the plant (Harbome 1998).
The aqueous and alcoholic extracts of this plant possess various medicinal properties, including anti-inflammatory, analgesic, antipyretic, antihyperglycemic, and anticancer activities (Shweta et al. 2007; Selim et al. 2006). The stem bark of the plant is known to have antimicrobial value (Khonde et al. 2009). Hepatoprotective value of ethanolic extract of the leaves is reported (Kharpate et al. 2007).
Ethanolic extract of bark is reported to have analgesic and anti-inflammatory activity against carrageenan and arachidonic acid-induced rat paw edema (Manna and Jenna 2009). Significant inhibition of acetic acid-induced writhing and tail clip-induced analgesia is observed, indicating the anti-inflammatory analgesic effect of the plant, which can block the histamine and serotonin pathways (Manna and Jenna 2009). Methanolic extracts of leaves also showed significant antipyretic activity at doses of 300 and 400 mg/kg in mice (Datta et al. 2011).
Traditional Uses
It has many traditional uses in curing leprosy, ear pain, stomachache, small pox, and tumors (Manna et al. 2009; Mehrotra and Shome 2009). The petals are crushed and used to cure chronic headache (Nandy et al. 2012). The flowers are mixed with sugars and applied locally to cure cancer (Balachandran and Govindrajan 2005).
## 7.9 Putranjiva roxburghii Wall.
* Vernacular names: Putranjiva
* Family: Putranjivaceae
Ecological Distribution and Morphological Characteristics
Putranjiva is native to Southeast Asia where it is widely grown due to its ecological, ornamental, and medicinal value.
It is a dioecious, evergreen tree which attains heights of up to 18 m and widths of 2 m with bark that is gray in color. The leaves are long, elliptic, and shiny in appearance (Fig. 7.8a–d).
Fig. 7.8
(a–d) The leaves and fruits of Putranjiva roxburghii (putranjiva) are used in cold and fever and the plant possesses antidiabetic activity. Important phytochemicals are saponins, mannitol, arachidic acid, linoleic acid, palmitic acid, glucoputranjivan, putranjivoside, glucosides and gallo-tannins, roxburghonic acid, putraflavone, putranjivanonol, and putranjic acid; (a) Tree; (b) Leaves; (c) Branch bearing flower bud; (d) Fruit
Important Phytochemicals and Medicinal Value
Phytochemical screening revealed the presence of saponins, mannitol, arachidic acid, linoleic acid, palmitic acid, glucoputranjivan, putranjivoside, glucosides, and gallo-tannins (Khandelwal et al. 1996; Khare 2007). A triterpene acid, roxburghonic acid, and putraflavone, a biflavonoid, are isolated from the alcoholic extract of leaves (Garg and Mitra 1968). However, the tree trunk comprises putranjivanonol and putranjic acid.
Antipyretic activity of ether extract of the leaves is reported at doses of 100, 200, and 400 mg/kg in acetic acid-induced writhing in mice. At a dose of 400 mg/kg, antipyretic activity is reported in mice induced with yeast fever (Reamongkol et al. 2009). Dose-dependent analgesic activity is observed in acetic acid-induced writhing rats; however, dose-independent activity is reported in the case of hot plate and tail flick models as compared with the standard drug indomethacin (Khobragade et al. 2013). Putranjiva is also known to possess antidiabetic activity (Amit et al. 2010).
Traditional Uses
The leaves, fruits, and stones of fruits are given for colds and fevers.
## 7.10 Salix babylonica L. (Syn. Salix japonica Thunb.)
* Vernacular name: Weeping willow
* Family: Salicaceae
Ecological Distribution and Morphological Characteristics
Salix babylonica is native to Northern China.
It grows as a medium-sized to large deciduous tree. The leaves are alternate and spirally arranged and have serrate margins and an acuminate tip. Dioecious flowers are catkins, borne on different trees. Flowering takes place in early spring (Fig. 7.9a–c).
Fig. 7.9
(a–d) More than 400 species of Salix spp. are medicinally important and many of them are antipyretic and analgesic. Main medicinal activities of willow plants are due to the presence of salicylate compounds. Important compounds are glucosides, trichocarpin, and salicin. Salicin is the main analgesic, antipyretic, and anti-inflammatory compound, which is converted to acetylsalicylate (aspirin) in the human body; (a) S. babylonica (Indian willow); (b) Catkin; (c) Fruit; (d) S. tetrasperma (Indian willow)
Important Phytochemicals and Medicinal Value
The main medicinal activities of willow plants are due to the presence of salicylate compounds (Ruuhola and Julkunen-Tiitto 2000; Abou-Zeid 2000; Salem et al. 2011). Compounds isolated from leaves include benzyl ester of gentisic acid 2'-Oacetyl β-D-glucosides, trichocarpin, salicin, kaempferol-7-O-glucoside, apigenin-7-O-galactoside, luteolin-4'-Oglucoside, and an ester of terephthalic acid (Khatoon et al. 1988).
The bark of the tree possesses salicin (3–4%), which has analgesic, antipyretic, and anti-inflammatory activities and is converted to acetylsalicylate (aspirin) in the human body. The leaves are antirheumatic and astringent. The plant possesses anthelmintic, antidote, antipyretic, antiseptic, fungicidal, insecticidal, and vermifuge activities.
Traditional Uses
In Chinese medicine, the leaf and root of the plant are known as Liu Ye. Decoction of the shoots and leaves is used to treat ulcer and skin diseases.
Young shoots and flower buds are edible parts of the plant. Bark infusion is used to treat diarrhea and fevers. The stem is used to make baskets. Bark decoction is used for hair growth. The seeds are used for treatment of fever, jaundice, and rheumatism. Andeans suggest that the plant can be used to whiten teeth. The bark can be used to treat fever and malaria. The Chinese use an infusion from the leaves, twigs, and bark for treating fever, jaundice, and rheumatism (Duke 2007). Traditionally, leaf and stem bark infusion is taken once daily for 4–5 days in order to relieve fever and pain (Chhetri 2004).
## 7.11 Salix tetrasperma Roxb.
* Vernacular name: Indian Willow
* Family: Salicaceae
Ecological Distribution and Morphological Characteristics
Genus Salix comprises over 400 species of the family Salicaceae. S. tetrasperma grows as a medium-sized tree in swampy areas. The plant is widely grown in many Southeast Asian countries due to its medicinal value and traditional uses.
The leaves are narrow and appear silky when young. The flowers are dioecious catkins without sepals and the petals are nectariferous. Female flowers produce two-lobed stigma, with a single ovary comprising many ovules. Flowering takes place in early spring. The fruit is a capsule and is found in groups of 3 or 4 (Fig. 7.9d).
Important Phytochemicals and Medicinal Value
Techniques like MS, 1D–NMR (1H and 13 C), and 2NMR have facilitated our knowledge about phytochemicals of many plants. Methanolic extract of stem bark comprises phytochemicals like beta-sitosterol acetate, friedelin, 3beta-friedelinol, beta-amyrin, and beta-sitosterol-O-glucoside as revealed through column chromatography. Ethanolic extract of leaves comprise salicin and derivatives like tremeloidin and 2'-O-p-(E)-caumaryl salicin. Catechol and tremulacin are extracted from dichloromethane fraction of the leaves (El-Shazly et al. 2012).
Hydroalcoholic extract of the plant possesses healing and antioxidant activities (Kishore et al. 2014). The bark exhibits diuretic and laxative properties (Mondal et al. 2010). Ethanolic extract of the plant possesses antimicrobial activities against E. coli, B. subtilis, and C. albicans (Mondal et al. 2010).
Traditional Uses
Analgesic and antipyretic properties of S. tetrasperma are reported in ancient texts from Egypt and Assyria. The plant is used to treat cough in children. The stem sap is consumed by females for treatment of dysmenorrhea. Hot water extract of the plant is used to induce abortion. The dried leaves are used to treat rheumatism, epilepsy, and stones in the bladder. The plant is also used for making ropes, baskets, handicrafts, and ornamental goods.
## 7.12 Tectona grandis L.
* Vernacular names: Teak
* Family: Lamiaceae
Ecological Distribution and Morphological Characteristics
T. grandis, commonly known as teak or sagwan, is one of the most famous timbers in the world. It is probably the most widely cultivated high-value hardwood (HVH) in the world and is native to India and Myanmar and Southeast Asian countries.
It is a large deciduous tree, 30–35 m tall with light brown bark. The leaves are simple, opposite, and broadly elliptical, with glandular dots. The flowers are white in color and small and have a pleasant smell (Fig. 7.10a–c).
Fig. 7.10
(a–c) Tectona grandis (teak) has antidiabetic, analgesic, and anti-inflammatory properties due toalkaloids, glycosides, saponins, steroids, and flavonoids. Important compounds like astectoquinone, 5-hydroxylapachol, tectol, betulinic acid, betulinic aldehyde, squalene, lapachol, acetovanillone, E-isofuraldehyde, evofolin, syringaresinol, medioresinol, balaphonin, lariciresinol, tectonoelin A, and tectonoelin B; (a) Young plant; (b) Tree; (c) Leaves
Important Phytochemicals and Medicinal Value
Many alkaloids, glycosides, saponins, steroids, and flavonoids have been reported in T. grandis (Asif 2011). Important compounds like astectoquinone, 5-hydroxylapachol, tectol, betulinic acid, betulinic aldehyde, squalene, and lapachol are also extracted from the plant (Aguinaldo et al. 1993; Gupta and Singh 2004).
In addition, acetovanillone, E-isofuraldehyde, Evofolin, syringaresinol, medioresinol, balaphonin, lariciresinol, zhebeiresinol, 1- hydroxypinoresinol together with two new compounds, i.e., tectonoelin A and tectonoelin B, are extracted from the leaves of T. grandis (Rodney et al. 2012).
Antipyretic activity of T. grandis is reported at doses of 250 and 500 mg/kg in yeast-induced pyrexia in Wistar rats due to the presence of lapachol, which is the main antipyretic compound (Nayeem and Karvekar 2010; Sharma et al. 2011).
Traditional Uses
The plant is used in the treatment of bronchitis, cold, and headache. It is used in the treatment of scabies, as a sedative, and also as a diuretic. It is reported to have antidiabetic, analgesic, and anti-inflammatory properties (Singh et al. 1996; Nayeem and Karvekar 2010; Gaisas et al. 2009; Diallo et al. 2008). The bark of sagwan is used as an astringent, an anthelmintic, and a depurative. It is used to treat bronchitis, hyperacidity, verminosis, burning sensation, diabetes, leprosy, and many skin diseases. The leaves are useful in inflammations, leprosy, skin diseases, pruritus, stomatitis, indolent ulcers, hemorrhages, and hemoptysis. The wood is acrid and used in the treatment of leucoderma. Oil extracted from the wood is best for headache, biliousness, and burning pains.
## 7.13 Zizyphus mauritiana Lam.
* Vernacular name: Chinese date, Jujube
* Family: Rhamnaceae
Ecological Distribution and Morphological Characteristics
Genus Zizyphus comprises over 40 species out of which Z. mauritiana is very common. It is native to Australia, China, Afghanistan, Pakistan, Malaysia, and India; however, as it can grow in a wide variety of climatic conditions, it is also found in other countries. More than 90 cultivars of plants are grown in India.
It grows as a medium-sized or small evergreen tree up to a height of 15 m. However, depending on climatic conditions it may grow as a shrub reaching to a height of a few meters tall. Plant is tolerant to dry conditions and cultivated due to its ecological, medicinal, and edible value. The trunk comprises drooping branches which are covered with spines. The leaves are alternate, are arranged in two rows, have oblong-elliptic shape, are 3-veined, and have hairless shiny lamina. Inflorescence is axillary cyme, with small yellow-green aromatic and pedicellate flowers with five petals. Pollination takes place through honeybees. The fruits are drupe, with globose to ovoid shape, have a rough surface, and ripen at different times, even when on the same tree; the seeds are tuberculate and brown, with a furrowed stone, and are oval shaped (Fig. 7.11a–e).
Fig. 7.11
(a–e) Zizyphus mauritiana (jujube) possesses antipyretic, anticancerous, antidiabetic, antifungal activities due to jujubosides, which are important saponins with anxiolytic, sedative, and sweet-binding properties. Cyclopeptides from the plant possess antimicrobial, antiplasmodial, antidiabetic, and analgesic activities; (a) Tree; (b) Leaves; (c) Flowers; (d) Young fruits; (e) Ripened fruits
Important Phytochemicals and Medicinal Value
Phytochemical screening has revealed the presence of many metabolites including terpenoids, alkaloids, saponins, and tannins. They contribute to anticancerous, antidiabetic, antioxidant, and antifungal activities of plants (Mishra et al. 2011; Jarald et al. 2009; Pandey et al. 2007).
Jujubosides are important saponins with anxiolytic, sedative, and sweet-binding properties (Matsuda et al. 1999). Cyclopeptides from the plant possess antimicrobial (Jossang et al. 1996; Kayser and Arndt 2000; Abalaka et al. 2010; Najafi 2013), antiplasmodial, antidiabetic, and analgesic activities (Goyal, et al. 2012; Balakrishman et al. 2012).
Z. jujuba is also reported to have antipyretic activity due to the presence of compounds like jujubosides, jubaine, cyclopeptides, and saponins. Antipyretic activity is evaluated against Brewer's yeast-induced pyrexia in rats (Balakrishman et al. 2012).
Traditional Uses
The plant is used as fodder and medicine and for construction purposes. The leaves are used to treat the liver and have antipyretic value. They are also cooked in Indonesia. The dried fruit pulp is used as a laxative. It is also used to treat wounds for rapid healing. The seeds are used with buttermilk to treat nausea and abdominal pains. They are also effective for rheumatism when mixed with oil. Bark decoction is also used to treat diarrhea and it relieves gingivitis. Bark extract is effective against gout and rheumatism. Infusion made from flowers is used as an eye lotion.
The fruit and flowers are the edible parts as they are a good source of calcium, vitamins, citric acid, and ascorbic acid. The ripe fruit is sweet and sour with white flesh and is a rich source of vitamin C. The leaves are also a source of minerals and nutrients. Honey is made from nectar of the flowers. The unripe fruits are used in making pickles. They may be consumed raw or stewed. The ripe fruit is crushed and served cold as a refreshing drink. In Africa, the dried pulp of the fruit is fermented, pressed, and used in making cakes. The bark of the tree is also used in dyeing of silk. The leaves are used to feed silkworms. The plant oil is being considered for use in making biodiesel.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_8
# 8. Aphrodisiac and Abortifacient Activities of Important Trees
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
Aphrodisiac and abortificient activities of many plants are known since long, and they have been used traditionally to improve sexual behavior and also as antinociceptives and abortifacients. This chapter includes aphrodisiac and abortifacient activities of trees like Albizia lebbeck, Broussonetia papyrifera, Dombeya rotundifolia, Jacaranda mimosifolia, Lantana camara, Myrtus communis, Ricinus communis, and Saraca indica.
## 8.1 Introduction
"Aphrodisiac" is derived from "Aphrodite," the name of the Greek goddess of love. Aphrodisiacs are substances which enhance sexual desire, and many of these substances are synthesized by plants. They are mentioned in traditional systems of medicine including Ayurveda, Rigveda, and Yajurveda. They have been used throughout history in many Asian, European, and African countries.
Many plants are known for their aphrodisiac activities, and there is sufficient scientific evidence coming from many experimental works which report that plants with aphrodisiacs improve sexual behavior and affect overall quality of life. However, they should be used with caution because many such plants can also be abortifacients when used without adequate knowledge. Some trees like Albizia lebbeck, Myristica fragrans, Punica granatum, and Juglans regia possess both aphrodisiac and abortifacient activities. Some parts are aphrodisiac while some parts possess abortifacient activities (Fig. 8.1a–e).
Fig. 8.1
(a–e) Hibiscus spp. possesses anticancer, antidiabetic, antimicrobial, and aphrodisiac activities; (a) Bush of Hibiscus moscheutos (rose mallow); (b) Flower; (c) Young leaves of Nyctanthes arobortristis are used traditionally as a female tonic and in alleviating gynecological problems; (d) Cuscuta reflexa, commonly known as dodder (in yellow) growing on Butea sp., is a medicinally important parasitic plant with antimicrobial, aphrodisiac, and hepatoprotective activities; (e) Seeds of Cascabela thevetia (yellow oleander) are used as antifertility agents; (f) flowers
Symptoms of sexual dysfunction are reported to occur in 10–50% of males and 25–63% of females. Aphrodisiacs are used for erectile dysfunction, to treat causes of infertility, and for sexual satisfaction. As sexual desires are controlled by the central nervous system, aphrodisiacs act through
1. (i)
altering the specific level of neurotransmitters or regulating the level of sex hormones like testosterone or progesterone within the body.
2. (ii)
providing a burst of nutritional value which improves the immediate health or well-being of the consumer and improves sexual efficiency.
3. (iii)
affecting the blood flow or increasing the duration of activity through numbing.
4. (iv)
crossing the blood-brain barrier and stimulating the area of sexual arousal like neurotransmitters or pheromones do.
Many plants with aphrodisiac and abortifacient activities have not yet been tested on humans; most of the experimental work is done on mice and rats. These animals are treated with sex-stimulating and abortion-causing natural compounds, and then their behavior is studied. Further, the exact mechanism through which many of these compounds act is still not properly understood. Therefore, there is a need to evaluate further the role of aphrodisiacs in humans and to make potential drugs in future.
## 8.2 An Account of Aphrodisiac and Abortifacient Activities of Economically Important Woody Plants
Although many herbaceous plants are evaluated for their aphrodisiac and abortifacient activities as compared with woody plants, experimental work has shown that some woody plants also possess such activities. A brief explanation of their aphrodisiac and abortifacient activities along with other medicinal properties is given in the following section.
## 8.3 Albizia lebbeck (L.) Benth.
* Vernacular names: Flea Tree, Lebbeck
* Family: Leguminosae (Fabaceae)
* Sub-family: Mimosoideae
Ecological Distribution and Morphological Characteristics
A. lebbeck is a deciduous tree which grows in the wild and is native to Indomalaya, New Guinea, Northern Australia, Pakistan, and Sri Lanka (Ali 1973). It is cultivated for its ornamental and economic value in many Asian countries.
The leaves are compound and bipinnate, with oblong and elliptic leaflets which are oblique at the base. The flowers are white, aromatic, in globose umbellate head, and have long thread-like stamens. The fruits are narrow, oblong, glabrous, persistent, green when young but turn golden brown on maturity, and produce oblong seeds (Fig. 8.2a–e).
Fig. 8.2
(a–e) Albizia lebbeck (flea tree) possesses both aphrodisiac and antifertility activities. Methanolic extract of the pod is reported to have antifertility activity; however, flowers and seeds are aphrodisiac; (a) tree; (b) leaves; (c) flowers; (d) young pods; (e) mature pods
Important Phytochemicals and Medicinal Value
The leaves are a rich source of many alkaloids, saponins, tannins, and flavonols (El-Mousallamy 1998). The leaves also contain echinocystic acid, flavon, vicenin II, and β-sitosterol. Some flavones and phenyl alaninol are isolated from fruit pods (Rashid et al. 2003). Bark comprises albigenic acid, which is a newly reported triterpenoid sapogenin. Other important compounds isolated from the bark include tannins, lebbecacidin, betulinic acid, and saponins (Faisal et al. 2012). The flowers are a source of triterponiods saponins labbekanin D and 4 saponins glycosides lebbckannins D, F, G, and H (Sharma and Dubey 2015). Important pharmacological activities of the plant are due to phytoconstituents like melacacidin, D-catechin, β-sitosterol, albiziahexoside, betulinic acid, echinocystic acid, and glycosides (Verma et al. 2013).
Methanolic extract of the pod is reported to have antifertility activity (Gupta et al. 2004, 2005). Saponins are isolated from methanolic extract of bark and pods, which have antispermatogenic effects (Sharma and Dubey 2015). Like pods, roots also possess spermatogenic properties (Kumar et al. 2007); however, the flowers and seeds are aphrodisiac (Sharma and Dubey 2015), and the flowers can also be used in hemicranias. The leaves possess antifertile and anticancer activities (Bobby et al. 2012).
The plant is reported to have antibacterial and antifungal activities due to the presence of lebbeckalysin (Faisal et al. 2012). Leaf extract showed inhibitory activities against E. coli, S. aureus, P. aeruginosa, and B. cereus. The bark is used effectively for toothache and for diseases of the gums.
Ethanolic extract of the leaves and bark revealed hepatoprotective activities in rats with induced hepatic injuries. Pods possess antimicrobial, antidiabetic (Zia-Ul Haq et al. 2013), and anticancer properties (Chulet et al. 2010).
Traditional Uses
Decoction of the leaves and bark is used in the treatment of bronchial asthma and other allergic disorders. The leaves are used for treating ophthalmic diseases, night blindness, syphilis, ulcer (Pathak et al. 2009), cold, cough and respiratory disorders, and also as analgesics (Saha and Ahmed 2009; Verma and Srivastav 2011). The bark is used for curing blood diseases, leucoderma, skin diseases, piles, paralysis, weakness, and to strengthen gums. All parts of plants are effective against snake bite. The wood is an important source of fuel.
## 8.4 Broussonetia papyrifera L. (Syn: Morus papyrifera)
* Vernacular names: Paper Mulberry, Wauke
* Family: Moraceae
Ecological Distribution and Morphological Characteristics
It is native to China, Japan, and Southeast Asia. Wauke has been used for making paper and therefore it is known as paper mulberry.
It grows as a medium-sized woody tree or shrub with ovate and toothed leaves like other members of Moraceae. Inflorescence is a catkin. Male and female flowers are on separate plants. Female flowers are solitary and red (Fig. 8.3a, b).
Fig. 8.3
(a, b) Fruits of Broussonetia papyrifera (paper mulberry) are used to treat infertility disorders; (a) leaves; (b) flower
Important Phytochemicals and Medicinal Value
Compounds isolated from root bark include Papyriflavonol A, a prenylated flavonol (5,7,3′,4′-tetrahydroxy-6,5′-di-(gamma,gammadimethylallyl)-flavonol) (Son et al. 2001; Kwak et al. 2003), uralenol, broussochalcone A (Cheng et al. 2001), broussoflavonols F and G, broussoflavan A and broussoaurone A (Ko et al. 1997), and broussoflavonols E (Fang et al. 1995).
Many isolated compounds are known to inhibit lipid peroxidation (Ko et al. 1997), have antiplatelet effects (Lin et al. 1996), and affect activities of the protein tyrosine phosphatase 1b (PTP1B) enzyme and aromatase (Chen et al. 2002). The fruit is used to treat impotence and ophthalmic disorders (Lee et al. 2001).
Traditional Uses
The flowers and leaves can be consumed due to their protein content and the presence of micronutrients. The leaves are used for treating blood in sputum, for uterine and menstrual bleeding, and for abdominal pain. The fibers from phloem are used in making textile fiber. The wood is used as a source for making furniture. In China, the leaves have been used in folk medicine against chronic prostatitis and as a tonic for kidney and liver purification and also to feed silkworms. In Hawaii, slim sap of paper mulberry is used as a mild laxative. It is also helpful against skin discoloration caused by pregnancy.
## 8.5 Butea monosperma Lam. (Syn B. frondosa Koenig Ex Roxb.)
* Vernacular name: Flame of the Forest
* Family: Leguminosae (Fabaceae)
Ecological Distribution and Morphological Characteristics
B. monosperma is a medium-sized deciduous tree with pinnate leaves and bright orange-red flowers.. It is native to tropical and subtropical regions of Southeast Asia. The flowers are large and arranged in racemes which are usually 15 cm long. Three flowers are arranged to form nodes of the rachis which are borne on long pedicels (Fig. 8.4a–e). Pedicels are about twice as long as the calyx. The bracts and bracteoles are deciduous. The pods are thin and flat.
Fig. 8.4
(a–e) Butea monosperma (Flame of the Forest) is known to improve sexual disorders including erectile dysfunction due to inhibition of Rho-kinase 2 activity or an increase in the ratio of smooth muscle to collagen level in penile tissue and also due to its potential to release nitric oxide and androgen levels. The root bark is an aphrodisiac; (a) tree; (b) leaves; (c) flowering branch; (d) close view of flowers; (e) petals fallen on the ground
Important Phytochemicals and Medicinal Value
Important metabolites include lupeonone, lupeol, medicarpin, calycosins, and cajanins. Bark contains kino-tannic acid, gallic acid, and pyrocatechin (Somani et al. 2006; Mishra et al. 2000; Nadkarni 2002). The seed oil is reported to have antibacterial and antifungal activities.
The plant is used to manage sexual disorders including erectile dysfunction. The root bark is used as an aphrodisiac, analgesic, and anthelmintic. Methanolic extract of the bark is reported to increase sexual behavior in both young and aged rats, which could be attributed to inhibition of Rho-kinase 2 activity or an increase in the ratio of smooth muscle to collagen level in penile tissue and also due to its potential to increase nitric oxide and androgen levels (Goswami et al. 2013).
Methanolic extract of B. monosperma bark possesses 2,2-diphenyl-1-picrylhydrazyl (DPPH), nitric oxide, and superoxide scavenging activity (Divya and Mini 2011). Researchers have shown that reactive oxygen species (ROSs) can cause morphological defects in sperm (Aziz et al. 2004), or they may activate Rho-kinase (Jernigan et al. 2008) and can also reduce the expression of soluble guanylyl cyclase (sGC) [38] which catalyzes conversion of guanosine triphosphate (GTP) to cyclic guanosine monophosphate (cGMP). Therefore, the antioxidant potential of methanol extract of bark might contribute in the management of erectile dysfunction.
The flowers are reported to have anticancerous, antidiabetic, and hepatoprotective properties. Ethanol extract of flowers is reported to show significant reduction in blood glucose level and serum cholesterol level, and improved glucose tolerance in alloxan-induced diabetic rats (Somani et al. 2006). Oral administration of ethanol extract of seeds is reported to have antidiabetic effects in noninsulin-dependent diabetes mellitus rats (Bavarva and Narasimhacharya 2008).
Traditional Uses
The plant is used as a source of timber and medicine, while the flowers are used for dyes due to their orange-red color. The bark fiber is used for making cordage. The wood pulp is used for newsprint manufacturing.
## 8.6 Dombeya rotundifolia Hocsht.
* Vernacular name: South African Wild Pear
* Family: Malvaceae
Ecological Distribution and Morphological Characteristics
Genus Dombeya comprises more than 250 species in the family Malvaceae. It is thought to originate in Africa.
It grows as an evergreen tree which is almost 7–8 m tall. The flowers are white, sweet-scented, and have a star-shaped stigma. The leaves are alternate, palmately lobed, 3 lobes prominent, cordate at the base and acuminate at the tip, and the upper laminal surface is pubescent and tomentose. Inflorescence is axillary and corymbose, and the bracts are ovate. The flower is monoecious. Epicalyx bracts are present at the base of the calyx. The petals are obliquely ovate. The stamens are 10–15 in groups of 3, the ovary is superior, and the style is 5-branched. The fruit is an ovoid to globose capsule which bears brown to black seeds (Fig. 8.5a–f).
Fig. 8.5
(a–f) Dombeya rotundifolia (wild pear) is effective in treating stomach disorders and can stimulate labor. It possesses cardioactive glycosides, saponins, lupeol, and beta-sitosterol. The plant is used due to its abortifacient activities; (a) young tree; (b) flowering buds; (c) branches bearing flowers; (d) pink flowers; (e) white flowers; (f) withered petals showing fruit development in green
Important Phytochemicals and Medicinal Value
The plant is known to possess cardioactive glycosides and saponins. Compounds isolated from the bark include lupeol and beta-sitosterol.
D. rotundifolia is used to treat nausea, headaches, dyspepsia, and stomach disorders and to stimulate labor (Reid et al. 2001; Semenya and Maroyi 2012). Methanol and hot water extracts are reported to cause contractions in rat uterus, revealing the role of the plant as an abortifacient (Ndwigah et al. 2004). Ethanolic extract of leaves and young shoots showed antibacterial activities against B. subtilis, S. aureus, and K. pneumoniae.
Traditional Uses
The plant is used for treatment of diarrhea in South Africa. It is also used to eradicate the evil effects of witchcraft in Tanzania. Traditionally, the plant is used for its antihelmintic, antidiarrheal, and abortifacient activities. The stem and root are used to treat syphilis. The wood and bark are used to treat intestinal ulcer, headaches, hemorrhoids, and diarrhea (Van Wyk et al. 2008; Thomas and Grant 1998).
## 8.7 Lantana camara L.
* Vernacular names: Shrub Verbenas
* Family: Verbenaceae
Ecological Distribution and Morphological Characteristics
L. camara, a local perennial shrub, is commonly known as wild or red sage and is the most widespread species of this genus. It is regarded both as a notorious weed and a popular ornamental garden plant. It is native to Central and South America.
The plant grows as a shrub with a tetrangular stem. The leaves are ovate, oblong, serrate, and covered with hairs. The flowers are small in clusters (umbel) with white, yellow, orange, pink, and red shades and change color upon maturity (Fig. 8.6a–g).
Fig. 8.6
(a–g) Hydroalcoholic extracts of the leaves of Lantana camara (shrub verbenas) have effects on fertility and teratology in rats. The roots are used as an oral contraceptive by women in South Africa; (a) bush; (b) pink flowers; (c) fruits (green); (d) yellow and pink flowers; (e) white flowers; (f) red flowers; (g) yellow flowers
Important Phytochemicals and Medicinal Value
L. camara contains lantadenes, the pentacyclic triterpenes which are reported to possess a number of useful biological activities and can also cause toxicity in cattle. However, sesquiterpenes like ß-caryophyllene, zingiberene, humulene, arcurcumene, gemacrene-D, and bisabolene are essential oil constituents of leaves and flowers (Nagassoum et al. 1999; Khan et al. 2002; Andersson and Dobson 2003).
Hydroalcoholic extracts of the leaves have shown an effect on fertility and teratology in rats (de Mello et al. 2003) and have potential to cause toxicity to an embryo (Kalita et al. 2012). L. scabiosaefolia is used in Northern Peru for reproductive problems and for female health (Bussmann and Glenn 2010). The roots of the plant have also been used as an oral contraceptive by women in South Africa (Dold and Cocks 2000).
Lantana spp. are known to possess antifungal, antiproliferative (Nagao et al. 2002), and antimicrobial activities (Juliani et al. 2002; Barreto et al. 2010). Leaves of Lantana are also reported to have antioxidant (Bhakta and Ganjewala 2009) and wound-healing properties (Abdulla et al. 2009).
Traditional Uses
The plant is used in folk medicine due to its antipyretic, antimicrobial, and antimutagenic activities. The leaves are used to treat ulcer, chickenpox, cancer, asthma, high blood pressure, fever, and cold. Lantana oil is used for treatment of skin and as an antiseptic for wounds.
## 8.8 Myrtus communis L.
* Vernacular names: Myrtle, Aas, Habb-ul-Aas
* Family: Myrtaceae
Ecological Distribution and Morphological Characteristics
Myrtus is an important genus of Myrtaceae with 16 known species. M. communis, commonly known as myrtle, is an evergreen aromatic perennial small tree. It is native to Europe, West Asia, and North Africa and grows throughout the Mediterranean region.
It grows up to 10 m. The leaves are aromatic, opposite, glossy, ovate to lanceolate, margined, coriaceous, and acuminate. The essential oil is secreted and released through many glands present in the leaves.
The flowers are white, star shaped, and pedicellate, with five petals, and sepals enclosing many stamens (Fig. 8.7a–d). The petals are soft and smooth, tomentose with margins covered with hairs, and contain glands. The fruit is a small handsome berry which becomes purple on ripening. Pollination takes place through insects. The seeds are kidney shaped.
Fig. 8.7
(a–d) Main activities of Myrtus communis (myrtle) are antioxidant, antifertile, and antihyperglycemic; (a) bush; (b) flowers; (c) young fruits; (d) mature fruits
Important Phytochemicals and Medicinal Value
Myrtus species are rich in essential oil, gallic and ellagic acids, and anthocyanin pigments (Romani et al. 1999; Cakir 2004). Myrtle has been a source of drug in the ancient Unani system of medicine since the Greek period. Important phytochemicals reported include tannins, flavonoids, coumarins, citric acid, and malic acid (Sumbul et al. 2011). Essential oil is also reported to have antifungal activity (Curini et al. 2003; Mohammadi et al. 2008).
The leaves are reported to have antiseptic, hyperglycemic, analgesic, and laxative properties. The root possesses antimicrobial properties. The main activities of the plant are antioxidant (Bouhle et al. 2008; Serce et al. 2010), anti-inflammatory, antifertile, abortifacient (Riddle 1994), and antihyperglycemic (Annalisa et al. 2004; Montoro et al. 2006; Alem et al. 2008; Akin et al. 2010; Nassar et al. 2010). Myrtle oil or essential oil released through the leaves is a mixture of monoterpenes, including 1,8-cineole, myrtenol, myrtenyl acetate, alpha-pinene, and limonene (Bradesi et al. 1997; Chryssavgi et al. 2008; Mimica-Dukić et al. 2010).); however, oil composition may vary depending upon geographical conditions. Myrtle oil finds wide application in the pharmaceutical industries due to its antimicrobial activities (Mansouri et al. 2001; Bouzouita et al. 2003; Appendino et al. 2006; Yadegarinia et al. 2006; Hayder et al. 2008), and in the cosmetic and perfume industries.
Myrtle berries are used as antiseptic, carminative, hair tonic, brain tonic, and cardiotonic (Montoro et al. 2006). They also possess antidiabetic activity. They contain resin, tannins, anthocyanin glucosides, kaempferol, quercitin, myricetin, rutinoside, aseculin, scopoletin, and caffeic acid. The leaves contain tannins, flavonoids, coumarins, myrtucommulone A and B, semimyrtucmmulone, ellagitannins, and ellagic acid (Annalisa et al. 2004; Feisst et al. 2005).
Traditional Uses
The leaves, twigs, and berries are used as a food source. In folk medicine, the plant is used to cure diarrhea; the leaves are used as mouthwash to treat candidiasis, for wound healing, and for urinary diseases (Mitra 1998). The plant parts are used to flavor meat and sauces in the food industry; they are also used in the cosmetic industries. Decoction of leaves is used as an antiseptic for cough and oral diseases. In France, the essential oil of the plant is also used for culinary purposes due to its antiseptic nature (Flaminia et al. 2004; Sumbul et al. 2011).
## 8.9 Ricinus communis L.
* Vernacular name: Castor Oil Plant
* Family: Euphorbiaceae
Ecological Distribution and Morphological Characteristics
R. communis is commonly known as castor oil plant. It is native to the Ethiopian region of tropical East Africa.
Castor oil plant grows as a perennial evergreen shrub but can reach the size of a small tree. It is widely cultivated for its oil seeds. The leaves are glossy, alternate, and palmately lobed with 5–11 lobes, with serrate margins. The flowers are monoecious and exhibit terminal panicle-like inflorescence (Fig. 8.8a–e). Female flowers are located above the male flowers and their stigma is feathery and lobed. The flowers have no petals. Female flowers comprise a small spiny ovary which develops into fruit. The fruit is a capsule covered with spines. The seeds are poisonous and considered to be one of the most deadly seeds among plants. The small structure on the seed end is known as a caruncle.
Fig. 8.8
(a–e) Seed extract of Ricinus communis (castor oil) plant possesses anticonceptive activity in female guinea pigs with estrogenic effects. Antiovulatory and antifertility activity is attributed to the presence of ricinoleic acid, which has spermicidal activity and can also be used as a male contraceptive agent. Oil from the plant is used for inducing labor; (a) bush; (b) red flowers; (c) flower buds; (d) male flowers; (e) fruits
Important Phytochemicals and Medicinal Value
Pharmacological activities of the plant are antiviral (Wang and Ng 2001), antioxidant (Choudhary et al. 2008), antibacterial (Kensa and Yasmin 2011), antipsychotic (Ferraz et al. 1999), antifertile (Isichei et al. 2000; Raji et al. 2006), convulsant, hepatoprotective, insecticidal (Upasani et al. 2003), and anti-inflammatory (Ilavarasan et al. 2006).
The leaves possess antilarval and ovicidal activity against mosquito larvae and contain isoquercetin, 2,5-dihydroxybenzoic, and epicatechin. Castor oil contains ricinoleic acid, dihydroxystearic, linoleic, oleic, and stearic acids (Ladda and Kamanthe 2014). The flowers contain apigenin, chlorogenin, rutin, and coumarin.
Seed extract of the plant is a mixture of phytochemicals like anthocyanin; vitamins A, C, and B6; calcium; iron; tannins; sterols; and essential oils. It possesses anthelmintic, cathartic, laxative, and purgative properties. Seed extract of the plant possesses anticonceptive activity in female guinea pigs with estrogenic effects (Makonnen et al. 1999). Ethanolic extract of the plant showed antidiabetic (Shokeen et al. 2008), antiovulatory (McNeil et al. 2003), and antifertile activities (Okwuasaba et al. 1999; Raji et al. 2006) in rats. Antifertility activity is attributed to the presence of ricinoleic acid, which has spermicidal activity, and therefore it has the potential to be used as a male contraceptive agent (Raji et al. 2006). Oil from the plant is used for inducing labor (Garry et al. 2000; Kelly et al. 2001).
Roots of castor oil plant exhibit anti-inflammatory and antioxidant activities (Iqbal et al. 2012). Crude seed extract possesses antimicrobial activities against C. albicans, P. aeruginosa, S. typhi, and E. coli. The presence of tannins indicates wide use of the plant in the leather and pharmaceutical industries. Castor oil is nontoxic, renewable, and biodegradable (Nagaraj 2009).
Traditional Uses
The plant is used traditionally to cure arthritis, paralysis, epilepsy, and urinary problems. The bark extract is applied to treat wounds and ulcers. The seed oil is a laxative and vermicide which is also used to treat ear infections as ear drops. However, due to the poisonous nature of the plant, it should be used carefully as a remedy for corns, deafness, fever, flu, epilepsy, headache, inflammation, stomachache, and toothache (Ladda and Kamanthe 2014). The flowers are used to treat glandular tumors. Fruit is known to possess hepatoprotective properties. Scalp massage with castor oil maintains hair health and removes dandruff. It is also used to treat skin infections, like scrofula, and dog bite. Castor oil is used in lubricating grease, polish, and plastics. On sulfonation, it becomes sulfated and is known as "Turkey Red Oil" and is used in cotton dyeing and in the printing industry to give luster. When dried, it is used in paint and varnishes. Due to its water-resistant property, it is used for protective covering, insulation, in food containers, and in gums. Sebacic acid is a castor oil product which is used in nylon production, resin, and fiber synthesis.
## 8.10 Saraca indica L. (Syn: Saraca asoca Roxb., De. Wild)
* Vernacular name: Ashoka
* Family: Leguminosae (Fabaceae)
Ecological Distribution and Morphological Characteristics
S. indica is a small, ancient evergreen tree, with conspicuous flowers of yellow-orange color. It is considered to be a sacred tree in countries like Nepal, India, and Sri Lanka.
The leaves are narrow, paripinnate, oblong, and lanceolate (Fig. 8.9a–c). Flowering takes place in early spring. The flowers are polygamous and apetalous, and the calyx is petaloid. Seeds are ellipsoid, oblong, and compressed.
Fig. 8.9
(a–c) Saraca indica (Ashoka) possesses aphrodisiac activities; (a) tree; (b) flowers; (c) flowers showing fruit development
Important Phytochemicals and Medicinal Value
The plant is known to contain many glycosides, flavanoids, tannins, and saponins. Antimicrobial properties are due to many phytochemicals such as alkaloids, steroids, tannins, phenols, flavanoids, steroids, fatty acids, and gums.
Phytochemical analysis of the bark showed the presence of epicatechin, procyanidin, cyanidin, catechin, leucopelargonidin, and leucocyanidin. The flowers contain oleic, linoleic, palmitic, stearic, linolenic, linoleic, and gallic acids; P-sitosterol; quercetin; pelargonidin; cyanidin; sitosterols; and leucocyanidin. However, the seeds and pods are a source of oleic, linoleic, palmitic, and stearic acids; catechol; (−) epicatechol; and leucocyanidin (Rastogi 2003).
The plant is known to have antibacterial, anticancer, and antitumor potential. Antimicrobial activities are due to the presence of catechins (Sadhu et al. 2007). It has long been used to treat gynecological disorders such as anodyne (Shirolkar et al. 2013).
Traditional Uses
S. asoca is useful in treating fever and burning sensations, and it has other pharmaceutical purposes. The bark of the plant is used as an astringent and refrigerant and to treat menstrual disorders (Pradhan et al. 2009). The dried roots are used in the treatment of paralysis and for skin diseases. Decoction of the plant is useful in treating rickets and calcium deficiency.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_9
# 9. Leguminous Trees and Their Medicinal Properties
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
This chapter gives an account of important legumes, their morphological characteristics, and compounds of medicinal value. Leguminosae comprises of over 17,000 species which constitute 1/12th of the world's flowering plants, with a diversity of herbaceous plants like peas, sweet peas, chick peas, and lentils to tall trees like golden shower and rosewood. Many of them are sources of food, fodder, and fuel, and some are medicinally important trees due to their anti-bacterial, anti-fungal, anti-microbial, anti-allergic, anti-diabetic, and anti-carcinogenic properties, which are mostly due to the presence of flavonoids (isoflavones), furanocoumarins, terpenoids, quinones, and xanthones.
## 9.1 Introduction
Leguminosae, also known as Fabaceae or legume family, comprises over 18,000 species, which constitute 1/12th of the world's flowering plants, with a diversity of herbaceous plants like peas, sweet peas, chick peas, and lentils to tall trees like golden shower and rosewood. Many ornamental trees like black and honey locust, cutch and coral, flame of forest, peacock flower, and mesquite belong to this family.
Many legumes produce boat-shaped or irregular-shaped flowers with an elongated fruit which bears seeds and is known as pod. Legumes are used industrially to produce biodegradable plastic. Isoflavones from many legumes are important anticancer compounds (Fig. 9.1a–f). Soyfood phytoestrogens are possible alternatives of hormone replacement therapy for postmenstrual women.
Fig. 9.1
(a–f) Calliandra surinamensis (syn. Acacia fasciculata), known as suriname, is an important legume against lung cancer cell lines. Important medicinal phytochemicals include β-sitosterol, β-sitosterol glucoside, β-amyrin, xanthone, and flavanol glycosides. (a) Flowering bush, (b) flower bud, (c) flower, (d) withered flowers, (e) young pods, (f) mature pods
Legume family is further divided into three subfamilies, which include:
1. (i)
Caesalpinioideae is also known as peacock family, which comprises more than 2200 species. It includes many economically important legumes like Bauhinia variegata, Cassia fistula, Caesalpinia pulcherrima, Delonix regia, Saraca indica, Tamarindus indica, and Parkinsonia spp.
2. (ii)
Faboideae have pea-shaped flowers and include genera like Cicer, Dalbergia, Erythrina, Lathyrus, Lupinus, Robinia, Sophora, and Wisteria spp.
3. (iii)
Mimosoideae includes over 3000 plants which produce bipinnately compound leaves and small-sized flowers which are composed of many stamens which are the conspicuous part of flowers, producing a capitulum or head. Medicinally important plants include Acacia spp., Albizia spp., and Mimosa spp.
Legumes like lentils, peas, and beans are a rich source of proteins due to the presence of nitrogen-fixing bacteria in their root nodules and are widely consumed worldwide. Edible and commonly used lentils include Lens culinaris, which are popular in Asian cuisine.
## 9.2 An Account of Medicinally Important Leguminous Trees
In the following section, important medicinal activities of some leguminous trees, including Acacia spp., Albizia spp., Erythrina spp., Dalbergia spp., and Prosopis spp., are discussed.
## 9.3 Acacia catechu L. (Syn: Senegalia catechu)
* Vernacular name: Cutch tree
* Subfamily: Mimosoideae
Ecological Distribution and Morphological Characteristics
Genus Acacia has more than 1000 species, and many of them are widely distributed in Africa, Asia, and Australia.
It grows as a deciduous medium-sized tree. Its leaves are bipinnately compound, having 9–30 pairs of pinnae on a glandular rachis. The flowers are white or yellow, pentamerous, with many stamens having yellow or white filaments, and appear in early summer. Pods develop in late summer, initially red in color and turn brown upon maturity (Fig. 9.2a–c).
Fig. 9.2
(a) Acacia catechu (cutch tree) possesses many pharmacological activities, which include antipyretic, antiinflammatory, hepatoprotective, and antimicrobial activities (Geetha et al. 2007). Important phytochemicals are tannic acid, catechu red, phlebotannin, cyanidol, tannins, flavonoids, and polyphenols. (b) Flowers, (c) pods, (d) A. longifolia (golden wattle) is antibacterial against S. aureus and P. aeruginosa
Important Phytochemicals and Medicinal Value
The phytochemical constituents present in different parts of A. catechu include catechuic acid, catechu, tannic acid, catechu red, and phlebotannin. Other compounds isolated are cyanidol, tannins, flavonoids, and polyphenols.
The extract of the plant possesses many pharmacological activities, which include antipyretic, anti-inflammatory, antidiarrheal, hepatoprotective, antioxidant, and antimicrobial activities (Singh and Lal 2006; Qadry 2008; Lakshmi et al. 2011).
Traditional Uses
The plant is an important source of food, timber, and leather tanning, and its heartwood is used in the dyeing industry. The extract from the wood is used as an astringent and is effective against spongy gums (Shinwari et al. 2007; Quattrocchi 2012).
## 9.4 Acacia modesta Wall. (Syn: Senegalia modesta)
* Vernacular name: Amritsar-gum
* Subfamily: Mimosoideae
Ecological Distribution and Morphological Characteristics
A. modesta is grown in many areas of Pakistan, India, and Afghanistan.
A. modesta grows as a medium-sized thorny tree with pinnately compound leaves. The flowers are white to pale yellow, fragrant, borne on auxiliary spikes, and appear in midsummer (June–August). The fruits are long pods with constrictions (Fig. 9.3a–b).
Fig. 9.3
(a–b) A. modesta (Amritsar-gum) is rich in many phytochemicals which are reported to have antibacterial activity. (a) Tree with pods, (b) pods
Important Phytochemicals and Medicinal Value
A. modesta is rich in many phytochemicals which are reported to have antibacterial activity. The most important product is the gum, which is used to treat backache (Sher et al. 2012). Its seeds are a rich source of amino acids and alkaloids (Chapman and Hall 1994). The different parts of the plant are reported to have antibacterial activity and are effective against dental diseases (Quattrocchi 2012).
Traditional Uses
The plant is used as a source of timber, food, fuel, and furniture, and for construction purposes (Sher et al. 2012). The bark of the plant is used traditionally for treating colds, bronchitis, and leukoderma. The pods and tender leaves are used in folk medicines to treat diabetes (Gilani et al.1999; Murad et al. 2011).
## 9.5 Albizia procera Roxb.
* Vernacular name: White siris
* Subfamily: Mimosoideae
Ecological Distribution and Morphological Characteristics
Genus Albizia contains almost 150 species. A. procera is a medium-sized tree which is also known as silk tree. It is native to tropical and subtropical regions of Asia and Africa.
Its leaves are bipinnately compound having leaflets 3 cm long. The flowers are aromatic, greenish yellow, appear in summer, and have long thread-like stamens that form a spike. The fruit is a narrow elongated pod (Fig. 9.4a–b).
Fig. 9.4
(a–b) Albizia procera (white siris) possesses hepatoprotective, CNS, and antioxidant activities. Many Albizia species are used for the treatment of anxiety, depression, and insomnia in traditional Chinese medicine. (a) Tree, (b) pods
Important Phytochemicals and Medicinal Value
Compounds of medicinal value isolated from different parts of the plant include many alkaloids, flavonoids, saponins, phenolic compounds, and triterpenoids (Gangwal et al. 2010). Phytochemicals isolated from the bark include 3-O-[β-dxylopyranosyl-(1 → 2)-α-L-arabinopyranosyl-(1 → 6)-2-acetamido-2-deoxy-β-D-glucopyranosyl] echinocystic acid, 5,2′,4′-trihydroxy-3,7,5′-trimethoxyflavonol-2′-O-β-D-galactopyranosyl-(1 → 4)-O-β-D-glucopyranoside (Sujatha et al. 2013).
The plant is reported to have potential hepatoprotective activity. Many pharmacological activities have been reported, including Central nervous system (CNS) activity, cardiotonic activity, and antioxidant activity (Mar et al. 1991). Petroleum ether extract of the plant possesses natural antioxidants and can be used to treat diseases associated with free radicals (Khatoon et al. 2013). A new cytotoxic compound echinocystic acid 3,16-O-bisglycosides from the bark is extracted, which is known to possess anticancer activity against HEPG2, A549, HT29, and MCF7 cell lines (Miyase et al. 2010).
Traditional Uses
The tree is used as a source of timber, nitrogen fixing, agricultural implements, shade, furniture, and tannin. Many Albizia species are used for the treatment of anxiety, depression, and insomnia in traditional Chinese medicine. The bark is used for treatment of wounds due to its antimicrobial properties (Chopda and Mahajan 2009). Due to the presence of albizzine, neurotoxicity is reported in mammals.
## 9.6 Cassia fistula L.
* Vernacular name: Golden shower tree
* Subfamily: Caesalpinoideae
Ecological Distribution and Morphological Characteristics
C. fistula is a tall deciduous tree, native to southern Pakistan, Thailand, and Sri Lanka. It is planted in the east of Indus in the plains to the north of Himalayas up to an elevation of 1200 m.
The leaves are compound, divided into four to six pairs of oval-shaped leaflets. Golden yellow flowers appear in early summer. The flowers are large and aromatic. They are borne on long and slender pedicels in pendant axillary racemes which are 30–60 cm long. The fruit is a legume with black seeds (Fig. 9.5a–e).
Fig. 9.5
(a–e) Cassia fistula (golden shower tree) possesses antidiabetic, anti-inflammatory, and antimicrobial activities. The fruit is edible and a rich source of iron and manganese. (a) Tree, (b) flowering buds, (c) inflorescence, (d) flower, (e) tree bearing pods
Important Phytochemicals and Medicinal Value
The leaves are a source of anthraquinone, oxyanthraquinone, rhein, and volatile oils. The fruit pulp comprises many amino acids like arginine, leucine, methionine, phenylalanine, tryptophan, aspartic, and glutamic acids (Rastogi and Mehrotra 2004). Other compounds of commercial value isolated include palmitic acid, oleic acid, linoleic acid, heptacosyl eicosanate, lupeol, anthraquinones, chrysophanol, physcion, citreorosein, rhein, rhein methyl ester, ziganein, 1,4,5-trihydroxyanthraquinone, coumarins, isoscopoletin, scopoletin, and two chromones 2,5-dimethyl-7-hydroxychromone and 2,5-dimethyl-7-methoxychromone. Three aromatic compounds isolated from aril are isovanillic acid, vanillic acid, and 2,4-dihydroxybenzaldehyde. The pulp of the plant is rich in tannins, albuminous starch, gluten, and gum.
The fruits are effective in the treatment of diabetes and also in reducing inflammation. The fruit is edible and is a rich source of iron and manganese (Lim 2012). However, some toxicity of the fruit pulp is also reported due to the presence of emodin glycoside (Nelson et al. 2007). The methanol extract of the seeds has also revealed some antimicrobial activity (Lachumy et al. 2010).
Traditional Uses
The plant is used as a source of fuel and the wood is used in making durable furniture (Sheikh 1993). The root of the plant is used traditionally in fever. The fruits are used as a cathartic and in snake bite. Antioxidant properties of its leaves, stem bark, pulp, and flowers are reported (Siddhuraju et al. 2002). The flowers and pods are used as purgatives and to treat asthma.
## 9.7 Dalbergia sissoo Roxb.
* Vernacular name: Rosewood tree
* Subfamily: Faboideae
Ecological Distribution and Morphological Characteristics
Dalbergia is an important member of the family Leguminosae, which has 300 species. It is native to the subcontinent and Southern Iran.
It grows as a tall perennial tree. The leaves are alternate, imparipinnately compound, having three to five leaflets on a long stalk. The flowers appear in axillary panicles from March to May and are white to pink in color. The pods are linear, oblong, indehiscent, having (one to four) kidney-shaped seeds (Fig. 9.6a–c).
Fig. 9.6
(a–c) Dalbergia sissoo (rosewood tree) is known to have anti-inflammatory, analgesic, antipyretic, and larvicidal activities. Important phytochemicals are dalbergenone, dalbergin, methyl dalbergin, phenyl chromene, dalbergichromene, nordalbergin, and isodalbergin. (a) Tree, (b) flowers, (c) pods
Important Phytochemicals and Medicinal Value
D. sissoo contains many important compounds like dalbergenone, dalbergin, methyl dalbergin, phenyl chromene, dalbergichromene, nordalbergin, and isodalbergin as minor constituents (Farag et al. 2001; Suraj et al. 2008; Bharath et al. 2013; Bhattacharya et al. 2014). Flavonoids are reported to be present in many species, which include prostaglandin synthetase inhibitors.
All parts of the plant are medicinally important and are known to have many anti-inflammatory, analgesic, antipyretic, and antimicrobial activities. The dried leaves are reported to have antibacterial, antiprotozoal, and anti-inflammatory activities (Hajare et al. 2001). The extract of aerial parts possesses antipyretic, analgesic, and estrogen-like activities (Kamboj 2000). The stem bark is abortifacient and antipyretic. Fresh flowers are also reported to possess estrogen-like activities (Bhattyacharya et al. 2014).
Traditional Uses
Dalbergia is a source of timber, fuel, furniture, and paper (Sheikh 1993). It is grown as a shade tree in tea plantations. The plant has various biological activities that are effective against blood diseases, scabies, syphilis, nausea, ulcers, and many skin and eye diseases. The oil from the plant possesses repellent activity (Bent 2004). The juice of the leaf is used to treat gonorrhea (Sath and Sharma 2004).
## 9.8 Delonix regia Raf.
* Vernacular name: Royal poinciana
* Subfamily: Caesalpinoideae
Ecological Distribution and Morphological Characteristics
Delonix regia is native to Malagasy and is widely planted in Africa and Asia.
It is an ornamental tree with fern-like leaves. The flowers are large and are scarlet in color, in axillary racemes. The seed pods are dark brown in color (Fig. 9.7a–b).
Fig. 9.7
(a–b) Delonix regia (royal poinciana) contains afzelin, astragalin, rhamnoside, rutinoside, neohesperidoside, quercetin 3-rutinoside, and quercetin 3-glucoside. The petals of flowers are known to have antimicrobial activities against S. aureus, S. typhimurium, S. paratyphi, S. typhi, and E. coli. (a) Leaves, (b) branches bearing flowers, (c) close view of flowers
Important Phytochemicals and Medicinal Value
Many important compounds are isolated from the leaf extracts, which include afzelin, astragalin, rhamnoside, rutinoside, neohesperidoside, quercetin 3-rutinoside, and quercetin 3-glucoside (isoquercetrin) (Azab et al. 2013).
The wood of D. regia contains high concentrations of polyphenolic compounds like anthocyanins and flavonols. The bark comprises bioactive natural compounds which have antioxidant and antibacterial properties (Fatmawaty and Astuti 2013; Teow et al. 2007).
The methanolic extract of D. regia stem bark showed the presence of lupeol, epilupeol, β-sitosterol, and stigmasterol (Jahan et al. 2010). Additionally, the bark extracts also showed gallic acid and other phenolic acids such as caffeic, ferulic, sorbic, sinapic, p-coumaric, m-coumaric, hydroxybenzoic, and hydroxycinnamic acids.
The anti-inflammatory properties of the leaves and bark might be attributed to many alkaloids and glycosides and terpenoids. Some studies have revealed the antifungal and anticancer properties of the plant. The leaves are also reported to have antibacterial and antimalarial properties (Parekh et al. 2005). The petals of the flowers comprise 29 carotenoids, and the flower is reported to have antimicrobial activities against S. aureus, S. typhimurium, S. paratyphi, S. typhi, and E. coli (Lim et al. 2012).
Traditional Uses
The plant is commonly grown for the beauty of its flowers and also as a shade tree. Many alkaloids, flavonoids, and phenolic compounds are isolated from the bark of D. regia, which have antibacterial properties (Salem 2013).
## 9.9 Erythrina suberosa Roxb.
* Vernacular name: Coral tree
* Subfamily: Faboideae
Ecological Distribution and Morphological Characteristics
Erythrina comprises more than 100 species which are distributed in tropical and subtropical regions worldwide.
E. suberosa is a medium-sized tree, with conical spines on its stem and branches. The bark is smooth when young and becomes corky with age. The flowers are bright red to scarlet in color (Fig. 9.8).
Fig. 9.8
Erythrina suberosa (coral tree) is reported to have antiplasmodial, anxiolytic, and cytotoxic activities against various cancer cell lines. Important phytochemicals include many phenolic compounds like isoflavones, chalcones, and pterocarpens
Important Phytochemicals and Medicinal Value
E. suberosa contains many phenolic compounds like isoflavones, chalcones, and pterocarpens. Other compounds reported include alkaloids like erysodienone, erythatine, and sterols like campesterol.
It is reported to have antiplasmodial and anxiolytic properties, and cytotoxic activity against various cancer cell lines. Flavonoids extracted from the stem bark (4′-methoxy licoflavanone (MLF) and Alpinumi soflavone (AIF)) showed cytotoxic effects in HL-60 cells (human leukemia). Due to the presence of squalene, a natural antioxidant, phytol, and other phenolic compounds, E. variegata possesses antimicrobial, anticancer, gastroprotective, hepatoprotective, and sunscreen properties (Sunitha et al. 2001; Amarowicz 2009; Ezhilan and Neelamegam 2012; Raman et al. 2012).
Traditional Uses
It is an important landscape management tree (Sheikh 1993). The plant is an important source of fuel. The bark is used in the treatment of fever, liver ailment, and rheumatism. The leaf extract has nematicidal properties. The bark is also used as an astringent and as a febrifuge. The decoction of the bark is effective in ophthalmia and other eye diseases. The leaves are used to cure arthritis, liver problems, and convulsions. The crushed leaf juice is effective in relieving pain and inflammation in rheumatic joints (Krtikar and Basu 1991). The juice of whole plant is used to cure chronic dysmenorrhea and sterility in fat women as it can reduce abdominal fats and can induce natural menstrual flow (Nadkarni et al. 1991).
## 9.10 Millettia ovalifolia (Syn: M. peguensis)
* Vernacular name: Rosewood tree
* Subfamily: Faboideae
Ecological Distribution and Morphological Characteristics
Genus Millettia consists of almost 150 species which are distributed in tropical and subtropical regions worldwide.
It is a tall ornamental deciduous tree and widely grown for its ecological and medicinal values. The leaves are imparipinnately compound, while the fruits are cylindrical pods 6.5–9 cm long (Fig. 9.9a–d).
Fig. 9.9
(a–d) Millettia ovalifolia (rosewood tree) comprises commercially important flavonoids and chalcones which have antibacterial activity. (a) Tree, (b) inflorescence, (c) flowers, (d) pods
Important Phytochemicals and Medicinal Value
M. ovalifolia comprises commercially important flavonoids and chalcones which have antibacterial activity (Rahman et al. 2013). Some of the species like M. pachycarpa are reported to inhibit activities of retroviral reverse transcriptase and human DNA polymerases. Other compounds like rotenone and 3α-hydroxyrotenone isolated from M. pervilleana extract are chemopreventive agents and have inhibitory effects on 12-O-tetradecanoylphorbol-13-acetate (TPA)-induced ornithine decarboxylase at the level of its m-RNA expression.
The ethyl acetate fraction of the bark of M. ovalifolia is reported to have antibacterial, antifungal, and insecticidal activities. Compounds isolated from M. pervilleana like prenylated isoflavanones are reported to have anticancer activity (Rahman et al. 2013).
Traditional Uses
It is well known for its insecticidal, molluscicidal, and pesticidal activities. Many species of Millettia are also used as potential inhibitors of different intestinal parasites and also as anticancer agents.
## 9.11 Parkinsonia aculeata L.
* Vernacular names: Mexican palo verde, Jerusalem thorn
* Subfamily: Caesalpinoideae
Ecological Distribution and Morphological Characteristics
Genus Parkinsonia comprises 15 species which are found in tropical America; however, 4 are found in Africa (Bibsy 1994). It is cultivated for its ornamental value. In Asia it is grown in Pakistan, Cambodia, India, Sri Lanka, Thailand, and Vietnam.
It is a small evergreen tree which can grow up to a height of 7–10 m. Flowering takes place from March to April and from September to October. The leaves are alternate, pinnate and consist of spiny-ended axis and resemble a blade of grass. The bark is smooth and yellow-green; the twigs are slender, slightly zigzag, and covered with spines. The flowers are in clusters, irregular, having yellow sepals that are fused at the base, five petals, and ten stamens. The fruit is an elongated pod having a hard texture and an oval shape (Fig. 9.10a–c).
Fig. 9.10
(a–c) Parkinsonia aculeata (Jerusalem thorn) possesses antidiabetic, hepatoprotective, antipyretic, abortifacient, and antimalarial activities due to phytochemicals like glycosides, flavonoids, sterols, orientin, vitexin, and iso-vitexin, Parkinsonin A, Parkinsonin B, and Parkintin. (a) Tree, (b) flowers, (c) spiny bark
Important Phytochemicals and Medicinal Value
Parkinsonia is known to possess antioxidant (Mruthunjaya and Hukkeri 2008; Sharma and Vig 2014), hepatoprotective (Hassan et al. 2008), antipyretic (Gupta et al. 2011), diaphoretic, abortifacient, and antimalarial activities. The flowers, leaves, and stems contain phytochemicals like glycosides, flavonoids, sterols, and minerals. The leaves also contain C-glycosyl flavones like orientin, vitexin, and iso-vitexin, Parkinsonin A, Parkinsonin B, and Parkintin (Nabil et al. 1997; Ali et al. 2005). The bark extract is known to possess antidiabetic activities (Saha et al. 2011).
The extract of the dried parts possesses CNS depressant activity. The aerial parts of the plant (leaves and flowers) possess antimalarial activity (Gupta et al. 2011). The antiamoebic activity is attributed to the presence of rotenoids (Kamal and Mathur 2007). Phytochemicals present in the dried parts include beta-amyrenone, beta-amyrin, daucosterol, palmitic acid, and sterol. The stem possesses antimicrobial activities due to the presence of chemicals like glycerol β-butanoate α,α′ 1-dipentanonate, β-Sitosterol-β-D-glucoside, β-Sitosterol, and glycerol α-heptanone kappa octanoate (Ali et al. 1999; Saha et al. 2011). Its antimicrobial activity is reported against E. coli, P. aeruginosa, S. typhi, and S. aureus, and antifungal activity against C. albicans, A. niger, M. canis, T. simii, and M. phaseolina (Divya et al. 2011).
Traditional Uses
The leaves, fruits, and stems are consumed orally to treat malaria and fever (Sharma and Vig 2014). The leaves and flowers are applied as poultice to cure rheumatism. The plant is used as a source of charcoal and as fodder for livestock.
## 9.12 Prosopis juliflora Swart.
* Vernacular name: Mesquite
* Subfamily: Mimosoideae
Ecological Distribution and Morphological Characteristics
Genus Prosopis comprises more than 44 species which are widely distributed in the dry regions of Southwest Asia and Africa. It is native to Mexico, South America, and Caribbean.
P. juliflora grows as a small tree and attains a height of up to 10–12 m. The leaves are pinnately compound and form a group of leaflets. The flowers are small, greenish yellow and formed as clusters (Fig. 9.11a–f). The fruits are pods containing almost 10–30 seeds. Prosopis is reported to be an invasive weed in many countries and difficult to remove because plants can regenerate through the roots.
Fig. 9.11
(a–f) Prosopis spp. (mesquite) are medicinally important legumes and possess antimicrobial, antidiabetic, anti-inflammatory, aphrodisiac, and insecticidal activities. (a) P. cineraria, (b, c) leaves of P. cineraria, (d) catkin, (e) pods, (f) P. juliflora leaves
Important Phytochemicals and Medicinal Value
Important compounds include flavonoids which occur in almost all parts of the plant, including the root, heartwood, sapwood, bark, leaf, fruit, and flower. Some alkaloids reported are known to have antifungal and antibacterial activities. Other constituents are thiamine, thiamine diphosphate, thiamine monophosphate, proteins, and propionic acid.
Saponins extracted from the plant are known to possess antimicrobial and insecticidal properties (Vincken et al. 2007; Ibrahim et al. 2013). The alkaloid fraction of the plant possesses antifungal activity (Ibrahim et al. 2013). The antifungal and antimicrobial activities of the leaves are due to the presence of juliflorine, julifloricine, juliprosopine, julifloravizole, Secojuliprosopinal, and mesquitol (Khursheed et al. 1986; Ahmad et al. 1992; Nakano et al. 2004; Sathiya and Muthuchelian 2008). The anti-inflammatory activity of the bark extract is reported due to compounds like juliprosopine and seojulirprsopinol (Sivakumar et al. 2009).
Traditional Uses
The pods contain antinutritional factors and are toxic. The phenolic compounds from different the parts of the plant are used in cardiovascular problems. It is an important nitrogen-fixing and soil stabilization tree. It is commonly used in folklore medicines. The crude fiber content is commonly used as a measure of the nutritive value of poultry and livestock feeds.
## 9.13 P. spicigera L. (Syn: P. cineraria)
* Vernacular name: Mesquite
* Subfamily: Mimosoideae
Ecological Distribution and Morphological Characteristics
P. spicigera is a deciduous, xerophytic, and drought-resistant tree which grows in the tropical regions of Asia. The plant is known as the "king of desert" or "wonder tree" due to its medicinal and economic values. It is also grown for its ecological role in soil improvement and stabilization of sand dunes.
It is a small thorny and irregularly branched tree. The bark is gray and thick, having deep fissures. The leaves are glabrous, petiolate, and stipulate (Fig. 9.11a–f). The flowers are small, yellow, and are produced in March–May. The fruits are in the form of pods.
Important Phytochemicals and Medicinal Value
Many flavones, fatty acids, and tannins, and spicigerin and prosogerins are obtained from phytochemical extracts. Tannins like pyrogallotanins and pyrocatecollic acids are found in the bark. The leaves are a source of linoleic and oleic acids (Malik and Khalidar. 2007). Many glucosides are extracted from flowers, and flavones from seeds (Rastogi and Mehrotra 1995; Gangal et al. 2009). Extracts from the unripen pods revealed the presence of many tannins, alkaloids, and glucosides (Khan et al. 2006). The seeds contain prosogerin-C, -D, and -E, patulitrin, luteolin, and rutin (Ukani et al. 2000). Patulitrin possesses significant cytotoxic activity against Lewis lung carcinoma, while luteolin is known to inhibit skin cancer (Lopez-Lazaro 2009).
Pharmacological activities of the plant are analgesic, antipyretic, antitumor, and nootropic (Bithu et al. 2012; Garg and Mittal 2013). The analgesic and antipyretic activities of ethanolic, aqueous, and petroleum ether extracts are reported through hot plate model and yeast-induced hyporexia in experimental rats (Ramasamy et al. 2009; Kumar et al. 2011; Joseph et al. 2011). The antihyperglycemic activity of 50% hydroalcoholic extract of the stem bark is reported at a dose of 300 mg/kg in alloxan-induced mice (Sharma et al. 2010). The antitumor activity of the hydroalcoholic extract of the leaf and bark is also reported at a dose of 200 and 400 mg/kg against Ehrlich ascites carcinoma tumor (Robertson et al. 2011).
Traditional Uses
The leaves are used as a source of fodder, and the wood is analgesic. The fruits are a rich source of vitamins. The bark of the tree is used in the treatment of leprosy, dyspepsia, asthma, scorpion sting, and earache. The pods are used as vegetables, and the flowers are used commercially for honey production.
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Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_10
# 10. Figs and Their Medicinal Value
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
This chapter includes medicinal properties of Ficus species like Ficus benghalensis, F. carica, F. elastica, F. lyrata, F. infectoria, F. religiosa, F. macrophylla, and F. virens. Figs are medicinally important fruits of many Ficus species, which are not only a source of many micronutrients but also possess many antidiabetic, anticancer, and antimicrobial activities due to the presence of compounds like alkaloids, triterpenoids, ascorbic acid, and flavonoids.
## 10.1 Introduction
Ficus is one of the large and important genuses of the family Moraceae which is known to comprise over 800 species and 2000 varieties of trees which grow in tropical and subtropical regions. The fruits of Ficus spp. are small, green, and round and are commonly known as figs. Some figs are sweet and edible like F. carica, but many of them are not edible. However, they are medicinally important fruits which are not only a source of many micronutrients but also possess many antidiabetic, anticancer, and antimicrobial activities (Lansky et al. 2008). The presence of many compounds like alkaloids (Baumgartner et al. 1990), triterpenoids, ascorbic acid (Khan et al. 1993), and flavonoids (Ilyas and Ilyas 1990) is detected in many species of Ficus. Phenolic compounds are major components of many figs (Veberic et al. 2008; Basudan et al. 2005; Sheu et al. 2005). Many Egyptian Ficus species are used in folk medicine to treat many respiratory disorders and skin diseases (Mousa et al. 1994).
## 10.2 Summary of Trees That Produce Medicinally Important Figs
Medicinal activities of important figs are described in the following sections.
## 10.3 Ficus benghalensis L.
* Vernacular name: Banyan tree
Ecological Distribution and Morphological Characteristics
F. benghalensis is a large evergreen tree. It is native to the Indian subcontinent, where it is considered to be a sacred tree and grown around temples due to its religious value in many cultures.
It grows as an epiphyte when young and develops from seeds which are dropped by birds on other trees and is therefore also considered destructive to forest trees (Fig. 10.1a–d). The fruits (receptacles) are sessile, axillary in pairs, and scarlet or brick red when ripe. Male flowers are dispersed with female flowers. Flowers appear during summer. The bark is smooth, light gray to white, and hard (Michael et al. 1997; Hayashi et al. 2007).
Fig. 10.1
(a–d) Ficus benghalensis (banyan) is an important medicinal fig with antimicrobial and antidiabetic activities. Important phytochemicals of medicinal value are leucoanthocyanin, friedelin, beta-sitosterol, and flavonols quercetin and rutin. (a) Growing leaf, (b) mature leaf, (c) young fruits, (d) red figs
Important Phytochemicals and Medicinal Value
F. benghalensis is an important medicinal plant. Many workers reported that almost every part of this tree possesses antimicrobial activities against a wide range of bacteria as revealed through aqueous, ethanolic, methanolic, and hexane extracts. These medicinal activities are due to different phytochemicals which are present within different parts of the plants. The bark of the plant contains important flavonoid compounds which are classified as A, B, and C, and are effective as hypoglycemic agents. Compounds A and C are characterized as leucoanthocyanidins, while B, as leucoanthocyanin (Khare 2004a, b). The leaves contain friedelin, β-sitosterol, flavonols, quercetin-3-galactoside, and rutin. The heartwood contains tiglic acid ester of si-taraxasterol.
The antimicrobial activity of the compounds isolated from the bark is very similar to commercial antibiotics, which indicates that the plant can be used for the production of new antibiotics. The bark and ethanolic root extract possess antibacterial activities against S. auerus, P. aeruginosa, and Klebsiella pneumoniae (Gayathri et al. 1998; Murti and Kumar 2011); however, the root extract is also reported to be more significant against S. aureus (Singh and Watal 2010). The bark extracts showed strong antibacterial activities against P. aeruginosa, E. coli, P. vulgaris, and E. faecalis (Manimozhi et al. 2012). The methanolic extract is revealed to be inhibitory against B. subtilis at 100 mg/ml (Jagtap et al. 2012). However, the hexane extract of the leaves showed more resistance against K. pneumoniae, P. aeruginosa, and M. luteus (Koona and Rao 2012). The plant is also reported to be effective in the treatment of diabetes due to the presence of leucopelargonins isolated from roots, barks, and fruits (Saravanamuttu and Sudarsanam 2012).
Traditional Uses
The latex and fruits are used to relieve pain and used to treat ulcers, soles of inflamed feet, toothaches, rheumatism, and lumbago.
## 10.4 F. benjamina L.
* Vernacular names: Fig, Weeping Fig
Ecological Distribution and Morphological Characteristics
F. benjamina is native to many regions of Southeast Asia. It is cultivated in many parts of the world, including American Samoa, French Polynesia, and Florida in the USA.
It is a medium-sized tree with a spreading crown and is also known as weeping fig (Fig. 10.2a–c).
Fig. 10.2
(a–c) Ficus benjamina (weeping fig) possesses antimicrobial, antipyretic, and antinociceptive properties due to the presence of quercetin, naringenin, cinnamic acid, naringenin, quercetin, caffeic acid, and stigmasterol. (a) Tree, (b) leaves, (c) figs
Important Phytochemicals and Medicinal Value
The leaves, fruits, and bark extracts of the plant revealed the presence of phytochemicals like cinnamic acid, naringenin, quercetin, caffeic acid, and stigmasterol (Almahy et al. 2003a). Antibacterial activities are due to the presence of quercetin and naringenin. Cinnamic acid revealed strong activities against gram-negative bacteria when compared with gram-positive bacteria. Hydrodistillation and Gas-chromatography mass spectrometry analysis (GC-MS) showed the presence of nine essential oils in its stem and roots. Essential oils of the stem include compounds such as 2-pentanone, hexadecanoic acid, palmitic acid, and 9,12-octadecadienoic acid. The roots contain eight compounds: methanamine, cyclopentanone, methyl-2 phenylindole, cyclopropaneoctanal, arsenous acid, hexadecanoic acid, palmitic acid, and 9,12-octadecadienoic acid (Imran et al. 2014). The roots and leaves are a source of antioxidants, while the stem exhibits antimicrobial activities (Dai et al. 2012). Ethanolic and aqueous extract of fruits revealed the presence of alkaloids which possess antimicrobial activities against S. aureus, E. coli, S. typhi, K. pneumoniae, and P. aeruginosa (Novelli et al. 2014). Ethanolic extract of the leaves showed antiviral activities against herpes simplex virus-1 and -2 (HSV-1 and HSV-2) and varicella-zoster virus (VZV), and fruit extracts inhibited only VZV activity (Yarmolinsky et al. 2009).
Traditional Uses
The plant possesses antimicrobial, antipyretic, and antinociceptive properties. The leaves and twigs are used as insect repellents (Parajuli 2000; Kanaujia et al. 2011). Traditionally, the plant is used as a stomachic, hyposensitive, and antidysentery agent. The latex and fruit extract are used for treating skin disorders, inflammation, piles, leprosy, malaria, and also cancer.
## 10.5 F. carica L.
* Vernacular name: Common Fig, Fig
Ecological Distribution and Morphological Characteristics
F. carica is an ancient plant and has been cultivated since long in human history. The nutritive value of figs is also mentioned in the holy Quran and in the Bible. It is native to Southwest Asia and eastern Mediterranean but is also grown in many tropical and subtropical countries due to its edible fruit which is medicinally important. Turkey, Egypt, Spain, Morocco, Greece, Italy, USA, Syria, and Brazil are major fig-producing countries.
F. carica grows as a deciduous tree up to a height of 10 m. Its leaves are petiolate, palmately lobed having a cordate base and is aromatic and serrate with reticulate venation. The inflorescence is pear shaped and bears male and female flowers on the inner surface. The syconium (edible fruit) is pear shaped, hollow, fleshy, and bears many seeds (Fig. 10.3a–e).
Fig. 10.3
(a–e) Ficus carica (common fig) is known to have antimicrobial, aphrodisiac, antipyretic, and antidiabetic activities. The leaves and fruits are rich sources of phenolics, proanthocyanidins, cyanidin and aglycone and pelargonidin derivatives. (a) Tree, (b) leaves with young figs, (c) young figs, (d) mature figs, (e) dried figs
Important Phytochemicals and Medicinal Value
The leaves and fruits of F. carica are a rich source of phenolics, proanthocyanidins, organic acids, and volatile compounds. Phytochemical screening showed the presence of psoralen, bergapten, umbelliferone, β-sitosterol, campestrol, and fucosterol.
Figs possess almost 15 anthocyanin pigments; most common among them are cyanidin, aglycone, and pelargonidin derivatives. The presence of many volatile compounds like benzyl alcohol, furanoid, linalool, cinnamic aldehyde, eugenol, angelicin, caryophyllene, and bergapten are detected in the pentane extract of the plant (Gibernau et al. 1997). Antipyretic and antituberculosis activities of the ethanolic extract of the plant are also reported (Patil et al. 2010; Khadabadi et al. 2007). The antimicrobial activity of the latex fraction of the plant is reported against human pathogens (Aref et al. 2010). The leaf extract of figs is also known to possess hepatoprotective properties (Gond and Khadabadi 2008).
The plant is used for various therapeutic benefits, including cardiovascular, respiratory, and anti-inflammatory problems (Guarrera 2005). The leaves are also used in Malaysia to cure anemia and tuberculosis (Khadabadi et al. 2007; Mohamad et al. 2011). Antimicrobial activities of the latex extract are reported against a wide range of microbial organisms (Jeong et al. 2009; Aref et al. 2010). The leaves are also known to possess antipyretic activities (Patil et al. 2010). The methanolic extract of the fig leaves possesses antibacterial activities against S. gordonii, S. anginosus, P. intermedia, and P. gingivalis.
Traditional Uses
In Unani medicine, F. carica is used as a mild laxative, expectorant, and diuretic. The wood is a source of timber, fuel, and coal. The plant is used traditionally to treat gastric disorders, inflammation, liver and spleen disorders, and also cancer. The dried fruit is known to have antidiabetic activities (Khare 2004a, b). The leaves are also used to cure jaundice (Nadkarnim and Nadkarni 1995). Figs are also used with walnuts in Unani medicine as an aphrodisiac tonic. Figs are consumed either raw or as dried fruits. The use of fig is very common in confectionary products, including jams, murabbas, cakes, jellies, and other desserts.
## 10.6 F. elastica (Roxb.)
* Vernacular name: Rubber Fig
Ecological Distribution and Morphological Characteristics
F. elastica is native to Nepal, India, and China. It is cultivated in Pakistan for to its economic value. It is a large tree with a stout trunk and oval leaves that produce white milky latex which is used to make rubber. The young juvenile leaves are curled, sheathed, and dark red in color, but become green upon maturity (Fig. 10.4a–c).
Fig. 10.4
(a–c) Ficus elastica (rubber fig) is an important antimicrobial and anti-inflammatory fig with important phytochemicals like chlorogenic acid, rutin, luteolin, coumarins, glucopyranosides, citroside, feroxidin, icarisides, kaempferin, myricitrin, syringin, roseoside, oleanolic acid, ursolic acid, and quercitrin. (a) Tree, (b) young sheathed leaf, (c) F. elastica "variegata"
Important Phytochemicals and Medicinal Value
The plant is a rich source of commercially important metabolites, which include chlorogenic acid, rutin, luteolin, and coumarins that have antimicrobial, anti-inflammatory, and antioxidant properties (Almahyl et al. 2003b; Kiem et al. 2012). Other important metabolites isolated from the different plant parts are benzyl O-β-D-glucopyranosides, citroside, feroxidin, icarisides, kaempferin, myricitrin, syringin, roseoside, oleanolic acid, ursolic acid, and quercitrin.
Traditional Uses
The leaf extract is used for the treatment of skin infections and skin allergies. It is a rich source of many antioxidants (Kiem et al. 2012). It is also reported to have wound-healing properties (Busso and Ea 2011; Armeena et al. 2013).
## 10.7 F. glomerata Roxb. (Syn: F. racemose L.)
* Vernacular name: Cluster Fig Tree
Ecological Distribution and Morphological Characteristics
F. glomerata grows as a medium-sized deciduous tree without aerial roots and is popular in indigenous system of medicines like Ayurvedic, Unani, and Homeopathy. It is native to Southeast Asia, Australia, and Malaysia. Figs grow very close to the tree trunk. In Hinduism, this tree is associated with prosperity and bringing peace in life. The leaves are ovate, elliptic, and petiolate. Figs are globose and become red on maturity (Fig. 10.5a, b).
Fig. 10.5
(a, b) Ficus glomerata (cluster fig) possesses antibacterial activity against B. subtilis, P. aeruginosa, E. coli, P. vulgaris, and S. aureus. The bark is abortifacient while latex is aphrodisiac. (a) Tree, (b) figs on trunk
Important Phytochemicals and Medicinal Value
The stem bark extract possesses antibacterial activity against B. subtilis (Mahato and Chudary 2005). The aqueous, methanolic, and ethanolic bark extracts showed mild antibacterial activities against P. aeruginosa, E. coli, P. vulgaris, and S. aureus (Manimozhi et al. 2012). The ethanolic root extract also revealed antibacterial activity against S. aureus (Murti and Kumar 2011).
Traditional Uses
The aqueous extract of the leaves of F. glomerata and F. retusa is mixed with sugar and honey and used for treating diarrhea. The extract from the bark, root, leaf, latex, and fruit is used for treating stomach disorders, diabetes, and used as a carminative and an astringent. The leaves and roots are used in dysentery and diabetes. Infusion of the leaves and bark is used as a mouthwash for spongy gums. The bark is an abortifacient while the latex is an aphrodisiac.
## 10.8 F. infectoria Miq. (Syn: F. virens)
* Vernacular name: White Fig
Ecological Distribution and Morphological Characteristics
F. infectoria is a medium-sized tree which grows up to 10–12 m in height. It is native to Nepal, India, and Pakistan, where it is grown for its ecological and economic value. The fruits are small and greenish white in color. The plant begins its life cycle as an epiphyte like many other Ficus species. The leaves are oblong, lanceolate, glossy, thick, and alternately arranged. The flowers are inconspicuous and white in color (Fig. 10.6a–d). The color of bark is brown and new leaves are reddish pink (Khare 2007).
Fig. 10.6
(a–d) Ficus infectoria (white fig) is reported to have many antibacterial, antifungal, and hyperglycemic activities. (a) Tree, (b) bark, (c) leaves, (d) figs
Important Phytochemicals and Medicinal Value
Phytochemical screening showed the presence of many phytoconstituents like alkaloids, glycoside, amino acid, phytosterols, tannin, and flavonoids (Chandira et al. 2010).
F. infectoria is reported to have many antibacterial, antifungal, and hyperglycemic properties. Some research suggests that the plant is also effective against diabetes (Kumar et al. 2012).
Traditional Uses
Traditionally, a decoction made from the bark is used for treating ulcers. It is grown as an avenue tree due to its ornamental value. The plant is used as a source of food, fuel, latex, and timber. The leaves are used in Thai cuisine. The tree is used as a shade tree in coffee plantations. Its young shoots and leaves are edible. The stem and bark are used in medicine especially for blood diseases. The timber is used for plywood, disposable chopsticks, molding, and in making other ornamental items.
## 10.9 F. lyrata Warb. (Syn: F. sycomorus)
* Vernacular name: Fiddle-leaf fig
Ecological Distribution and Morphological Characteristics
F. lyrata is an evergreen tree which is indigenous to tropical, central, and west Africa. It is used as a shade tree and is suitable for indoor growing. It is grown for decoration in Europe and in North America. The leaves are bright, fresh, pinnate, having a lyre-shaped blade. The fruits are solitary, fleshy, round, and green (Fig. 10.7).
Fig. 10.7
Ficus lyrata (fiddle-leaf fig) possesses potent antibacterial activities against several bacterial strains, including P. aeruginosa, S. aureus, Shigella spp., C. freundii, P. vulgaris, and Klebsiella. Phytochemical screening showed the presence of compounds such as arabinose, β-amyrins, β-carotenes, glycosides, β-setosterols, and xanthotoxol
Important Phytochemicals and Medicinal Value
The plant possesses potent antibacterial activity against several bacterial strains, including P. aeruginosa, S. aureus, Shigella spp., C. freundii, P. vulgaris, and Klebsiella, and its aqueous extract is more potent than its ethanolic extract. It also possesses diuretic and Central nervous system (CNS) depressant activities. Phytochemical screening showed the presence of compounds such as arabinose, β-amyrins, β-carotenes, glycosides, β-setosterols, and xanthotoxol (Jeong and Lachance 2001; McGovern 2002; Vaya and Mahmood 2006). The leaves contain chromone, flavonoids, and β-sitosterol.
Traditional Uses
The plant is used as a source of food, fuel, latex, and timber. Figs are used to control diabetes.
## 10.10 F. macrophylla L.
* Vernacular names: Moreton Bay Fig, Strangler Fig
Ecological Distribution and Morphological Characteristics
F. macrophylla is a large evergreen tree which is native to Australia. The plant produces buttress roots, elliptic dark green leaves, and monoecious flowers (Fig. 10.8a–d).
Fig. 10.8
(a–d) Ficus macrophylla (strangler fig) is used for gastrointestinal, respiratory, inflammatory, and cardiovascular disorders. (a) Tree, (b) bark (author with her nephew under the tree), (c) aerial roots, (d) leaves with figs
Important Phytochemicals and Medicinal Value
All Ficus species are well known for their flavonoids, glycosides, phenolic acids, steroids, saponins, coumarins, tannins, triterpinoids, α-hydroxy ursolic acid, protocatechuic acid, and maslinic acid. Many furanocoumarins are also isolated, which include psoralen and bergapten.
Traditional Uses
Various extracts from different parts of the plant showed many biological activities. The fruit, root, and leaves are used in medicine for different disorders such as gastrointestinal, respiratory, inflammatory, and cardiovascular disorders.
## 10.11 F. religiosa L.
* Vernacular name: Sacred Fig
Ecological Distribution and Morphological Characteristics
F. religiosa is a large perennial tree. It is native to Nepal, Pakistan, India, and China. It is commonly found in sub-Himalayan plains (Shinwari et al. 2007). The plant produces large cordate leaves having an elongated narrow tip (Fig. 10.9a–d).
Fig. 10.9
(a–d) Ficus religiosa (sacred fig) is known to possess many anticarcinogenic, anti-inflammatory, and antibacterial activities. Hypoglycemic activities of plant are due to beta-sitosterol and glycosides. The fruit, young leaves, root, and bark are cooked with milk, mixed with honey, and used as an aphrodisiac; however, the powder made from fruit is used for promoting conception. (a) Tree, (b) cordate leaves, (c) young figs, (d) mature figs
The flowers appear in April and May. The fruits of the tree are figs which are purple in color. F. religiosa is grown in Pakistan for its ecological, economic, and medicinal importance. It is also considered to be a sacred tree in many cultures.
Important Phytochemicals and Medicinal Value
The stem bark of the plant is a rich source of important metabolites like alkaloids, flavonoids, phenols, tannins, and steroids like lanosterol, sitosteryl-glucoside, stigmasterol methyl oleanolate, and vitamin K (Swami and Bisht 1996; Sheetal et al. 2008). The fruit is a rich source of proteins and essential amino acids, i.e., isoleucine and phenylalanine.
The methanolic extract of the bark showed the presence of flavonoids, saponins, steroids, wax, terpenoids, cardiac glycosides, and tannins (Babu et al. 2010; Uma et al. 2009). However, quercetin is the most abundant flavonol found in the bark (Taskeen et al. 2009). The bark extract also comprises compounds like bergapten, bergaptol, lanosterol, stigmasterol, and lupeol (Joseph and Justin 2010; Margareth and Miranda 2009). The leaves are a source of campestrol, stigmasterol, lupeol, and tannic acid (Suryawanshi et al. 2011).
The plant is known to possess many anticancer, anti-inflammatory, and antibacterial activities (Mousa et al. 1994; Uddin et al. 2009). Its antidiabetic potential is due to the presence of compounds like β-sitosteryl-d-glucoside which is extracted from the bark (Saravanamuttu and Sudarsanam 2012). Seventy percent of its aqueous-ethanol extract completely inhibits H. pylori at 500 μg/ml (Zaidi et al. 2009). In another experiment, the aqueous extract is reported to possess antimicrobial activities against B. subtilis and P. aeruginosa (Preeti et al. 2010). The ethanolic extract of the leaves inhibited the growth of B. subtilis, S. aureus, E. coli, and P. aeruginosa (Farrukh and Ahmad 2003). However, its chloroform extract inhibits the growth of S. typhi and S. typhimurium (Hemaiswarya et al. 2009). Different extracts of the bark also showed antibacterial activities against E. coli, P. vulgaris, B. subtilis, and S. aureus (Uma et al. 2009; Manimozhi et al. 2012). Phytosterolin from the bark is a strong CNS stimulant. The hypoglycemic activities of the plant are due to β-sitosterol and glycosides. The stem bark also possesses antiprotozoal activity against E. histolytica and antiviral activity against Ranikhet disease virus (Khare 2004a, b).
Traditional Uses
In folk medicine, a decoction made from the bark is used for treating whooping cough and asthma. The plant is used as a source of fodder and is also grown as an avenue tree. The decoction of the bark is used to treat hemorrhages. Its leaves are used for covering wounds, and the root paste is used for skin infections. The fruit, young leaves, root, and bark are cooked with milk, mixed with honey, and used as an aphrodisiac; however, the powder made from the fruit is used for promoting conception (Khare 2004a, b). Different parts of the plant are used in combination with herbs to treat a wide range of ailments (Aiyegoro and Okoh 2009).
## 10.12 F. retusa L. (Syn F. microcarpa)
* Vernacular names: Malayan banyan, Taiwan banyan, Curtain fig
Ecological Distribution and Morphological Characteristics
F. retusa grows as an evergreen tree up to a height of 15 m. Its leaves are dark green, oval, and alternate. Aerial roots surround the tree trunk (Fig. 10.10a–d).
Fig. 10.10
(a–d) Ficus retusa (curtain fig) possess antibacterial activities against B. cereus, B. subtilis, S. aureus, E. coli, P. aeruginosa, S. marcescens, and A. tumefaciens. Important phytochemicals are retusaphenol, afzelechin, luteolin, catechin, vitexin, ß-sitosterol acetate, ß-amyrin acetate, moretenone, friedelenol, ß-amyrin, and ß-sitosterol. (a, b) Tree, (c) close view of bark, (d) leaves with figs
Important Phytochemicals and Medicinal Value
Methanolic extracts from the wood, bark, and leaves possess antibacterial activities against B. cereus, B. subtilis, S. aureus, E. coli, P. aeruginosa, S. marcescens, and A. tumefaciens. The extract from the aerial parts of F. retusa "variegata" possesses antifungal activities against C. albicans and Mucor spp. (Sarg et al. 2011). Antibacterial properties of the leaves against many gram-positive and gram-negative bacteria are due to the presence of compounds like 1, 2-benzenedicarboxylic acid-dibutyl ester and phenol,4-(2-aminopropyl) as revealed through the aqueous-ethanolic extract of the leaves (Aly et al. 2013; Beerse et al. 2002). Further, the ethanolic extract of the aerial parts of F. retusa "variegata" revealed the presence of many polyphenolic compounds like retusaphenol, afzelechin, luteolin, catechin, vitexin, ß-sitosterol acetate, ß-amyrin acetate, moretenone, friedelenol, ß-amyrin, and ß-sitosterol (Sarg et al. 2011).
Traditional Uses
The bark is used for the treatment of diabetes, ulcers, burning sensation, hemorrhages, leprosy, itching, liver disease, and toothaches.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_11
# 11. Nuts and Their Nutritive and Medicinal Value
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
This chapter includes medicinal properties of important nuts like Anacardium occidentalis, Juglans regia, Pistacia vera and Prunus amygdalus.
## 11.1 Introduction
Nuts are, botanically, fruits with edible seeds, i.e., kernels which are enclosed in their hard shells. All nuts are rich in micronutrients and oils and are considered to be healthy and an instant source of energy. However, 'nut' is used in the culinary world in a wide sense, and many seeds which are not nuts are also considered as nuts. Almonds, acorns, betel nuts, cashew nuts, chestnuts, cola nuts, hazelnuts, macadamia nuts, nutmeg, peanuts, pecans, pistachios, pine nuts, quinoa, and walnuts are consumed worldwide as snacks and are also mixed with desserts due to their taste and nutritive and medicinal value (Fig. 11.1).
Fig. 11.1
Macadamia nuts are a rich source of monounsaturated fatty acids and many antioxidants like tocols and tocopherols. They are known to reduce plasma LDL-cholesterol and also reduce the risk of coronary heart diseases
Nuts have dense nutrients with complex matrices which are rich in unsaturated fatty acids and other bioactive compounds having high-quality protein, fibers, minerals, tocopherols, phytosterols, vitamins, and phenolic compounds. Nuts contain almost 4–16% of saturated fatty acids and almost half of the total fat is made up of unsaturated fatty acids, monounsaturated fatty acids, and polyunsaturated fatty acids (Ros 2010). Proteins constitute almost 25% of energy in nuts having a high content of L-arginine (Brufau et al. 2006). Nuts like almonds and pistachios contain many antioxidants in their outer soft shell and more than half of them are lost when the skin is removed or bleached (Milbury et al. 2006; Blomhoff et al. 2006; Seeram et al. 2006).
## 11.2 An Account of Medicinal Properties of Some Nuts
Nuts are medicinally important food and known to reduce the risk of cardiovascular diseases, diabetes, hypertension, cancer, and inflammation. Their consumption is associated with the lowering of cholesterol, oxidative stress, and reducing the risk of coronary heart disease as reported in Adventist Health Study 1992 and Nuts and Health Symposium. They are included in the American Heart Association dietary metrics for 2020 due to their cardioprotective properties and coronary disease reduction (Lloyd-Jones et al. 2011). Almost 4–11% reduction is reported in LDL-cholesterol with the consumption of almonds (Jenkins et al. 2008), hazelnuts (Mercanligil et al. 2007), pistachios (Gebauer et al. 2008), and macadamias (Griel et al. 2008). In Mediterranean diet enriched with nuts, improvement in insulin levels and fasting glucose levels is reported in diabetic and non-diabetic people (Estruch et al. 2006).
Medicinal activities of some popular nuts are discussed in the next section.
## 11.3 Anacardium occidentale L.
* Vernacular name: Cashew nut
* Family: Anacardiaceae
Ecological Distribution and Morphological Characteristics
It grows as a medium-sized tree up to a height of 12 m. Although, cashew nut is native to Brazil, its distribution is widespread in tropical South and Central America, Mexico and West Indies, South Africa, and South Asia. It was cultivated in 1600 by Portuguese traders to prevent soil erosion (Panda 2013).
Leaves are alternate, ovate, almost 15–20 cm long, with a leathery texture. Flowers are red panicles which are present at the end of branches. These panicles may be male or female. Fruits are kidney shaped and attached on to the fleshy swollen stalk (Fig. 11.2). The fruit stalk is known as the cashew apple which bears a hanging nut.
Fig. 11.2
Anacardium occidentale (cashew) nuts are a rich source of energy due to many nutrients including copper, zinc, magnesium, iron, selenium, and vitamin B-6 which help in the formation of red blood cells and maintain bone health. They possess many anxiolytic and antiaging properties
Important Phytochemicals and Medicinal Value
Ethanol and aqueous extracts of leaves revealed the presence of saponins, flavonoids, tannins, and phenolic compounds (Jaiswal et al. 2012).
Cashew nuts possess antimicrobial (Doss and Thangavel 2011), antidiabetic, anti-inflammatory (Akinpelu 2001), and antifungal properties (Dahake et al. 2009).
Hexane extracts from cashew nut leaves are shown to exhibit antihyperglycemic and renal protective effects in streptozotocin-induced diabetic rats (Tédong et al. 2007). However, methanolic extracts of stem bark possess antimutagenic and antigenotoxic effects on Chinese hamster lung fibroblast V79 (Barcelos et al. 2007). Leaf extracts exhibit antimicrobial activities against Porphyromonas gingivalis and Prevotella intermedia (Varghese et al. 2013). Anacardic acid in cashew nuts is an anti-oxidant which possesses cardioprotective, anticancer, and antiaging activities.
Traditional Uses
Cashew nuts are one of the popular and most expensive nuts used all over the world due to their taste and energetic, nutritive, and medicinal properties. They have good fats and are therefore considered to be a healthy food for heart patients. They are a rich source of copper, zinc, magnesium, iron, selenium, and vitamin B-6. Copper and iron help in the formation of red blood cells while phosphorus maintains bone health. Due to the presence of zinc, cashew nuts strengthen the immune system.
Wine is made from cherries Rosa roxburghii and cashew apples in rural and urban communities of Tanzania. Juice, syrup, jams, sweet pickles, candies and masala cookies, doughnuts, porridge mix, and sponge cakes made from cashew apples are important products which are used in food industries (Panda 2013). Cashew apple juice is also prepared by mixing it with lime juice or pineapple juice. Wine prepared from cashew apples is used in Asia and Latin America.
Different parts of the plant are used in a variety of ways in different cultures. In Brazil, hot water extract of the leaf is used to treat diabetes while in Jamaica hot water extract of the bark is used for diabetes. In Thailand, hot water extract of the dried leaf is used to treat diabetes. In Columbia, the seeds are considered as an aphrodisiac. The fruit is also an aphrodisiac and has antipyretic properties. It is used traditionally for curing piles, dysentery, and skin diseases; however, the root is purgative. The plant extract is applied externally in leprosy and it is rubefacient (Aiswarya et al. 2011). It is also used to treat gastrointestinal disorders, mouth ulcers, throat problems, and hypertension (Gonçalves et al. 2005; Taylor 2005).
## 11.4 Juglans regia L.
* Vernacular name: Walnut
* Family: Juglandaceae
Ecological Distribution and Morphological Characteristics
It is native to Central Asia, India, Nepal, and China. Walnut trees are grown commercially in many countries including USA, France, Italy, Spain, Turkey, Iraq, northeastern regions of China, Karakoram and Himalayan ranges of Pakistan, India, Nepal, Bhutan, and Bangladesh due to the nutritive and medicinal values of their nuts.
Walnut is a deciduous and monoecious tree. It is one of the oldest nut-growing species. It grows up to height of 30 m. The leaves are compound, and the leaflets are oval having pointed tips and smooth margins. The male flowers are catkins which are borne laterally on a year-old wood while the female flowers are borne terminally. The female flowers are small, pubescent, and green without any prominent sepals and petals and appear earlier than the male flowers. The female flowers have one ovary which bears a feathery stigma. The wall of the ovary is fused to bract or involucre which develops into the fleshy husk of the fruit. The husk can easily be separated from the nutshell which is wrinkled and rough and bears furrows. The nuts are round to ovoid in shape, with two kernels which are separated by a central plate. Commercial nut production takes place at the age of 10 years. Fruit matures after 4–5 months and harvested during September to October (Fig. 11.3).
Fig. 11.3
Juglans regia (walnuts) are healthy nuts and a source of omega-3-fatty acids, juglone, and gamma-tocopherol. They possesses antidiabetic and cardioprotective properties
Important Phytochemicals and Medicinal Value
Walnut possesses a broad spectrum of antimicrobial activities against E. coli, S. aureus, S. mutans, and P. aeruginosa (Alkhawajah 1997). Nuts and other plant parts contain volatile aromatic oil. Juglone is an allelopathic compound which possesses anticancer properties (Sugie et al. 1998; Segura-Aguilar et al. 1992). It is extracted from the husk of walnut. It is an inhibitor of peptidyl-propyl isomerase Pin1 which is overexpressed in many types of cancer (Thakur 2011). Juglone also possesses cardioprotective properties (Erdemoglu et al. 2003). However, the allelopathic effect of juglone is reported on tomato and alfalfa. It is also a chemopreventive agent for human intestinal neoplasia.
Walnuts are known to be healthy diet due to the presence of unsaturated fatty acids, antioxidants, tannins, gamma-tocopherol, juglone, copper, manganese, and phenolic contents and they help to maintain cardiovascular health (Erdemoglu et al. 2003).
Traditional Uses
The bark of the walnut tree is used as a toothbrush and is also used as a dye lip cosmetics. Powder made from the bark of the tree possesses aphrodisiac properties. Brushing with a walnut bark protects the teeth from gingival and periodontal infections. Juglone is used in herbal remedies and commercial hair dyes (Erdemoglu et al. 2003; Thakur 2011).
## 11.5 Pistacia vera L.
* Vernacular name: Pistachio
* Family: Anacardiaceae
Ecological Distribution and Morphological Characteristics
Pistachio is native to many Asian countries like Lebanon, Palestine, Syria, Iran, Iraq, and India and also Southern Europe. The family Anacardiaceae comprises 70 genera and over 600 species. Twenty species of the genus Pistacia have been reported, among which the important ones are P. vera, P. atlantica, P. terebinthus, P. khinjuk, and P. lentiscus. P. vera is generally grown for commercial use due to the medicinal properties of its fruits.
It is a deciduous tree with pinnately compound leaves. It can grow up to a height of 8–10 m and can survive longer periods of drought. The plant is dioecious with the male flowers on one plant and the female flowers on other plants. The female flowers are without any petals and nectaries. Pollen dispersal takes place through wind.
Pistachio nuts are drupes botanically, and like all other drupes, they comprise an outer exocarp, a fleshy mesocarp, and an endocarp which encloses the seed. However, drupes differ in their edible mesocarp, and as in almonds and pistachios, the seeds are edible while in stone fruits it is the mesocarp which is edible. Pistachio drupe comprises a kernel or nutmeat, enclosing a thin hard shell which is the endocarp of the fruit. The exocarp surrounds a fleshy hull, which is the mesocarp, and the endocarp of fruit (Fig. 11.4).
Fig. 11.4
Pistacia spp. (pistachios) are rich in monounsaturated and polyunsaturated fatty acids, proteins, vitamins, and minerals. They are known to protect cardiovascular health and possess antidiabetic and anti-obesity properties
Pistachios achieve maximum fruit production after 10 years.
Important Phytochemicals and Medicinal Value
Ethanolic extracts from the leaves and flowers are reported to exhibit antioxidant properties. Pistachio nuts are rich in phenolic compounds which possess many strong antioxidants (Souri et al. 2004; Halvorsen et al. 2006). Leaves, kernel, hull, and gums possess antimicrobial, cardioprotective, anti-inflammatory (Giner-Larza et al. 2000), antinociceptive (Hosseinzadeh et al. 2011), and insecticidal properties (Pascual-Villalobos and Robledo 1998) due to their flavonoids and phenolic compounds. The green hull of the pistachio also possesses antimicrobial, antioxidant, and antimutagenic properties (Rajaei et al. 2009). Gum extract has anti-inflammatory and antinociceptive properties (Parvardeh et al. 2002). The resin of P. lentiscus possesses antimicrobial properties. Gum and essential oil from P. atlantica are antibacterial against H. pylori (Ramezani et al. 2004; Paraschos et al. 2011).
Traditional Uses
Different species of pistachios are used worldwide for treating disorders related to digestion, wounds, respiratory problems, hemorrhoids, jaundice, tooth diseases, and cold and flu symptoms. P. lentiscus is more commonly used due to its resin. It is an ancient plant which has been used for 5000 years.
Pistacia species are used in food industry due to their nutritive and medicinal value. P. vera nuts are used as food additives (derMarderosian and Beutler 2010). The fruit of P. terebinthus is used as a snack and also in making a coffee-like drink (Gogus et al. 2011). P. lentiscus is used as a food colorant (Longo et al. 2007). Pistacia fruit consumption dates back to 7000 B.C. (derMarderosian and Beutler 2010).
## 11.6 Prunus amygdalus L. Batsch
* Vernacular name: Almond
* Family: Rosaceae
Ecological Distribution and Morphological Characteristics
Almonds are native to Asia and North Africa. California is the world's major producer of almonds. Almond tree is an ancient plant which is mentioned in Greek mythology, Bible, and Shakespeare's writings.
It grows as a small deciduous tree with a height of 10 m. The leaves are long and petiolate with serrate margins. The twig is green when young, turns purple after exposure to sunlight, and later becomes gray. The flowers are pink and have five petals. They may appear as single flowers or in pairs. Fruit formation takes place after 7–8 months of flowering in autumn. Almond fruit is a drupe comprising an outer hull and a hard shell having the seed inside. Drupe formation takes place in many members of the Rosaceae family.
Not all almonds trees produce edible fruits. Almond varieties which produce edible fruits are known as sweet varieties like P. amygdalus dulcis Mill.; however, some are poisonous like P. amygdalus Batsch amara (D.C) (Focke). Some varieties are a mixture of sweet and poisonous varieties. They can be differentiated on the basis of their flower colors: P. amygdalus dulcis produces white-colored flowers, while P. amygdalus amara produces flowers which are pink in color (Fig. 11.5a–d). Bitter almonds are wider and shorter than sweet almonds. Almost 4–9 mg of hydrogen cyanide is yielded per almond. They are native of Iran and Asia Minor and cultivated in Italy, Spain, Sicily, South France, and Morocco. Bitter almonds contain amygdalin which on hydrolysis yields benzaldehyde and hydrocyanic acid which is a poisonous compound.
Fig. 11.5
(a–d) Prunus amygdalus (almond) is rich in palmitic, palmitoleic, stearic, oleic, linoleic, alpha linoleic, arachidic, eicosanoic, behenic, and erucic acids along with other fatty acids and nutrients. Almonds are known to possess aphrodisiac, antiaging, cardioprotective, and neuroprotective properties (a) Tree in bloom (b, c) Flower (d) Fruits
Important Phytochemicals and Medicinal Value
Almonds are a rich source of nutrients like fats, minerals, and proteins which possess antioxidants activities. They contain calcium, magnesium, potassium, arginine, vitamins B and E, and phytosterols that can be helpful in reducing cholesterol levels.
Almonds contain globulins as active constituents including amandine, albumin, and amino acids such as arginine, histidine, lysine, phenylalanine, leucine, valine, tryptophan, methionine, and cystine. Cholesterol-lowering properties are due to phytosterols which constitute 187 mg/100 mg of almonds (Phillips et al. 2005). Almond oils contain 62% of monounsaturated oleic acids (omega-9-fatty acid), 24% of linoleic acid (polyunsaturated omega 6 essential fatty acid), and 6% of palmitic acid (Berry et al. 1992). Sweet almond oil is extracted from dried kernels. They possess emollient properties.
Almonds are well known for maintaining neuroprotective health and also for making muscles strong. They possess various pharmacological properties, including antioxidant (Pinelo et al. 2004), antistress (Bansal et al. 2009), and immunostimulant (Puri et al. 2000) activities. Skin of almonds contains flavonols like isorhamnetin, kaempferol, quercetin, catechin, epicatechin, flavanoids, anthocyanins, and phenolic acids including caffeic acid, ferulic acid, P-coumaric acid, and vanillic acid (Frison-Norrie and Sporns 2002).
Antioxidant activities of almonds are attributed to phenolic compounds which are higher in the hull as compared with the shell. Antiviral activities of almonds are reported against herpes simplex virus (HSV-2) replication (Arena and Bisignano 2010). Almonds contain many nutrients which regulate the enzymes involved in cholesterol synthesis and bile acid production (Berryman et al. 2011). Almonds possess memory-enhancing properties, and its role in treating Alzheimer's disease is being explored due to its memory-restoring properties. In a study conducted on rats, oral administration of ethanolic extract of leaves, flowers, and seeds of almonds showed significant reduction in blood glucose level of diabetic mice, which suggests antidiabetic activity of almonds leaves, flowers, and seeds (Shah et al. 2011). Almond consumption is also associated with improvement in serum lipid profile values (Phung et al. 2009).
Traditional Uses
Traditionally almonds are used for treating gastroenteritis infections and diabetes. A polyherbal formulation (Tenex Royal) is known to improve sexual behavior in males (Gopumadhavan et al. 2003). Almond fruits are used as a source of dietary proteins, alpha-tocopherol, and fibers. However, almond nuts may cause some allergic reactions.
Sweet almonds are used in making desserts and confectionaries, sweets and chocolates. They are also used in making halwas. Almond drinks and almond teas are also common in markets. A European-based candy marzipan is made by crushing of sweet almonds. Almond extract is used as a substitute of vanilla extract. Raw almonds are consumed due to their taste, health values, and memory-enhancing activities. Almond oil 'Roghan-e-badam' is used externally for skin massage and to remove skin dryness. It possesses antiaging properties and reduces dark circles around the eyes, black heads, and pimples. Almond milk is known to reduce weight. Benefits of taking soaked almonds in water is known to prevent cancer, diabetes, and skin inflammation.
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© Springer International Publishing AG 2017
Aisha Saleem KhanMedicinally Important Trees10.1007/978-3-319-56777-8_12
# 12. Medicinally Important Edible Fruits
Aisha Saleem Khan1
(1)
Department of Biological Sciences, Forman Christian College University, Lahore, Pakistan
Abstract
Many edible fruits contain important nutrients and phytochemicals of medicinal importance including antioxidants, flavonoids, and glycosides, which are important anticarcinogenic, antimicrobial, antidiabetic, and antiaging substances. Tree strees included for their medicinal value are Citrus spp., Phoenix spp., Malus spp., Mangifera spp., Prunus spp., Punica granatum, and Psidium guajava.
## 12.1 Introduction
Many edible fruits, vegetables and seeds are sources of nutrients and are medicinally important. They are widely consumed all over the world due to their sweet taste and nutritive and medicinal values. All flowering plants produce fruits with seeds either exposed or enclosed within ovary walls; however, not all of them are edible. Many plants produce fruits which are not sweet, poisonous or have no nutrition and are very small, and therefore they cannot be consumed. However, edible fruits are mostly sweet, nutritious, and medicinally important as many of them have antidiabetic, antiaging, antimicrobial, antioxidant, and anticarcinogenic potential.
Many antioxidants which are present in fruits serve as inhibitors of initiation and formation of tumors. Fruits rich in anthocyanins are sources of many antioxidants and possess anticancer activities. Natural anticarcinogens are present in many edible fruits like cherries, jambul, lemon, mango, cranberries, mulberries, peach, plum, pomegranate, orange, raspberries, and strawberries (Fig. 12.1a–g). They act to scavenge free radicals present within cells which can damage lipids and proteins and some mutagens which can bind with DNA and can initiate tumor formation and convert carcinogens into anticarcinogenic compounds.
Fig. 12.1
(a, b) Prunus avium, sweet cherry is a source of important phytonutrients which are responsible for anticancer, antiinflammatory and antimicrobial activities (c) Diospyros kaki (persimmon) is an medicinally important edible fruit with antidiabetic, antibacterial and cardioprotective properties (d) Pyrus communis (common pear) possesses important skin whitening, antiinflammatory, antibacterial, antidiabetic and analgesic activities due to compounds like arbutin, quercetin, kaempferol, fredielin, sterols, isoquercitrin, ursolic acid, phloridzin and tannins (e) Prunus domestica, plum is known to possess antidiabetic, anticancer, laxative, antiosteoporosis, anxiolytic and antimicrobial activities (f) Flower (g) Ripened fruits
## 12.2 Medicinal Properties of Important Fruits
Although many culinary herbs and shrubs also produce medicinally important fruits, in the following section only edible fruits of trees will be discussed. However, due to the wide variety of trees with edible medicinal fruits, it is not possible to describe them in one chapter, so the medicinal value of only a few fruits is described here, which include apples, date palms, lemon, pomegranate, guava, mangoes, and orange.
## 12.3 Citrus x sinensis L.
Vernacular name: Orange
Family: Rutaceae
Ecological Distribution and Morphological Characteristics
Citrus spp. grow as evergreen trees up to a height of almost 33 m. Citrus sinensis (sweet orange) is one of the widely grown hesperidia for commercial species as compared with bitter orange and mandarin orange. Brazil, U.S, Mexico, Italy, and China are major producers of oranges. Brazil is one of the largest orange-producing countries and exports orange juice to the USA and earns 1.6 billion dollars in foreign exchange by exporting only frozen concentrated orange juice (FCOJ). The USA is the world's second largest exporter of oranges.
Common orange, acidless orange, pigmented orange, and navel orange are the main varieties of orange grown all over the world. Among common oranges, Valencia, Pineapple, Hamlin, Parson Brown, Jaffa, and Shamouti are grown in various parts of the world. The main cultivars of pigmented orange are Doblefina, Eutrifina, Moro, Tarocco, Ovale, Sanguinello Commun, Ruby blood, and Maltese blood.
The leaves are small, aromatic, alternate, and oval shaped. White aromatic flowers are axillary and contain nectar. Exocarp or peel or flavedo is the outer-most layer of orange fruit, mesocarp is known as albedo, and edible pulp is endocarp which comprises vesicles filled with juice (Fig. 12.2a–h).
Fig. 12.2
(a–h) Flavonoids of Citrus spp. peel possess anticarcinogenic activities against skin, colon, prostate, lung and liver cancer (a) Tree (b) Leaves (c) Flower buds (d) Flower (e) Ripened fruit (f–h) Fruits in market
Important Phytochemicals and Medicinal Value
Hesperidin is a major flavonoid glycoside found in orange peel (Kanaze et al. 2009) and used in the pharmaceutical industry due to its chemopretentive activity (Al-Ashaal and El-Shelawy 2011; Rawsona et al. 2014). In addition to hesperidin, citrus peel comprises other polyhydroxyflavones like neohesperidin, naringin, nobiletin, and polymethoxyflavones, i.e., nobiletin, tangeritin, sinesetin, 3,5,6,7,8,3′,4′-heptamethoxyflavone (PMFs), and 3,5,6,7,3′,4′-hexamethoxyflavone (Li et al. 2006; Londoño-Londoño et al. 2012; Pan et al. 2012). Pericarp of fruit releases essential oil comprising limonene and beta-myrcene, aliphatic aldehydes, and linalool.
Sweet orange oil is high in carotenes and is widely used in the beverage industry. Bitter orange oil, also known as Bigarade orange (C. aurantium), contains almost the same oils as found in sweet orange oil: limonene, myrcene, linalool, and pinene. Bitter orange flower oil contains linalool (28–44%), linalyl acetate (3–15%), limonene (9–18%), and beta-pinene (7–17%). In addition, it also contains geranyl acetate, neryl acetate, nerolidol, and farnesol. Mandarin orange oil (C. reticulata) contains a low quantity of limonene and a high concentration of teripinene. However, tangerine, an American variety, comprises 90% limonene.
Ethanol extract is known to have antibacterial activities against E.coli (Ekwenye and Edeha 2010). Aqueous and ethanolic extract of orange peel revealed the presence of alkaloids, terpenoids, flavonoids, glycosides, steroids, tannins, and oils (Chede 2013).
PMFs can inhibit activity of human leukemic cell (HL-60) lines. Tangerin can inhibit the metastasis stage, thus preventing adhesion and invasion, and it inhibits cyclin dependent kinases (Cdk) and causes cell arrest in G1 phase. It also enhances Cdk, and inhibits extracellular-signaling-regulated kinase phosphorylation. This causes reduction in breast cancer cells and cytolysis and repression in human lung cancer cells. Nobiletin inhibits proliferation and migration of human umbilical endothelial cells of prostate cancer cells, and skin, breast, and colon carcinoma cell lines (Kawaii et al. 1999; Chen et al. 2007). Flavonoids of citrus peel also possess anticarcinogenic activities against skin, colon (Janakiram and Rao 2008), prostate (Hanley 2010; Obertova et al. 2012), lung (Luo et al. 2008), and liver cancer (Ma et al. 2014).
Orange peel contains citral and possesses strong antioxidant and cytotoxic activity against human carcinoma cell lines (Al-Ashaal and El-Shelawy 2011). Orange peel is also known to have anti-inflammatory (Gosslau et al. 2014; Wang et al. 2014; Li et al. 2014) and antihypolipidemia activities (Kurowska and Manthey 2004) and is effective in delaying Alzheimer's disease (Nakajima et al. 2013; Yabuki et al. 2014).
Traditional Uses
Sweet orange oil is produced through pressing the orange. It is used to add flavors to food, in perfume industry and also for aromatherapy. Sweet orange varieties like blood, navel and white-fleshed oranges yield more juice. Mandarin orange Lipton green tea and tangerine orange zinger herbal tea are also available in the market. Orange peel is used in many skin care products for facial masks. In Pakistan, orange peel is crushed, soaked in milk, and applied to tone skin.
## 12.4 Citrus x limon L.
Vernacular name: Lemon
Family: Rutaceae
Ecological Distribution and Morphological Characteristics
Lemon is native to Asian countries including China, Pakistan, and India. The word lemon is derived from the Arabic word "Linum". Eureka, Lisbon, and Meyer are common varieties of lemon.
Lemon grows as a small evergreen perennial tree with thorny branches. The leaves are small, simple, elliptic, and lanceolate. The flowers are small, white, and aromatic with many stamens. The fruit is hesperidium with thick rind and turns yellow when mature (Fig. 12.3a–e).
Fig. 12.3
(a–e) D-limonene in lemons is a compound with anticancerous properties and it is widely used in clinical applications. Lemon fruit are a good source of citric acid, limonoids, flavonoids, and many vitamins. Lemon is also known to have antidiabetic value due to presence of quercetin which stimulates insulin formation (a) Tree (b) Bark (c) Flower (d) Developing fruit (e) Ripened fruit
Important Phytochemicals and Medicinal Value
Lemon fruit are a good source of citric acid, limonoids, flavonoids, and terpenoid compounds and folic acid, vitamin C, B1, B2, B3, B5, B6 and B9, fibers, calcium, iron, magnesium, manganese, phosphorus, potassium, and zinc. The main essential oils of lemon are citral, pinene, thujene, and limonene.
Lemon fruit, leaves, stem, and flowers possess antimicrobial activities due to the presence of flavonoids against many bacterial strains (Kawaii et al. 2000; Burt 2004; Ortuno et al. 2006). These flavonoids act as antioxidants, scavenge free radicals, modulate enzyme activities, and inhibit cell proliferation (Hanahan and Weinberg 2000; Duthie and Crozier 2000; Song and Bae 2013; Arul and Subramanian 2013; Kou et al. 2013; Li et al. 2014; Sultana et al. 2014).
Anticancer activity of lemon peel is reported due to the presence of polymethoxyflavones (PMFs) which inhibit tumor growth by blocking metastasis cascade, reducing mobility of cancer cells in the circulatory system, proapoptosis, and antiangiogenesis (Wang et al. 2014). Approximately six PMFs and three 5-demthoxyflavones are extracted from lemon peel with increased serum antioxidant potential against lipid peroxidation and effectiveness against oxidative stress. These PMFs exhibit anti-inflammatory, antitumor, and antiatherosclerosis activities (Mulvihill and Huff 2012; Assini et al. 2013). They are also known to have neuroprotective activity and are effective for chemotherapy (Hwang et al. 2012). Bark of lemon is known to possess antipyretic properties (Kumar 2007). D-limonene in lemon is also known to have anticancer properties and it is widely used in clinical applications.
Lemon zest green tea is very effective in reducing weight and for maintaining cardiovascular health. Lemon is known to have antidiabetic value due to the presence of quercetin, which stimulates insulin formation (Aruoma et al. 2012). Due to vitamin C, lemon also maintains teeth and bone health as vitamin C helps in calcium absorption. It also prevents scurvy. Vitamin C maintains the body's immune system by producing interferon protein, which protects cells from bacterial and viral infections. The presence of vitamin C and quercetin in lemon also neutralizes free radicals and reduces risk of cataract by almost 80% (Kumar 2007). Quercetin also protects beta cells of the pancreas, which produce insulin.
Traditional Uses
Pectin compounds in lemon are good for maintaining liver health by stimulating bile formation (Kumar 2007). Lemon juice is squeezed in Asian cuisine to facilitate digestion. Lemon tea with honey and ginger is used to cure cough. Lemon peel is used in skin care products as a facial toner. Limonene found in essential oil finds numerous applications in the food, pharmaceuticals, cosmetics, and beverage industries. Lemon oil is also used as a flavoring agent due to its curative and carminative properties. It also stimulates the salivary glands and facilitates digestion. It helps in wound healing due to its antiseptic nature. Use of lemon oil is recommended while cooking food to prevent infection from Cholera and Salmonella. It is also used in aromatherapy. Exposure with lemon oil is beneficial for killing Pneumococci, which cause lung infection. Regular use of lemon can also reduce the chances of osteoporosis in women.
The homemade lemon drink "lemonade" is made by squeezing lemon in cold water and then adding sugar. It is also available in the market. Lemon is also used in making lemon cakes, lemon pies, lemon ice, lemon iced tea, and for garnishing and dressing.
Use of lemon is very common in dish washing soap bars and washing detergents as an antiseptic. A number of cosmetic products by well-known industries like L'Oreal, Olay, Garnier, Aesop, Estee Lauder, Ponds, Revlon, Avon, and The Body Shop use "limonene" in skin care, hair care, body care, and healthcare products. Use of lemon is also very common in everyday products including perfumes, shampoos, soaps, detergents, and body sprays. Lemon is used in many homemade recipes to clean and tone skin. It is also used as a stain remover and bleaching agent.
One tablespoon of lemon juice before a meal is recommended for asthmatic patients. Taking 1 teaspoon of salt with a few drops of lemon oil in boiled water and inhaling it thee times a day can relieve throat irritation. Mixing a teaspoon of lemon juice in a glass of water and then adding a pinch of sodium bicarbonate reduces acidity in the stomach. Lemon juice mixed with an equal amount of honey stops excessive saliva formation. Adding a teaspoon of rock salt in a hot water tub with a few slices of lemon provides foot relaxation. Lemon is also a natural detoxifier and removes toxins from the body. A tablespoon of lemon juice mixed with olive oil naturally cleanses the body in the morning.
## 12.5 Malus domestica L.
Vernacular name: Apple
Family: Rosaceae
Ecological Distribution and Morphological Characteristics
Over 40 species of Malus are grown worldwide. Seven species are similar to Malus domestica. It is one of the cash crops of many Asian countries and grown widely in the Himalayan regions of Kashmir. More than 1000 cultivars of apple were developed; however, some of them have been lost. Around 100 cultivars are grown for commercial uses.
M. domestica is grown as a fruit tree. It is a small and deciduous tree which is native to Central Asia. The leaves are 5–12 cm long and 3–6 cm wide. The flowers are white, with inferior ovary, having five cavities mostly with two seeds. Inflorescence is cymose comprising a cluster of 4–8 flowers. Flower in the center is also known as the "King blossom" which opens first and is capable of producing relatively larger fruit which is a pome with a fleshy part derived from the floral cup not from the ovary. The seeds are small, black, and are present within cavities (Fig. 12.4a–h).
Fig. 12.4
(a–h) Malus spp. are known to maintain cholesterol level and effective against Alzeheimer's disease, cancer and microbial infections. Apples are rich in phenolic compounds and polyphenols which maintain cardiovascular health (a) Tree (b) Leaves (c) Flower buds (d) Flowering branch (e) Flower (f–h) Fruits in market
Important Phytochemicals and Medicinal Value
Apples are rich in phenolic compounds, i.e., catechin, epicatechin, phloridzin, quercetin, anthocyanidins, chlorogenic acid, and hydroxycinnamates, which are widely present in the peel, flesh, and seeds (Cuthbertson et al. 2012; Vrhovsek et al. 2004). Catechin, epicatechin, and procyanidin B1 polyphenols maintain cholesterol levels and are effective for cardiovascular health (Serra et al. 2012).
Apple juice delays the aging process and is effective against Alzeheimer's disease (Chan and Shea 2009). The presence of phloretin-O-glycosides, phloretin-2′-O-glucoside, and phloretin-2′-O-(2″-O-xylosyl) glucosides in apples is beneficial for treatment of diabetes mellitus. They inhibit sodium-dependent glucose transporters in intestinal lumen and decrease the absorption of glucose (Marks et al. 2009). Apple fruits have strong antioxidant, antiinflammatory, antiviral, and antimicrobial activities (Fratianni et al. 2007; Martin et al. 2007).
Fruit is a rich source of sugars, vitamins, dietary fibers, and phenolic compounds, which possess anticarcinogenic, antidiabetic, and cardioactive properties (Boyer and Liu 2004). Phytochemicals in fruits like procyanidins have anticancer activities (Gerhauser 2008). Apple phenols extract protects gastric epithelial cells (Graziani et al. 2005). Regular consumption decreases risk of lung and colon cancer (Hyson 2011). Anticancer activity of apple fruit is also reported against skin, breast, and colon cancer. Apple pomace is a source of natural polyphenol so it is recommended that it should also be consumed.
Traditional Uses
Apples are consumed all over the world, more than any other fruit. They are widely used in desserts like pies, cakes, jams, jellies, beverages, and apple butter. In Pakistan, apples are used in making murrabbas, which is consumed for breakfast due to its health benefits. Apple cider vinegar is a very popular and healthy vinegar made from apple.
Avicenna (AD 980–1037) mentioned medicinal properties of vinegar in his famous book Al-Qanoon fit Tibb (The Canon of Medicine). Apple cider vinegar is prepared from apple must and cider. It is used in salad dressing, food preservatives, and commercial beverages.
It is effective against microbial infections, acts as a natural detoxifier, is useful against muscle soreness and aching joints, for arrhythmia and for heart strengthening, for improving digestion and weight loss, for combating gallstones, for prostate shrinking, for identifying cervical cancer, for treating arthritis, for maintaining blood pressure and for delaying premature aging, for treating skin problems including skin burns, and for removing dirt from the skin by using with rose water and aloe gel (Bragg and Bragg 2008). Gargling with raw organic vinegar is recommended and can soothe the throat. Applying a bandage soaked with Apple cider vinegar (AVC) can remove corns and callouses and brings comfort to the feet. It can be applied to treat cold and genital sores caused by the herpes virus. It possesses antiallergy activities against poisonous plants like poison oak and ivy (Bragg and Bragg 2008).
Apple cider vinegar is also used to rejuvenate skin and can be used as a facial treatment. It is recommended to wash the face with warm water (without soap) and then treat the face with a hot water-soaked cloth for a few minutes. Then soak a thin cloth in warm apple cider vinegar (1tbsp of vinegar per cup of water) and apply to the face again and cover the vinegar-soaked cloth with a towel wrung out in hot water. Then lie down for about 10 min in a position where the feet are raised on a couch against the wall so more blood is circulated toward the face for better circulation. Remove both cloths and gently rub the skin upwards with a coarse towel. This facial treatment can remove dirt from deep in the skin and skin will appear much younger and will shine like a polished apple. This can be repeated on a weekly basis. It is also effective for treating dandruff, baldness, and itchy scalp.
## 12.6 Mangifera indica L.
Vernacular names: Mango
Family: Anacardiaceae
Ecological Distribution and Morphological Characteristics
M. indica L. is a large evergreen, long-living tree, with a heavy crown. It is native to tropical Asia and grows in the wild in Pakistan (Sheikh 1993). It is the national fruit of Pakistan. More than 100 cultivars of mango are grown all over the world. The leaves are alternate, lanceolate, and spirally arranged. The mango tree bears unisexual and bisexual flowers on the same plant. The flowers are small, greenish-yellow, and appear in March (Fig. 12.5a–f). The edible fruit of M. indica is drupe, which is very delicious and can be consumed in a variety of ways.
Fig. 12.5
(a–f) Mangifera indica (mango) possesses antiviral, antifungal, anticancer and antidiabetic properties. Young leaves of mango, stem bark and fruits are rich source of mangiferin which is a pharmacologically active compounds (a) Tree (b) Panicles (c–e) Young fruits (f) Ripened fruits
Young leaves of mango, stem bark and fruits are a rich source of mangiferin, a flavonoid which is a pharmacologically active compound and widely distributed in the leaves and bark of Anacardiaceae (Yoshimi et al. 2001). The bark comprises protocatechic acid, catechin, mangiferin, alanine, glycine, and amino-butyric acid (Ross 1999; Scartezzini and Speroni 2000). A natural C-glucoside xanthone, mangiferin 6,7-tetrahydroxyxanthone is also reported in various parts of M. indica (Muruganandan et al. 2002).
Important Phytochemicals and Medicinal Value
Mango wood is used for construction purposes (Sheikh 1993). M. indica is medicinally important due to its antiviral (Zhu et al. 1993; Guha et al. 1996), antifungal (Stoilova et al. 2005), anticancer, and antidiabetic properties (Miura et al. 2001; Muruganandan et al. 2005). Other pharmacological activities from different plant parts include antioxidant (Leiro et al. 2003), radioprotective (Jagetia and Baliga 2005), immunomodulatory (Leiro et al. 2004), and antiallergic (Rivera et al. 2006).
## 12.7 Prunus persica L. Batsch
Vernacular name: Peach
Family: Rosaceae
Ecological Distribution and Morphological Characteristics
Prunus persica, commonly known as peach, grows in temperate regions and it is thought to originate in China. China, Italy, the USA, and Spain are major peach-producing countries. The most common varieties of peach in the USA are Elberta, Red Haven, and Halford.
Peach grows as a deciduous tree up to 6 m in height. The leaves are glossy, green, and lens shaped. The flowers are borne in leaf axils with five petals, which are pink to white in color (Fig. 12.6a–d). Stamens are present in the form of tubes known as hypanthia. The flowers are monoecious and can self-pollinate. Flowering starts in April and seeds ripen in summer. Fruit formation takes place from a single ovary which becomes fleshy upon ripening. The hard interior part of the fruit is a stone or pit which encloses seeds. Fruits with smooth skin are known as nectarines.
Fig. 12.6
(a–d) Prunus persica (peach) possess hepatoprotective and anticancer activities. Peaches are rich source of vitamins A and C and ß-carotene (a) Tree (b) Flowers (c) Close view of flower (d) Fruits
Important Phytochemicals and Medicinal Value
The main phytochemicals present in peaches include carotenoids, anthocyanins, and phenolic compounds (Radi et al. 1997; Gao and Mazza 1995; Gil et al. 2002; Cevallos-Casals et al. 2005).
The leaves of the peach plant have astringent, diuretic, and laxative activities and their consumption can cause hallucination. Ethanolic extract of peach leaves possesses hepatoprotective activity (Chaudhary et al. 2015).
Peaches are a rich source of vitamins A and C and betacarotene. They are alkaline to the body and act as a natural detoxifier to remove toxins from the body. Peaches with orange flesh contain carotenoids, beta-carotenes, and beta-cryptoxanthin (Tourjee et al. 1998). In addition, chlorogenic acid, neochlorogenic acid, catechins, epicatechin, and quercitin-3-rutinoside are also present in peaches (Kim et al. 2003; Tomás-Barberán et al. 2001).
Traditional Uses
Peach leaves are used traditionally for treatment of cough, bronchitis, and gastritis. The flowers have diuretic, sedative, and vermifuge activities. Peach tea is made from flowers and leaves and acts as a detoxifier for the kidneys. It is made by adding 1/4 cup of peach flowers in boiling water, simmered for a few minutes, and consumed with honey.
## 12.8 Psidium guajava L.
Vernacular name: Guava
Family: Myrtaceae
Ecological Distribution and Morphological Characteristics
Guava is widely grown in tropical and subtropical countries and is native to tropical South America. The genus Psidium is known to comprise approximately 150 species of small trees and shrubs, out of which 20 produce edible fruit while the rest produce wild fruit of relatively low quality (Mani et al. 2011). P. guajava is the most common and popular cultivated species of guava.
Guava is a small evergreen tree up to 10 m in height. The leaves are simple, opposite, and oblong to elliptic, with prominent veins, 2–6 inches long and 1–2 inch wide. They release fragrance when crushed and turn red in winter and during senescence due to anthocyanin accumulation. Monoecious flowers are white and fragrant, with 4–6 petals and numerous anthers. The fruit is round, many seeded, and comprises fleshy pericarp (Fig. 12.7a–f).
Fig. 12.7
(a–f) Essential oils of Psidium guajava (guava) possess anticancer and antimicrobial activities against cervical cancer cells. Presence of lectin in guava allows it to bind to E.coli and prevents infection. Leaf extract is reported to have antidepressant activity on CNS (a, b) Developing leaves (c) Flower (d–f) Developing fruits (g) Mature fruits
Important Phytochemicals and Medicinal Value
Guava leaves and fruits possess many pharmacological activities. Guava is mainly known for its antimicrobial and antispasmodic activities (Abdelrahim et al. 2002; Gutierrez and Mitchell 2008; Biswas et al. 2013).
Metabolites of medicinal importance are phenolic compounds, flavonoids, carotenoids, and terpenoids. The leaves are a source of essential oils like cineol and eugenol, alpha-pinene, beta-pinene, limonene, menthol, caryophyllene oxide, beta-copanene, farnesene, humulene, selinene, cardinene, curcumene, nerolidiol, beta-sitosterol, crategolic and guayavolic acids, quercetin, avicularin, and prenol and other phytochemicals of medicinal importance including tannins, flavonoids and terpenoids, resins, and mineral salts (Burkill 1997; Nadkarni and Nadkarni 1999; Ncube et al. 2008; Joseph and Priya 2011). Guajonic acid, beta-sitosterol, uvaol, oleanolic, and ursolic acid are also extracted from the leaves (Begum et al. 2004). Extract from guava leaves forms a colored complex with iron used to treat African trypanosomiasis (Shruthi et al. 2013).
Common methods for isolation of extracts include maceration, infusion, percolation, digestion, decoction, fermentation, Soxhlet extraction, microwave and ultrasound extraction, supercritical fluid, and phytonic extraction (Biswas et al. 2013).
Guava is a rich source of vitamin A and C, thiamin, riboflavin, niacin, dietary fibers, folic acid, calcium, phosphorus, copper, and manganese. It comprises antioxidants and vitamins and compounds like leutein, zeaxanthin, and lycopenes. Guava fruits are recommended for lowering of blood pressure and blood cholesterol (Singh et al. 1992& Singh et al. 1993).
Essential oils from guava are known to possess anticarcinogenic activities against cervical cancer cells (Joseph and Priya 2011). The leaf extract is used as a cough sedative in pharmaceutical preparations (Metwally et al. 2011). Antitumor activity of guava is reported against three cell lines (HeLa, RKO-AS45–1, Wi-26VA4) due to quercetin (Braga et al. 2014).
Antibacterial activity is reported against Shigella, Salmonella, Staphylococcus, E.coli, Clostridium, and Pseudomonas (Shruthi et al. 2013). The presence of lectin in guava allows it to bind to E.coli and prevents infection (Rodriguez et al. 2001). Leaf extract is reported to have antidepressant activity on the Central nervous system (CNS) (Shaheen 2000). Antiproliferative activity of guava is reported to be 4.37 times higher than vincristine (Manosroi et al. 2006). Anti-HIV activity of guava is also reported due to its inhibitory effect on proteases by saponins (Mao et al. 2010).
Traditional Uses
Use of guava fruit is very common traditionally due to its nutritive, antioxidant, antimicrobial, antiinflammatory and analgesic (Ojewole 2006), antidiabetic, anticough (Jaiarj et al. 1999), antiallergy hepatoprotective, gastroprotective, nephroprotective (Lin and Yin 2012), and cardioactive activities. It is also used to treat hypertension and obesity (Begum et al. 2004; Sunagawa et al. 2004).
Guava tea is useful for bronchitis, asthma attacks, cough, and pulmonary diseases (Khan and Ahmad 1985). Fruit paste and cheese are popular in Florida, the West Indies and South America. Guava leaves are crushed and applied to wounds traditionally in Pakistan as they are known to possess antimalarial and antiamebic activities (Morton 1981; Tona et al. 1999). Decoction is made from guava root and used to treat diarrhea, dysentery, toothache, and acne. In South Africa and in Fiji, root decoction is used as an astringent for ulcers and wounds (Gutierrez and Mitchell 2008). Guava bark comprises 11–27% tannins and is used for tanning and dyeing purposes.
## 12.9 Punica granatum L.
Vernacular name: Pomegranate
Family: Lythraceae
Ecological Distribution and Morphological Characteristics
Pomegranate is widely grown in Southeast Asia including China, Pakistan, Iran, and India, in California and Arizona in the USA, and throughout the Mediterranean region. It is native from the Middle East to the Himalayas. More than 764 cultivars are grown in Iran (Rahimi et al. 2012). It is an ancient plant and is mentioned in books from different religions. It is cultivated up to altitudes of 2000 m in the western range in Pakistan. Pomegranate was introduced by the Spanish in 1796 in Latin America and California.
Pomegranate grows as a small tree up to a height of 30 m, which requires exposure to sunlight, and it can survive for hundreds of years. Pomegranate is deciduous with opposite, oblong, entire, and glabrous leaves. Flowering takes place from March to September; however, depending upon cultivar, pomegranate trees can flower many times in spring and summer. The flowers are large, monoecious with six petals, obovate, and wrinkled with many stamens (Fig. 12.8a–h). The ovary is inferior with one style, and the stigma is bilobed. Both cross- and self-pollination can take place. Insects and birds help in pollination. The fruit is a reddish edible berry which comprises around 600 seeds enclosed in white spongy pulp which is astringent (Stover and Mercure 2007). Aril is the edible part of the fruit. The red color of the fruit is due to the presence of delphinidin, pelargonidin, and cyanidin (Noda et al. 2002).
Fig. 12.8
(a–h) Fruit juice of Punica granatum (pomegranate) is known to have antihypertension, anti-HIV and anticancer activity due to presence of protein and fats, vitamin C, vitamin B5, polyphenols, ϐ- sitosterol, ellagic acid and ellagitannins (a) Young leaves (b) Flower buds (c) Flowers (d) Young fruit (e) Ripened fruit (f–h) Fruits in markets
Important Phytochemicals and Medicinal Value
Antioxidant activity of P. granatum is three times greater than green tea extract, and therefore consuming the fruit maintains cardiovascular health by reducing Low-density lipoproteins (LDL) and cholesterol (Fuhrman et al. 2005; Rosenblat et al. 2010). The fruit juice is known to have antihypertension activity by reducing oxidative stress (Mohan et al. 2010).
Pomegranate fruit is a rich source of protein and fats as compared to the seeds, which comprise 10% aril weight. It is rich in vitamin C, which can fulfill 16% of the individual daily requirement. In addition, the fruit is a good source of vitamin B5 (pantothenic acid). It is reported that administration of 50 mg/kg of fruit extract for 28 days can result in reduction of malondialdehyde (Toklu et al. 2007).
Anticancer activity of pomegranate is attributed to estrogenic compounds, i.e., luteolin, quercetin, kaempferol, ellagic acid (E), ellagitannins, caffeic acid (C), punicic acid (P), punicalagin, and polysaccharide (PSP 001), in fruit peel extract. It is reported to have an inhibitory effect on lung (Khan et al. 2007), breast (Sturgeon and Ronnenberg 2010), colon (Hong et al. 2008; Khan 2009), and prostate cancer (Malik and Mukhtar 2006; Seeram et al. 2007; Rettig et al. 2008; Koyama et al. 2010). The seed oil, fruit juice, and peel extract are also helpful in breast cancer.
Ellagitannins have anticancer activity against colon cancer. They hydrolyze in stomach and release ellagic acid (EA) and further form urolothin A through microbial infection. They inhibit Wnt signaling, which can produce carcinogens. Punic acid from pomegranate (Omega-5 long chain polyunsaturated fatty acid) can induce apoptosis in estrogen in sensitive breast cancer cell line. Pomegranate is reported to improve epidermal sperm concentration and sperm motility and decreases the number of abnormal sperm cells (Türk et al. 2008).
Neuroprotective activities are due to neonatal hypoxia-ischemic brain injury (West et al. 2007). Tannins found in pericarp of pomegranate possess antiviral activity against genital herepes simplex virus (HSV-2) by inhibiting viral replication, killing and blocking their absorption to cells. Methanolic extract of pomegranate is reported to have antibacterial activity against S. aureus and S. epidermidis.
Pomegranate juice is also reported to possess possible anti-HIV activity due to the presence of polyphenols, ϐ- sitosterol, ellagic acid, and sugars and is beneficial in influenza, herpes virus, and poxvirus. A combination of fulvic acid and pomegranate juice is known to inhibit many strains of influenza including H5N1as polyphenols from pomegranate can cause damage to virion structure as revealed through electron microscopy. Pomegranate rind extract possesses skin-whitening activity. This is due to the inhibition of melanocyte proliferation and inhibition of melanin synthesis by tyrosinase in melanocytes. The flowers are known to improve learning and memory in diabetic rats.
Traditional Uses
All parts of the pomegranate plant including root, leaves, flowers, seeds, bark, and fruit are medicinally important and are used traditionally. The seeds are known to comprise estrogenic compounds like estrone and estradiol. The fruit is used traditionally for treating digestive disorders, and is effective against microbial infections and respiratory problems (Kim and Choi 2009). The dried pulp is useful for treatment of headache, oral diseases, acne, and skin allergies (Ricci et al. 2006). The fruit is used to treat acidosis. Dried pomegranate seeds are used as "anardana" in Asian cuisine.
## 12.10 Phoenix dactylifera L.
Vernacular names: Date palm
Family: Arecaceae
Ecological Distribution and Morphological Characteristics
Date palm is cultivated in tropical and subtropical regions due to its ornamental and nutritive value. It is a medium-sized tree which forms a clump comprising many stems from the main root. The leaves are pinnate with spines on petioles. Male and female plants are separate. The fruits are oval and cylindrical with a single stone as the seed. The fruits are sweet berries due to their high sugar content (Fig. 12.9a–f).
Fig. 12.9
(a–f) Fruits of Phoenix dactylifera date are also known to possess antioxidants and anticancerous compounds with antifungal, nephroprotective and hepatoprotective activities. Leaves are aphrodisiac (a, b) Date palm trees (c) Leaves (d) Male inflorescence (e) Young dates (f) Ripening dates
Important Phytochemicals and Medicinal Value
Date pulp is rich in sugars like glucose and fructose, and dietary fibers which facilitate digestion (Al Farsi and Lee 2008). Many vitamins like riboflavin, biotin, thiamine, ascorbic, and folic acid are found in date fruits. The pulp of the fruit is a rich source of many nutrients like calcium, iron, copper, magnesium, fluorine, manganese, phosphorus, and potassium. Furthermore, phytochemicals like sterols, phenolics, carotenoids, procyanidins, anthocyanins, and flavonoids are also reported through phytochemical analysis. Approximatey 13 flavonoid glycosides are reported to exist in fruit upon maturity. The most notable of these are leuteolin, quercetin, and apigenin (Bilgari et al. 2009).
Traditional Uses
Date palm is an ancient plant and has religious importance for Muslims because it is mentioned in the Quran in many places. Due to its high nutritional value, great yields, and its long life the date palm has been mentioned as the "tree of life" (Augstburger et al. 2002). The fruits of date are also known to possess antioxidants due to their high polyphenol content (Al-Turki 2008). The leaves are aphrodisiac and the flowers are used as a tonic for the liver. It is also known that fluorine is useful against teeth decay (Al Farsi and Lee 2008). Dates are reported to have anticancer (Ishurda and John 2004), antifungal (Bokhari and Kahkashan 2012), nephroprotective (Al Qarawi et al. 2008), and hepatoprotective potential.
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Index
A
Abacetus candidus
Abietane
Abortifacient
Acarbose
Acetic acid writhing
Acetophenone
Acetovanillone
Acetylation
Achrassaponin
Acquired immunodeficiency syndrome (AIDS)
Acuminate
Acute
Adaptogenic
Adenocarcinoma
Adenophyllone
Adenovirus type 2
Adipocytes
Adrenaline
Aegeline
A549 cells
Aflatoxin
Afzelechin
Afzelin
Agar well diffusion method
Aglycone oleandrigenin
Aglycones
AIDS
SeeAcquired immunodeficiency syndrome (AIDS)
Ailanthinol B
Ailanthinone
Ailanthone
Albafuran
Albanol
Albedo
Albino rats
Albizzine
Aleosin
Alexipharmic
Alkaloids
Allamandin
Allamcin
Allelopathic
Alliin
Allohimachol
Alloxan
α-amyrin acetate
α-spinasterol
Alpinumi soflavone
Alterante arrangement
Alternaria alternate
Alzheimer's disease
Amentoflavone
Amnesia
Amoebicide
Amygdallin
Amyrin
Anacardiaceae
Analgesic
Andrographolide
Angina
Annonaceae
Anolignan
Anthelmintic
Anthocyanins
Anthraquinone(s)
Antiaging
Antiallergic
Antiasthmatic
Antiatherosclerosis
Antibacterial
Antibiotic
Anticarcinogenesis
Anticarcinogenic
Anticirrhosis
Anti-dengue
Anti-depressing
Antidiabetic
Antieczema
Antifertile
Antifungal
Antihepatitis
Anti-HIV
Antihyperlipidemic
Anti-inflammatory
Anti-influenza
Anti-malarial
Antimellin
Antimicrobial
Antinociceptive
Antioxidants
Antiplasmodial
Antiproliferative
Antipyretic
Antiseptic
Antispasmodic
Antispermatogenic
Antistaphylococcal
Antithrombotic
Antitumor
Antiviral
Anvirzel
Anxiolytic
Apetalous
Aphrodisiac
Apigenin(s)
Apocynaceae
Apoptosis
Apotirucallane
Appetizer
Aqueous extracts
Arboran
Arcurcumene
Areoles
Aril
Arjunolic acid
Aromatherapy
Arterial macrophage
Arthritis
Artocarpanone
Artocarpetin
Artocarpins
Asclerodanediterpenes
Aspergillus sp.
A. fumigatus
A. niger
A. parasiticus
Aspirin
Asteraceae
Asthma
Astragalin
Astringent
Atherogenic
Atherosclerosis
Axillary
Ayurvedic
Azadirachitin
Azafluorenealkaloids
B
Bacillus sp.
B. anthracis
B. cereus
B. subtilis
Balaphonin
Balsam
Barringtogenol
B-cells
Bee glue
Bengalinoside
Benzyl isothiocyanate
Berberine
Bergapten
Berry
β-lactam
β-setosterol
β-sitosteryl glycoside
Betacyanins
Betalain
Betulin
Betulinic acid
BHA
SeeButylated hydroxy anisole (BHA)
BHT
SeeButylated hydroxy toluene (BHT)
Bicyclogermacrene
Bicyclomahanimbicine
Biflavonoids
Bignoniaceae
Bilobed
Biochanin
Biodegradable
Biofuel
Bioindicator
Biopesticides
Bipinnately compound
Bisabolene
Bis-indole alkaloid
Blennorrhea
Bombacaceae
Bombamalones
Bombaxquinone B
Borneol
Botrytis cinerea
Bracts
Brassicaceae
Brassinosteroids
Breast cancer
Brewer's yeast
Bromelain
Bronchitis
Brosimone
Broth dilution assays
Broussoflavonols
B16f10 melanoma
Buchapine
Busseins
Butylated hydroxy anisole (BHA)
Butylated hydroxy toluene (BHT)
C
Cactaceae
Cadabicine
Cadinane sesquiterpenoids
Cadinene
Caducous
Caesalpinoideae
Caffeic acid
Caffeoylquinic acid
Cajanins
Calanolide
Calycosins
Calystegin
Campestrol
Camphene
Camphor
Camptothecin
Candida albicans
Candidiasis
Capillary zone electrophoresis
Capitulum
Capparaceae
Capsaicin
Capsule
Carbazole
Carbon tetrachloride (CCl 4 )
Carcinogens
Cardenolides
Cardioactive glycosides
Cardioprotective
Caricaceae
Caricin
Carminative
Carnphor
Carotene
Carotenoids
Carpaine
Carposide
Carrageenan
Casuarinaceae
Casuarine
Catechins
Catechols
Cathartic
Catkins
CCl 4
See Carbon tetrachloride (CCl 4 )
Cedrelons
Centdarol
Central nervous system (CNS)
Cerebrosides
Cervical carcinoma
CGA
SeeChlorogenic acid (CGA)
Chalcones
Chaparrine
Chemopapain
Chemotherapy
Chickenpox
Chloroform extracts
Chlorogenic acid (CGA)
Chlorpyrifos
Choline
Chronic dysmenorrhea
Chrysin
Chrysoeriol
Chukrasins
Chymopapain
1,8-cineole
Cinnamyl cinnamate
Ciprofloxacin
Citronellal
Citronellol
Cladode
Clerodanediterpenes
Clotrimazole
CMW
SeeCytomegalovirus (CMW)
CNS
SeeCentral nervous system (CNS)
Cold
Collagen
Colon carcinoma
Combretaceae
Combretastatin
Complete
Compositae
Compound
Condensed tannins
Cones
Conjunctivitis
Contraceptives
Cordate
Coriaceous
Cornusins
Coronary heart disease
Corosolic acid
Corrugated
Corymb
Cough
Coumaric acid
Coumarin
Coumarylplumieride
Cruciferae
Cryptococcus neoformans
Cryptoxanthin
Cucurbitaceae
Cupressaceae
Cupressuflavone
Curcain
Curcumin
Cyanidin
Cyanogenic glycosides
Cyathia
Cycadaceae
Cycloartanes
Cycloartocarpin
Cyclomorusin
Cyclooxygenases
Cyclooxygenases-2
Cyclopeptides
Cyme
Cymene
Cynomacurins
Cytokines
Cytomegalovirus (CMW)
Cytotoxic
D
DAL
SeeDalton's ascitic lymphoma (DAL)
Dalbergenone
Dalbergichromene
Dalbergin
Dalton's ascitic lymphoma (DAL)
DCCC
SeeDroplet counter-current chromatography (DCCC)
Deciduous
Decoction
Dehiscent
Dehydrocarpain I
Delphinidin
Dementia
Dentate
Deoxyjasminigenin
Depo Provera
Depression
Dermatitis
Diabetes mellitus
Diarrhea
Digitate
Diglucoside
Dihydrojasminine
Dihydromorins
Dilactones
Dilapachone
Dilleniaceae
Dioecious
Diosgenin
Diosindigo
Diospyrin
Disk diffusion method
Distillation
Diuretic
Diverin
Docetexel
Docosane
Doxorubicin
Droplet counter-current chromatography (DCCC)
Drupe
Dysentery
Dyslipidemia
Dyspepsia
E
Ebenaceae
Echinocystic acid
Echitamine
EGCG
SeeEpigallocatechin-3-gallate (EGCG)
Ehrlich ascites carcinoma
11β-HSD1
11β-HSD2
Ellagic acid
Ellagitannins
Ellipsoid
Elliptic
Emblicanin
Emmenagogues
Emollient
Endocarp
Endothelial dysfunction
Entamoeba histolytica
Enterobacter cloacae
Epicalyx
Epicorosolic acid
Epidermoid cancer
Epigallocatechin-3-gallate (EGCG)
Epilepsy
Epiphyte
Epipodophyllotoxins
Epitaraxerol
Epoxide
Erectile dysfunction
Erthrodiol
Erysodienone
Erythatine
Erythrodiol monocaprylate
Escherichia coli
ESI mass spectrometry
Estradiol
Ethanolic extracts
Ethnopharmacopeia
Ethylacetate fraction
Etoposides
Eugenols
Euphol
Euphorbiaceae
Evergreen
Evofolin
Exocarp
Expectorant
Extraction
F
Fabaceae
Faboideae
Farnesene
Farnesol
Fast atom bombardment mass spectroscopy
Febrifuge
Federal Drug Administration (FDA)
Fenchor
Ferulates
Ferulic acid
Feruloylquinic acid
Fever
Fibrosarcoma
Figs
Five flower herbal tea
Flavanoids
Flavedo
Flavellagic acids
Flavopiridol
Folinic acid
Follicles
Fourier transform infrared spectroscopy (FTIR)
Fragilin
Frederine
Friedelin
FTIR
SeeFourier transform infrared spectroscopy (FTIR)
Fucosterol
Fulvic acid
Fulvoplumierin
Fungicidal
Furanocoumarins
Fusarium sp.
F. oxysporum
F. solani
G
Galactagogue
Galactotannins
Galangin
Gallacatechins
Gallate
Gallic acid
Gallicin
Galucarubinone
Gamatay
γ-selinene
Gastroenteritis
Gastroprotective
Gemacrene-D
Genetically engineered
Genistein
Genotoxic
Geraniol
Germacrene
Gingerol
Girinimbin
Glabrin
Glabrosaponin
Glabrous
Glibenclamide
Globose
Glossy
Glucoputranjivan
Glucosides
Glucosinolates
Glutathione
Glycogenesis
Glycosides
Gold nano particles
Gossypetin
Gylcyrrhizin
Gynecological cancer
H
Harmane
Harmine
HDL
SeeHigh density lipid (HDL)
Headspace-solid phase micro extraction
Hemicranias
Hemorrhoids
Hentriacontane
Hentriacontanol
Hepatic fibrosis
Hepatitis
Hepatitis B virus
Hepatocellular carcinoma
Hepatocytes
Hepatoma
Hepatoprotective
Hep2 cells
Heptacosanoate
Herpes simplex virus (HSV-1)
Hesperidin
Hespiridium
Hexenol
High density lipid (HDL)
High performance liquid chromatography (HPLC)
Himachalol
Himadarol
Histone deacytylases
HIV
SeeHuman immunodeficiency virus (HIV)
Honokiol
HPLC
SeeHigh performance liquid chromatography (HPLC)
HSV-1
SeeHerpes simplex virus (HSV-1)
Hull
Human immunodeficiency virus (HIV)
Human leukemic cell (HL-60)
Humulene
Husk
Hydrodistillation
Hydrolyzed tannins
Hydroxybenzoic acid
Hydroxymaprounic acid
Hydroxytyrosol
Hypanthium
Hyperaccumulators
Hyperglycemic
Hypericin
Hyperlipidemia
Hypertension
Hypoglycemic
Hypolipidemic
Hypophyllanthin
Hypothalamus
I
Icarisides
Immunomodulating
Immunosuppressive
Imparipinnate
Imperatorin
Impressed
Indomethacin
Inflorescence
Influenza
Infrared, 1D/2D (dimensional) nuclear magnetic resonance (NMR)
Infusion
Ingenol
Ingol derivative
Insecticidal
Insecticides
Insomnia
Insulin
Inumakiols
Iridoids
Irinotecan
Ischemic mitral regurgitation
Isodalbergin
Isodiospyrin
Isoflavones
E-isofuraldehyde
Iso-mahanimbin
Isomurrayafoline
Isoorientin
Isophytol
Isopinnatal
Isoplumericin
Isoproterenol
Isorhamnetin
Isostrictinin
Isothiocyanates
Isotocarpin
Isovitexin
J
Jacalin
Jacaranone
Jacraninoside
Jambolin
Jambosine
Jasminine
Jasmone
Jatropham
Jatrophine
Jaundice
Juglandaceae
Juglone
Jujubosides
Julifloravizole
Julifloricine
Juliflorine
Juliprosopine
Juvabional
K
Kaempferol
Kankone
Kanugin
Karangin
Keampferide
Kernel
Kigelinone
Klebsiellapneumoniae, 89
Koenidine
Koenigin
Koenigine
Koenimbine
Koenine
Kolavenic acid
Kutkoside
Kuwanol
L
Labiatae
Lacinilene
Lactones
Lamiaceae
Lanceolate
Lapachone
Laphachol
Lariciresinol
Larvicidal
Latex
Lauric acid
Laxative
LDL
SeeLow density lipoprotein (LDL)
Leaflets
Lebbecacidin
Lectins
Lecythidaceae
Legumes
Leguminosae
Lenticels
Lentils
Leprosy
Leucoanthocyanidins
Leucocyanidin
Leucoderma
Leucopelargonidin
Leucopelargonins
Leukemia
Lewis lung carcinoma
Liliaceae
Limonene
Limonoids
Linalool
Lipolysis
Lipoproteins
Liriodenine
Lithospermic acid
Lonchocarpol
Longaniaceae
Longifolene
Low density lipoprotein (LDL)
Lucenin-I
Lumbago
Lung cancer
Lupane acid
Lupenone
Lupeol
Lupeolacetate
Lupeonone
Luteolin
Lycopene
Lymphoblastoid
Lymphocytic leukemia
Lymphoma
Lythraceae
M
Maceration
Macrophominaphaseolina, 226
Magnoliaceae
Magnolol
Mahanimbicine
Malaria
Mallatojaponin
Malondialdehyde
Malvaceae
Malvidin
Mangiferin
Margins
Maxoderm ™
MCF-7
Measles
Medicarpin
Medioresinol
Melanin
Melanoma
Meliacarpin
Meliaceae
Meliacin
Melianol
Meliartenin vanillin
Menorrhagia
Menoxenia
Menthol
Mesocarp
Mesquitol
Methanolic extracts
4'-methoxy licoflavanone (MLF)
Methylation
Methyl dalbergin
Methyltransfereases
Michellamines
Micrococcus luteus
Microphyllon
Microsatellite markers
Microsporumcanis
Microtubules
Mimosoideae
Mimusopfarnanol
Mimusopic acids
Mimusopins
Mimusopsides
MLF
See4'-methoxy licoflavanone (MLF)
Molluscicidal
Monoecious
Moraceae
Moran A
Moranoline
Moretenol
Moretenone
Morin
Morusin
MRC-5 cancer cell line
Mucilage
Mulberroside F
Murrayaline
Murrayastine
Myeloblastosis virus
Myeloid leukemia
Myocardial
Myocytes
Myrcene
Myricetin
Myricetol
Myrtaceae
N
Nanoparticles
Naphthoquinones
Naringenin
Nectarines
Neoandrographolide
Neoglabrin
Neohesperidoside
Neoplasia
Nephroprotective
Nephropathy)
Nerifolin
Neriin
Nerol
Nerolidol
Neuroprotective
Neurotoxic
NF-kappaB transcription
Nicotinic acid
Nictoflorin
Nimbidin
Nimbin
Nitidine
Nobiletin
Nonadecane
Norartocarpetin
Nordalbergin
Norlimonoids
Noroliveroline
Norplant ©
N-nitrosodiethylamine
Nutraceutical
Nyctaginaceae
Nyctanthin
O
Obovate
Obtuse
Oleaceae
Oleandrin
Oleanolic acid
Oleanolic acid caprylates
Oliveroline-β-N-oxide
Opposite arrangement
Opthalmopathy
Organic extraction
Organophosphorus
Orientin
Osteoporosis
Osteosarcoma
Ovate
Ovicidal
Oxalic acid
Oxidative stress
Oxyresveratrol
Ozone depleting industrial solvents
P
Paclitaxel
Palmately compound
Pancreatic cancer
Papain
Papaverine
Paracetamol
Parkinsonin
Parkinson's disease
Parkintin
Patulitrin
Pedicel
Pedunculagin
Pelargonidin
Pelvic nerve ganglia
Pendulous
Penicillium chrysogenum
Pentamerous
Peonidin
Peptides
Peptidyl-propyl isomerase pin1
Pericarp
Periplogenin
Peruvosides
Petiolate
Petroleum ether extracts
Phebalosin
Phellandrene
Phenolates
Phenyl chromene
Phenylethanoid
Pheromones
Phlobatannin
Phloridzin
Phosphorylation
Photoaging
Photoperiodism
Phyllanthin
Phyllotaxy
Physcion
Phytoestrogens
Phytol
Phytoremediation
Phytosterolin
Phytosterols
Picroside
Pinaceae
Pinene
Pinitol
Pinnatal
Pinnate
Pinnately compound
Pinnatin
Pinoresinol
Piperamides
Piperidine
Piperine
Plants functional products
Plants secondary metabolites
Platanaceae
Plumericin
Plumieride
PMFs
SeePolymethoxyflavones (PMFs)
Pneumoniae
Podocarpaceae
Podolactones
Poliovirus
Polyacetylenes
Polyalthialdoic acid
Polyamines
Polygamous
Polymethoxyflavones (PMFs)
Polyphenols
Pomiferins
Pongamol
Pongapin
Porphyromonas gingivalis
Poultices
Prevotella intermedia
Proanthocyanidins
ProEnhance ™
Progesterone
Prosogerin
Prosogerine
Prostaglandin
Prostaglandin synthase
Prostanoids
Prostate
Protease inhibitors
Protein tyrosine phosphatase (PTP)
Proteolytic
Proteus vulgaris
Protocatechuic acid
Protolimonoids
Protoplumericine
Pseudocarpain
Pseudomonas sp.
P.aeruginosa
P. pseudomallei
Psoralen
Psychedelic
Pterocarpens
P-388 lymphocytic leukemia system
PTP
SeeProtein tyrosine phosphatase (PTP)
Pubescence
Punicalagin
Punigluconin
Purgative
Putranjivaceae
Putranjivoside
Pypayafolinecarbazole alkaloids
Pyrexia
Pyrocatecollic acids
Pyrogallol
Pyrogallotanins
Pyrogens
Q
Quassinoid alkaloids
Quercetin
Quercitrocides
Quinones
R
Raceme
Rachis
Radical-scavenging
Raman microspectroscopic techniques
Ranikhet disease virus
Ranunculaceae
Rauscher murine leukemia virus
Receptacles
Replication
Resin
Resveratrol
Retusaphenol
Reverse phase HPLC
Reverse transcriptase
Rhamnosides
Rhazmanine
Rheumatism
Rhizopus stolonifer
Rho-kinase
Ricinoleic acid
Rind
Robinetin
Rosaceae
Rotenone
Rubiaceae
Rubifacient
Rubrinol
Rutaceae
Rutin
Rutinoside
Rutoside
Ruvosides
S
Salicaceae
Salicin
Salicortin
Salicylates
Salmonella sp.
S. enterica ser typhi
S. typhi
Sambacin
Sambacolingoside
Sambacoside
Sapindaceae
Saponin
Sapotaceae
Sapotin
Sapotinin
Saraca indica
Scabies
Scopolamine-induced amnesia
Scutellarin
Secoiridoids
Selinene
Semimyrtucmmulone
Seojulirprsopinol
Serrate
Serratia marcescens
Sesamin
Sessile
Shallomin
Shamimin
Shigella flexneri
Silica gel column chromatography
Silvestrol
Silymarin
Simaroubaceae
Simmering
Sitosterol
Sitz bath
Small pox
Soaking
Sohumbertiol
Solanaceae
Solitary
Spathulenol
Spermatogenesis
Spermicidal
Spicigerine
Spines
Squalene
Staphylococcus sp.
S. aureus
S. epidermidis
Steam bath
Stem bark
Sterculiaceae
Sterols
Stimulants
Stipulate
Streptococcus sp.
S. anginosus
S. faecalis
S. gordonii
S. mitis
Streptozotocin
Strictinin
Succulents
Sulforaphanes
Swertifrancheside
Swiss rats
Syringaresinol
Syringic acid
T
Tabularin
Tail immersion test
Tangeretin
Tannins
Taraxerol
Taxanes
Taxol(s)
Techomine
Tecomanine
Tecostanine
Tectonoelin
Teratology
Termilignan
Terpenoids
Terpineol
Testosterone
Thannilignan
Thevetin
Thevetoxin
Thiosulfinates
Thujone
Thyroid
Tirucallanes
Tocopherol
Tomentose
Topotecan
Totarol
Trans-linalool oxide
Tremulacin
Triacanthosides
Trichocarpin
Trichomes
Trichomonas vaginalis
Trichophyton simii
Trichosporon beigelii
Tricosane
Trifoliate
Trifolin
Trigonelline bases
Triterpenoids
Trypanosomiasis
Tubercles
Tuberculosis
Tubotaiwine
Turkey red oil
Type 1 diabetes
Type 2 diabetes
Tyrosine kinase
U
Ulcers
Umbelliferone
Umbles
Ursolic acids
V
Vanillic acid
Vapor therapy
Varicella-zoster virus (VZV)
Verbenaceae
Vermifuge
Verminosis
Vesicular stomatitis virus
Vicenin-II
VigRX Oil ™
Vinblastine
Vincristine
Vindesin
Vindoline
Vinylpyridine
Virility patch and rxtm
Virility pills
Vitexin
VLDL cholestrol
Vleurosine
VZV
SeeVaricella-zoster virus (VZV)
W
Warfarin
Wheezing
Wistar rats
X
Xanthones
Xanthotoxol
Xanthoxylin
Y
Yohimbe
Z
Zeaxanthin
Zeta-carotene
Zhebeiresinol
Zingiberene
| {
"redpajama_set_name": "RedPajamaBook"
} | 9,381 |
(34077) Yoshiakifuse (2000 OV68) – planetoida z grupy pasa głównego asteroid okrążająca Słońce w ciągu 4,69 lat w średniej odległości 2,8 j.a. Odkryta 30 lipca 2000 roku.
Zobacz też
lista planetoid 34001–35000
lista planetoid
Linki zewnętrzne
Planetoidy pasa głównego
Nazwane planetoidy
Obiekty astronomiczne odkryte w 2000 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,683 |
https://www.gameinformer.com/b/news/archive/2013/02/07/rumor-sonys-ps4-priced-at-over-400-dollars.aspx
Report: Sony's PS4 Priced At Over $400
by Matthew Kato on Feb 07, 2013 at 03:11 AM
According to a newspaper article from Japan, Sony's upcoming system will be north of $400, but less than the initial PS3 offerings.
Japanese newspaper Asahi Shimbun reports that the PlayStation 4 will retail for 40,000 yen, which works out to be about $428. This would be a bit less than the initial price of the PS3 back when it released in Japan in 2006 with 20GB ($499) and 60GB ($599) configurations. Speaking of which, the Asahi reporting of a single price would seem to indicate that Sony will launch with only one version of the console this time, but that has not been officially confirmed.
Also mentioned in the report is that the system's controller will be similar in shape to the current one, but will have touchpad.
We expect to find out more about the system at Sony's February 20 event.
[via The Verge] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,181 |
Q: How do I watch a window.variable change with a callback? I have a global variable and its type is String:
window.my_global_var = 'string';
It might be changed by some external-loaded JavaScript files or an AJAX request. So I want to watch it and invoke a callback when it's updated.
I searched a while but found Object.observe is already deprecated. I found an answer but it is better used to observe an object, not a String window.variable.
The worst approach would be using a setInterval to watch it but I guess it's too stupid.
Is there any good way to do this?
A: You can use Object.defineProperties on window:
function onValueUpdate(my_global_var) {
// Some arbitrary logic you want to execute on callback
console.log(`my_global_var was updated: ${my_global_var}`);
}
Object.defineProperties(window, {
_my_global_var: {
value: 'string',
writable: true
},
my_global_var: {
get: function() {
return this._my_global_var;
},
set: function(val) {
this._my_global_var = val;
onValueUpdate(this._my_global_var);
}
}
});
window.my_global_var = 'New string';
When you access window.my_global_var it behaves exactly as the property of type String would. But when you set it you can adjust it to use any additional logic.
Function onValueUpdate needs to be public (or you can use a public method instead).
There's a warning against this approach in the answer you've found though:
I'd not go with getters/setters solution - it's complicated, not scalable and not maintainable.
So if you need scalability you probably should look for some library that can do that. Otherwise this should work just as well.
A: You could wrap the global object in a proxy with a set handler. You would need to pass the proxy around your program, rather than relying implicitly on the global object, however.
const handler = {
set(obj, prop, value) {
if (prop === 'foo')
console.log(`Property updated with value: ${value}!`)
return Reflect.set(...arguments)
}
};
const proxy = new Proxy(window, handler);
proxy.foo = 1 // "Property updated with value: 1!"
proxy.bar = 2
console.log(foo) // 1
console.log(bar) // 2
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,588 |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-solve-log-3-x-log-3-x-5-3","text":"# How do you solve log_3 x-log_3(x+5)=3?\n\nSep 11, 2016\n\n$x = - \\frac{135}{26} = 5 \\frac{5}{26}$\n\n#### Explanation:\n\n\"If the log terms are being subtracted, then the numbers are being divided\"\n\n${\\log}_{3} \\left(\\frac{x}{x + 5}\\right) = 3$\n\n\"Both sides must be logs, or both sides must be numbers\"\n\n${\\log}_{3} 3 = 1 \\text{ } \\rightarrow 3 {\\log}_{3} 3 = 3 \\times 1 = 3$\n\n${\\log}_{3} \\left(\\frac{x}{x + 5}\\right) = 3 {\\log}_{3} 3$\n\n${\\log}_{3} \\left(\\frac{x}{x + 5}\\right) = {\\log}_{3} {3}^{3}$\n\n$\\therefore \\frac{x}{x + 5} = \\frac{27}{1} \\text{ } \\left({3}^{3} = 27\\right)$\n\n$27 \\left(x + 5\\right) = x$\n\n$27 x + 135 = x$\n\n$26 x = - 135$\n\n$x = - \\frac{135}{26}$","date":"2019-06-16 19:19:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 10, \"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\": 0.880058765411377, \"perplexity\": 4302.967656195585}, \"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-2019-26\/segments\/1560627998291.9\/warc\/CC-MAIN-20190616182800-20190616204800-00205.warc.gz\"}"} | null | null |
\section{Introduction and summary}
A \emph{finite category}
over a fixed field $k$, that we will assume to be algebraically closed throughout,
is an Abelian category enriched over finite-dimensional $k$-vector spaces which has enough projective objects and finitely many isomorphism classes of simple objects; moreover, one requires that every object has finite length.
Any finite category $\cat{C}$ comes with a right exact endofunctor
\begin{align}
\catf{N}^\catf{r} : \cat{C}\longrightarrow\cat{C}\ , \quad X \longmapsto \catf{N}^\catf{r} X:=\int^{Y\in \cat{C}} \cat{C}(X,Y)^* \otimes Y \ ,
\end{align}
the \emph{(right) Nakayama functor}, where $\otimes$ denotes the tensoring of objects in $\cat{C}$ with finite-dimensional vector spaces.
This Morita invariant description was given in \cite{fss}, and it reduces to the usual definition of the (right) Nakayama functor
for the category of finite-dimensional modules over a finite-dimensional $k$-algebra.
As a consequence of the coend description of $\catf{N}^\catf{r}$,
we obtain in Corollary~\ref{cortheiso}
natural isomorphisms
\begin{align} \cat{C}(P,X) \cong \cat{C}(X,\catf{N}^\catf{r} P)^* \quad \text{for}
\quad X\in\cat{C}\ , \quad P\in \operatorname{\catf{Proj}} \cat{C} \
\end{align}
turning the subcategory $\operatorname{\catf{Proj}}\cat{C}\subset \cat{C}$
into an $\catf{N}^\catf{r}$-twisted Calabi-Yau category.
Through the correspondence between
(twisted) Calabi-Yau structures and (twisted) traces, one obtains the trace
\begin{align}\catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \longrightarrow k \quad \text{for}\quad P\in \operatorname{\catf{Proj}} \cat{C} \ . \label{eqngentrace} \end{align}
It is now an obvious task to relate this relatively generically constructed
trace (or rather family of traces) to the
modified trace \cite{geerpmturaev,mtrace1,mtrace2,mtrace3,mtrace,bbg18}. Modified traces are
not only a concept of independent algebraic interest, but can also
be used for the construction of invariants of closed three-dimensional manifolds, see e.g.\ \cite{cgp,bcgpm}.
In such constructions, they serve as a non-semisimple replacement for quantum traces.
In this article,
we prove in Theorem~\ref{thmmtrace} that, for a finite tensor category in the sense of Etingof-Ostrik \cite{etingofostrik}, i.e.\ a finite category with rigid monoidal product and simple unit,
the trace~\eqref{eqngentrace} on the tensor ideal of projective objects, indeed produces a twisted modified trace.
\begin{reptheorem}{thmmtrace}
For any finite tensor category $\cat{C}$,
the twisted trace
$ (\catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \longrightarrow k)_{P\in \operatorname{\catf{Proj}}\cat{C}}$
from \eqref{eqngentrace}
is (twisted) cyclic, non-degenerate and satisfies a generalized partial trace property.
Under the additional assumption that on the finite tensor category $\cat{C}$ a pivotal structure has been chosen,
the twisted trace
$ (\catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \longrightarrow k)_{P\in \operatorname{\catf{Proj}}\cat{C}}$
can be naturally identified with a right modified $D$-trace,
where $D\in\cat{C}$ is the distinguished invertible object of $\cat{C}$.
\end{reptheorem}
This uses crucially that by \cite[Theorem~4.26]{fss} the Nakayama functor of a finite tensor category can be expressed as
\begin{align}
\catf{N}^\catf{r} \cong D^{-1}\otimes -^{\vee \vee}
\end{align}
using the distinguished invertible object $D\in \cat{C}$ \cite{eno-d} and the double dual functor $-^{\vee \vee}$.
Note that we do not require a pivotal structure to define our traces because the double dual functor can be conveniently
absorbed into the Nakayama functor. If $\cat{C}$ is pivotal, however, we recover the usual definitions.
Our motivation for unraveling the connection between the Nakayama functor and the modified trace
is topological, but does not directly come from invariants of closed three-dimensional manifolds. Instead, we are motivated by two-dimensional topological conformal field theory, a certain type of differential graded two-dimensional open-closed topological field theory:
Suppose that we are given a finite tensor category $\cat{C}$ and
a \emph{symmetric Frobenius structure}, by which we mean
a certain trivialization of the right Nakayama functor as right $\cat{C}$-module functor relative to a pivotal structure (we give the details in Definition~\ref{defsymfrob}; it will amount to a pivotal structure and a trivialization of the distinguished invertible object).
Then the trace coming from \emph{this}
particular trivialization of $\catf{N}^\catf{r}$
produces, as discussed above,
a Calabi-Yau structure on the tensor ideal $\operatorname{\catf{Proj}} \cat{C}\subset \cat{C}$.
To this Calabi-Yau structure on $\operatorname{\catf{Proj}} \cat{C}$,
Costello's Theorem \cite{costellotcft} associates a
topological conformal field theory $\Phiit_\cat{C}$
that we refer to as the \emph{trace field theory}
of the finite tensor category $\cat{C}$
with symmetric Frobenius structure.
On a technical level, $\Phiit_\cat{C}:\catf{OC}\longrightarrow\catf{Ch}_k$ is a symmetric monoidal functor from a certain differential graded version of the open-closed bordism category to chain complexes, we recall the details in Section~\ref{sectracefieldtheory}.
If we evaluate $\Phiit_\cat{C}$
on the \emph{open} part of the two-dimensional bordism category, $\Phiit_\cat{C}$ provides topological tools to compute with traces, but only captures information that one could have obtained by hand. This is drastically different for the \emph{closed} part of the two-dimensional bordism category: On a closed boundary component, i.e.\ on the circle, we obtain, following again \cite{costellotcft}, the Hochschild complex of $\cat{C}$, i.e.\ the homotopy coend $\int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X)$ over the endomorphism spaces of projective objects. On this complex, we have an action of the prop provided by the chains on moduli spaces of Riemann surfaces with closed boundary components. Phrased differently,
the trace field theory $\Phiit_\cat{C}$
captures the higher structures induced by the modified trace on the Hochschild complex of $\cat{C}$ while, at the same time, being very accessible
through the tools available for computations with Nakayama functors.
These higher structures will be developed in detail elsewhere. For the present article, no homotopy theory is needed, and we focus entirely on the purely linear consequences, i.e.\ on structures induced in homological degree zero.
\begin{reptheorem}{thmtracefieldtheory}
Let $\cat{C}$ be a finite tensor category with symmetric Frobenius structure and $\Phiit_\cat{C}:\catf{OC}\longrightarrow \catf{Ch}_k$ its trace field theory.
The evaluation of $\Phiit_\cat{C}$ on the disk
with one incoming open boundary interval whose complementing free boundary carries the label $P\in\operatorname{\catf{Proj}}\cat{C}$
\begin{equation} \Phiit_\cat{C} \left( \tikzfig{diskohne} \right)\ : \ \cat{C}(P,P)\longrightarrow k
\end{equation}
is a right modified trace, while the evaluation of $\Phiit_\cat{C}$ on the cylinder with one incoming open boundary interval with complementing free boundary label $P\in\operatorname{\catf{Proj}}\cat{C}$ and one outgoing closed boundary circle
\begin{align}
\Phiit_\cat{C} \left( \tikzfig{ht} \right)\ : \ \cat{C}(P,P)\longrightarrow \int_\mathbb{L}^{P\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(P,P)
\end{align}
agrees, after taking zeroth homology, with the Hattori-Stallings trace of $\cat{C}$.
\end{reptheorem}
By evaluation of $\Phiit_\cat{C}$
on the pair of pants, we obtain a non-unital multiplication $\star$
on the Hochschild complex
(it will generally not have a unit because the bordism that would normally give us a unit is not admitted in Costello's category $\catf{OC}$).
From results of Wahl and Westerland \cite{wahlwesterland}, we can conclude that this multiplication is supported, up to homotopy, in degree zero. Moreover, it is homotopy commutative by construction.
Besides the connection between the Nakayama functor and the modified trace,
the construction of this multiplication or rather its degree zero remnant is one of the main results of this short article and will be one of the key ingredients for future work.
In the present article, we prove that the product $\star$ is block diagonal (Proposition~\ref{propdiag}) and provide a formula for $\star$ (when evaluated on identity morphisms)
involving the \emph{handle elements}
\begin{align}
\xi_{P,Q} := \Phiit_\cat{C}\left( \tikzfig{handleelement} \right)\in\cat{C}(P,P)
\quad\text{for}\quad P,Q\in\operatorname{\catf{Proj}}\cat{C} \ . \label{eqnhandleelementintro}
\end{align}
of the trace field theory.
\begin{reptheorem}{thmhandleelement}
Let $\cat{C}$ be a finite tensor category with symmetric Frobenius structure.
\begin{pnum}
\item Let $P,Q\in\operatorname{\catf{Proj}}\cat{C}$.
Up to boundary in the Hochschild complex of $\cat{C}$,
the $\star$-product of $\mathrm{id}_P$ and $\mathrm{id}_Q$ is the handle element $\xi_{P,Q}$ of $P$ and $Q$:
\begin{align} \mathrm{id}_P \star \mathrm{id}_Q \simeq \xi_{P,Q} \ .
\end{align}
\item
All handle elements in the sense of~\eqref{eqnhandleelementintro} are central elements in the endomorphism algebras of $\cat{C}$.
\item
The modified trace of the handle element is given by
\begin{align} \catf{t}_P \xi_{P,Q}=\mathrm{dim} \,\cat{C}(P,Q) \ . \label{eqntraceformulai}
\end{align}
\end{pnum}
\end{reptheorem}
Formula~\eqref{eqntraceformulai} tells us that the modified trace of the handle elements recovers the entries of the Cartan matrix of $\cat{C}$.
If $P$ is simple, the handle element can be identified with the number
\begin{align}
\xi_{P,Q} = \frac{\mathrm{dim}\, \cat{C}(P,Q)}{d^\text{m} (P)} \in k \ ,
\end{align}
where $d^\text{m} (P):=\catf{t}_P(\mathrm{id}_P)\in k^\times$ is the modified dimension of $P$.
If we denote for an endomorphism $f:P\longrightarrow P$ of a projective object $P$
the Hattori-Stallings trace by $\catf{HS}(f)\in HH_0(\cat{C})$, we obtain the following statement in homology:
\begin{repcorollary}{corhs}
For any finite tensor category $\cat{C}$ with symmetric Frobenius structure,
\begin{align}
\catf{t} (\catf{HS} (\mathrm{id}_P) \star \catf{HS} (\mathrm{id}_Q) ) = \mathrm{dim}\, \cat{C}(P,Q) \quad \text{for}\quad P,Q \in \operatorname{\catf{Proj}} \cat{C} \ .
\end{align}
\end{repcorollary}
Here we denote the map on $HH_0(\cat{C})$
induced by the modified trace
again by $\catf{t}$.
\subparagraph{Conventions.} As already mentioned above, for the entire article, we fix an algebraically closed field $k$ (which is not assumed to have characteristic zero).
Concerning the convention on left and right duality, we follow \cite{egno}:
In a \emph{rigid} monoidal category $\cat{C}$, every object $X \in \cat{C}$ has \begin{itemize}
\item a \emph{left dual} $X^\vee$
that comes with an evaluation $d_X:X^\vee \otimes X\longrightarrow I$ and a coevaluation $b_X:I\longrightarrow X\otimes X^\vee$ subject to the usual zigzag identities,
\item and a \emph{right dual ${^\vee \! X}$} that comes with an evaluation $\widetilde d_X : X\otimes {^\vee \! X} \longrightarrow I$ and a coevaluation $\widetilde b_X : I \longrightarrow {^\vee \! X}\otimes X$ again subject to the usual zigzag identities.
\end{itemize}
\subparagraph{Acknowledgments.}
We would like to thank
Jürgen Fuchs,
Lukas Müller
and Nathalie Wahl
for helpful discussions.
CS is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation)
under Germany's Excellence Strategy -- EXC 2121 ``Quantum Universe'' -- 390833306.
LW gratefully acknowledges support by
the Danish National Research Foundation through the Copenhagen Centre for Geometry
and Topology (DNRF151).
\subparagraph{Note.}
While finalizing this manuscript,
the preprint \cite{ss21}
appeared which provides a proof for a connection between the Nakayama functor and modified traces very similar to the one afforded by Theorem~\ref{thmmtrace}.
\section{Traces on finite categories\label{sectracesfin}}
For any finite category $\cat{C}$, the (right) Nakayama functor $\catf{N}^\catf{r} : \cat{C}\longrightarrow\cat{C}$ is given by
\begin{align} \catf{N}^\catf{r} X:= \int^{Y\in \cat{C}} \cat{C}(X,Y)^* \otimes Y \ . \label{defeqnnaka}
\end{align}
This is the Morita invariant description
given in \cite{fss} for the usual Nakayama functor
for finite-dimensional modules over a finite-dimensional algebra $A$
which is given by
\begin{align}\label{eqnnakayamaalg} \catf{N}^\catf{r} X= \mathrm{Hom}_A(X,A)^*\cong A^* \otimes_A X \end{align}
for any finite-dimensional $A$-module $X$.
The Nakayama functor sends projective objects to injective objects.
\begin{proposition}\label{propdualhom}
For any finite category $\cat{C}$, there is a canonical isomorphism of chain complexes
\begin{align} \cat{C}(X,Y_\bullet)^* \cong \cat{C}(Y_\bullet, \catf{N}^\catf{r} X ) \end{align} natural in objects $X,Y\in\cat{C}$, where $Y_\bullet$ is a projective resolution of $Y$.
\end{proposition}
\begin{proof}
Since every finite category can be written as finite-dimensional modules over a finite-dimensional algebra, we conclude from the comparison of~\eqref{defeqnnaka} and~\eqref{eqnnakayamaalg} that the right hand side of~\eqref{defeqnnaka} can be modeled as a \emph{finite} colimit.
Since $\cat{C}(Y_\bullet,-)$ is exact, the finite colimit used to define $\catf{N}^\catf{r} X$ is preserved, which leads to
\begin{align}
\cat{C}(Y_\bullet, \catf{N}^\catf{r} X )\cong \int^{Z\in\cat{C}} \cat{C}(Y_\bullet,\cat{C}(X,Z)^*\otimes Z)\cong \int^{Z\in\cat{C}} \cat{C}(Y_\bullet, Z)\otimes \cat{C}(X,Z)^*\cong \cat{C}(X,Y_\bullet)^* \ .
\end{align}
All coends are computed degree-wise here.
In the last step, we have used the Yoneda Lemma.
\end{proof}
The isomorphisms from Proposition~\ref{propdualhom} can be used to obtain a twisted Calabi-Yau structure on $\operatorname{\catf{Proj}} \cat{C}$.
\begin{definition}
An \emph{$(F,G)$-twisted Calabi-Yau category} is a linear category $\cat{A}$ with endofunctors $F,G:\cat{A}\longrightarrow\cat{A}$ and isomorphisms $\cat{A}(F(X),Y)\cong \cat{A}(Y,G(X))^*$ natural in $X,Y\in \cat{A}$.
\end{definition}
In order to avoid overloaded notation,
we call a twisted Calabi-Yau category \emph{left $F$-twisted} and \emph{right $G$-twisted} if the twist datum $(F,G)$ is given $(F,\mathrm{id}_\cat{A})$ and $(\mathrm{id}_\cat{A},G)$, respectively.
By a \emph{Calabi-Yau category} (without any mention of twists) we will understand an untwisted Calabi-Yau category in the sense that $F=G=\mathrm{id}_\cat{A}$.
A Calabi-Yau category with one object is a symmetric Frobenius algebra.
\begin{corollary}\label{cortheiso}
For a finite category $\cat{C}$,
there are canonical isomorphisms
\begin{align} \cat{C}(P,X) \cong \cat{C}(X,\catf{N}^\catf{r} P)^* \label{eqntheiso}
\end{align}
natural in $X\in\cat{C}$ and $P\in \operatorname{\catf{Proj}} \cat{C}$.
In particular, $\operatorname{\catf{Proj}} \cat{C}$ is a right $\catf{N}^\catf{r}$-twisted Calabi-Yau category.
\end{corollary}
\begin{proof
For $X\in \cat{C}$ and $P\in \operatorname{\catf{Proj}} \cat{C}$,
we find (we denote equivalences of chain complexes aka quasi-isomorphisms by $\simeq$ and isomorphisms, as before, by $\cong$)
\begin{align}\begin{array}{rclll}
\cat{C}(X,\catf{N}^\catf{r} P)^* &\simeq & \cat{C}(X_\bullet,\catf{N}^\catf{r} P)^* && \text{(because $\catf{N}^\catf{r} P$ is injective)} \\ &\cong& \cat{C}(P,X_\bullet)&& \text{(Proposition~\ref{propdualhom})}\\& \simeq &\cat{C}(P,X) && \text{(because $P$ is projective)} \ . \end{array}
\end{align}
\end{proof}
\begin{definition}[Trace of a finite category]\label{deftracefc}
For any finite category $\cat{C}$, we
define the pairings
\begin{align}
\label{eqntheparings} \spr{ -,- } : \cat{C}(P,X)\otimes\cat{C}(X,\catf{N}^\catf{r} P) \ra{ \eqref{eqntheiso} } \cat{C}(X,\catf{N}^\catf{r} P)^*\otimes \cat{C}(X,\catf{N}^\catf{r} P) &\ra{\text{evaluation}} k \\ \quad \text{for}\quad &X\in\cat{C}\ , \ P\in\operatorname{\catf{Proj}}\cat{C}
\end{align}
and, by considering the case $X=P$ in~\eqref{eqntheparings},
the \emph{twisted trace} \begin{align} \catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \ra{ \spr{\mathrm{id}_P,-} } k \quad \text{for}\quad P\in\operatorname{\catf{Proj}} \cat{C} \label{eqnthetrace} \end{align} on $\cat{C}(P,\catf{N}^\catf{r} P)$.
We refer to the family of maps~\eqref{eqnthetrace},
where $P$ runs over all projective objects, as the \emph{twisted trace on $\cat{C}$}.
By an \emph{untwisting} of the twisted trace, we mean a trivialization $\catf{N}^\catf{r} \cong \mathrm{id}_\cat{C}$ of $\catf{N}^\catf{r}$ (if there exists any) and the resulting identification of the maps \eqref{eqnthetrace} with maps $\cat{C}(P,P)\longrightarrow k$ that we then refer to as \emph{untwisted trace}, or just \emph{trace} for brevity.
\end{definition}
It is important to note that the twisted trace is canonical while the untwisting (if possible) will involve choices.
An untwisting of the trace is equivalent to an untwisting of the twisted Calabi-Yau structure from Corollary~\ref{cortheiso}.
The usual correspondence between Calabi-Yau structures and traces can be adapted to the present situation and leads to the following:
\begin{lemma}\label{lemmatrace}
For any finite category $\cat{C}$, the twisted trace
\begin{align}\catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \longrightarrow k \quad \text{for}\quad P\in \operatorname{\catf{Proj}} \cat{C} \label{eqnthetraces} \end{align}
has the following properties:
\begin{pnum}
\item Cyclicity: For $P,Q\in\operatorname{\catf{Proj}} \cat{C}$, $f: P\longrightarrow \catf{N}^\catf{r} Q$
and $g :Q\longrightarrow P$, we have
\begin{align}
\catf{t}_Q(fg)=\catf{t}_P(\catf{N}^\catf{r}(g)f) \ .
\end{align}
\item Non-degeneracy: The trace is non-degenerate in the sense that the pairings
\begin{align}
\cat{C}(P,X)\otimes \cat{C}(X,\catf{N}^\catf{r} P) \longrightarrow k \ , \quad f\otimes g \longmapsto \catf{t}_P (gf) \label{eqnsecondpairing}
\end{align}
are non-degenerate.
In fact, they agree with the pairings~\eqref{eqntheparings}.
\end{pnum}
\end{lemma}
\begin{proof}
Let $X,Y \in \cat{C}$ and $P,Q\in\operatorname{\catf{Proj}}\cat{C}$. Naturality of \eqref{eqntheiso} in $X$ means for $a:P\longrightarrow X$, $b :Y\longrightarrow \catf{N}^\catf{r} P$ and $c: X\longrightarrow Y$
\begin{align}
\spr{ {a},bc}=\spr{ {ca},b} \ . \label{hateqn1}
\end{align}
Naturality of \eqref{eqntheiso} in $P$ means for $a:Q\longrightarrow X$, $b:P\longrightarrow Q$ and $c:X\longrightarrow \catf{N}^\catf{r} P$
\begin{align}
\spr{ {a},\catf{N}^\catf{r}(b)c}=\spr{ ab,c} \ . \label{hateqn2}
\end{align}
This implies for $f: P\longrightarrow \catf{N}^\catf{r} Q$
and $g :Q\longrightarrow P$
\begin{align}
\catf{t}_Q(fg)\stackrel{\eqref{eqnthetrace}}{=}\spr{ {\mathrm{id}_Q},fg} \stackrel{\eqref{hateqn1}}{=} \spr{ {g},f} \stackrel{\eqref{hateqn2}}{=} \spr{ {\mathrm{id}_P},\catf{N}^\catf{r}(g)f} \stackrel{\eqref{eqnthetrace}}{=} \catf{t}_P(\catf{N}^\catf{r}(g)f) \ .
\end{align}
This proves cyclicity.
Non-degeneracy holds by construction because it follows easily from~\eqref{hateqn1} that the pairing~\eqref{eqnsecondpairing} agrees with $\spr{-,-}$.
\end{proof}
\needspace{5\baselineskip}
\section{Traces on finite tensor categories and connection to modified traces\label{tracesonfinitfc}}
We will now turn to finite \emph{tensor} categories and
connect the construction from
Definition~\ref{deftracefc} to modified traces \cite{mtrace1,mtrace2,mtrace3,mtrace}.
A (twisted, right) modified trace on the tensor ideal of projective objects in a pivotal finite tensor category is a cyclic, non-degenerate trace that satisfies the right partial trace property as we will discuss in detail below. The first two properties hold very generally for traces constructed from linear trivializations of the Nakayama functor thanks to Lemma~\ref{lemmatrace}.
The partial trace property makes use of the monoidal structure. In order to understand when we can formulate and prove such a property for the trace from Definition~\ref{deftracefc}, one needs to understand the Nakayama functor of a finite tensor category:
Let $\cat{C}$ be any finite tensor category.
We denote by $\cat{C}_\cat{C}$ the finite category $\cat{C}$ as regular right module category over itself and by $\cat{C}^{\vee \vee}$ the finite category $\cat{C}$ as $\cat{C}$-right module with action given by $X.Y:=X\otimes Y^{\vee \vee}$ for $X,Y\in\cat{C}$.
\begin{theorem}[{\cite[Theorem~3.26]{fss}}] \label{thmnakamodule}
For any finite tensor category $\cat{C}$, the (right) Nakayama functor is an equivalence $\catf{N}^\catf{r} : \cat{C}_\cat{C} \ra{\simeq} \cat{C}^{\vee \vee}$ of right $\cat{C}$-module categories; in particular, comes with canonical isomorphisms $ \catf{N}^\catf{r}(-\otimes X)\cong \catf{N}^\catf{r}(-)\otimes X^{\vee\vee}$ for $X\in\cat{C}$. Moreover, $\catf{N}^\catf{r} I\cong D^{-1}$, where $D\in\cat{C}$ is the distinguished invertible object and $D^{-1}$ its dual, and hence
\begin{align} \catf{N}^\catf{r} \cong D^{-1}\otimes -^{\vee\vee}
\end{align}
by a canonical isomorphism.
\end{theorem}
Together with Proposition~\ref{propdualhom},
this implies:
\begin{corollary}\label{cordualhom}
For any finite tensor category $\cat{C}$, there are canonical isomorphisms
\begin{align} \cat{C}(X,Y_\bullet)^* \cong \cat{C}(Y_\bullet, D^{-1} \otimes X^{\vee \vee} ) \end{align} natural in objects $X,Y\in\cat{C}$, where $Y_\bullet$ is a projective resolution of $Y$.
In particular, any pivotal structure on $\cat{C}$ provides canonical isomorphisms
\begin{align} \cat{C}(X,Y_\bullet)^* \cong \cat{C}(Y_\bullet, D^{-1} \otimes X ) \ . \end{align}
\end{corollary}
We now propose a generalization of the partial trace property that does not need a pivotal structure (from our perspective, this will turn out to be more natural):
Let $\cat{C}$ be a finite tensor category.
For $X\in\cat{C}$ and $P\in\operatorname{\catf{Proj}}\cat{C}$, we may use Theorem~\ref{thmnakamodule} to define a
map
\begin{align} \cat{C}\left(P\otimes X, \catf{N}^\catf{r}(P)\otimes X^{\vee \vee} \right) \longrightarrow \cat{C}(P, \catf{N}^\catf{r} P ) \label{eqnptpart2}
\end{align} sending $f: P\otimes X \longrightarrow \catf{N}^\catf{r}(P)\otimes X^{\vee \vee}$ to
\begin{align}
P \ra{P\otimes b_X} P \otimes X \otimes X^\vee \ra{f\otimes X^\vee} \catf{N}^\catf{r}(P)\otimes X^{\vee \vee} \otimes X^\vee \ra{\catf{N}^\catf{r}(P)\otimes d_{X^\vee}} \catf{N}^\catf{r} P \ .
\end{align}
\begin{definition}
Let $\cat{C}$ be a finite tensor category, $P\in \operatorname{\catf{Proj}} \cat{C}$ and $X\in\cat{C}$.
Then we define the
\emph{right partial trace} as the composition
\begin{align}
\catf{tr}_\catf{r}^X: \cat{C}(P\otimes X, \catf{N}^\catf{r}(P\otimes X)) \ra{\text{Theorem~\ref{thmnakamodule}}} \cat{C}(P\otimes X, \catf{N}^\catf{r}(P)\otimes X^{\vee \vee} ) \ra{\eqref{eqnptpart2}} \cat{C}(P, \catf{N}^\catf{r}(P) ) \ . \label{eqnpartialtrace0}
\end{align}
\end{definition}
All of this crucially uses that $P\otimes X$ (and also $X\otimes P$) is projective if $P$ is, i.e.\ the ideal property property of $\operatorname{\catf{Proj}} \cat{C}$.
\begin{remark}\label{rempartialtrace}
A pivotal structure is not needed for the definition given here because the double dual is absorbed into the Nakayama functor. In presence of a pivotal structure $\omega : -^{\vee \vee}\cong \mathrm{id}_\cat{C}$, however, our definition specializes to the usual partial trace property for a \emph{right $D$-trace} in the terminology of \cite{mtrace}
in the sense that the composition
\begin{align}
\cat{C}(D\otimes P\otimes X, P\otimes X) \cong
\cat{C}(P\otimes X, \catf{N}^\catf{r}( P\otimes X)) \ra{\catf{tr}_\catf{r}^X} \cat{C}(P, \catf{N}^\catf{r}(P) ) \ ,
\end{align}
where the first isomorphism uses duality, Theorem~\ref{thmnakamodule} and the (inverse) pivotal structure,
is the usual partial trace.
\end{remark}
\begin{proposition}\label{proppartialtrace}
For any finite tensor category $\cat{C}$,
the twisted trace
\begin{align}\catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \longrightarrow k \quad \text{for}\quad P\in \operatorname{\catf{Proj}} \cat{C} \end{align}
from Definition~\ref{deftracefc}
satisfies the right partial trace property: For $X\in\cat{C}$ and $P\in\operatorname{\catf{Proj}} \cat{C}$ and any morphism $f:P\otimes X\longrightarrow \catf{N}^\catf{r}(P\otimes X)$
\begin{align} \catf{t}_P \catf{tr}_\catf{r}^X (f)=\catf{t}_{P\otimes X}( f) \ .\label{eqnpartialtrace} \end{align}
\end{proposition}
\begin{proof}
For $X,Y\in\cat{C}$, $P\in\operatorname{\catf{Proj}}\cat{C}$ and a projective resolution $Y_\bullet$ of $Y$,
consider the following diagram in which all maps are isomorphisms (we explain all parts of the diagram and its commutativity afterwards):\small
\begin{equation}
\begin{tikzcd}
\cat{C}(P\otimes X,Y_\bullet)^*\ar[d,swap,"\vee"] & \ar[l,swap,"\text{YL}"] \ar[d,swap,"\vee"]\ar[rr,"(\diamond)"] \int^{Z\in\cat{C}} \cat{C}(Y_\bullet,Z) \otimes \cat{C}(P\otimes X,Z)^* && \cat{C}(Y_\bullet,\catf{N}^\catf{r} (P\otimes X))\ar[dd,swap,"\catf{N}^\catf{r}(-\otimes X)\cong \catf{N}^\catf{r}(-)\otimes X^{\vee\vee}"] \\ \cat{C}(P,Y_\bullet \otimes X^\vee)^* & \ar[l,swap,"\text{YL}"] \int^{Z\in\cat{C}} \cat{C}(Y_\bullet,Z)\otimes\cat{C}(P,Z\otimes X^\vee)^* \ar[d,"\text{relabeling}"] \\ & \ar[lu,"\text{YL}"]\int^{Z' \in \cat{C}} \cat{C}(Y_\bullet\otimes X^\vee,Z')\otimes \cat{C}(P,Z')^* \ar[r,"(\diamond)"] & \cat{C}(Y_\bullet \otimes X^\vee,\catf{N}^\catf{r} P ) \ar[r,"\vee"] & \cat{C}(Y_\bullet,\catf{N}^\catf{r} P \otimes X^{\vee \vee}) \ .
\end{tikzcd}
\end{equation}\normalsize
The isomorphisms labeled `YL' and `$\vee$' come from the Yoneda Lemma and duality, respectively.
The isomorphisms $(\diamond)$
pull the coend and the tensoring with vector spaces out of the hom functor using exactness of $\cat{C}(Y_\bullet,-)$
(they follow essentially from the definition~\eqref{defeqnnaka} of the Nakayama functor).
The `relabeling' isomorphism $\int^{Z\in\cat{C}} \cat{C}(Y_\bullet,Z)\otimes\cat{C}(P,Z\otimes X^\vee)^*\longrightarrow \int^{Z' \in \cat{C}} \cat{C}(Y_\bullet\otimes X^\vee,Z')\otimes \cat{C}(P,Z')^*$
sends
$
f \otimes \alpha \in \cat{C}(Y_\bullet,Z)\otimes\cat{C}(P,Z\otimes X^\vee)^*
$
living in the summand indexed by $Z$ of the coend $\int^{Z\in\cat{C}} \cat{C}(Y_\bullet,Z)\otimes\cat{C}(P,Z\otimes X^\vee)^*$
to $(f\otimes X^\vee) \otimes \alpha$ living in the summand indexed by $Z\otimes X^\vee$ of the coend $\int^{Z' \in \cat{C}} \cat{C}(Y_\bullet\otimes X^\vee,Z')\otimes \cat{C}(P,Z')^*$.
The vertical isomorphism on the very right uses Theorem~\ref{thmnakamodule}.
In fact, the isomorphism $\catf{N}^\catf{r}(-\otimes X)\cong \catf{N}^\catf{r}(-)\otimes X^{\vee\vee}$ can be \emph{obtained} by extracting the isomorphism $\cat{C}(Y_\bullet,\catf{N}^\catf{r} (P\otimes X))\longrightarrow \cat{C}(Y_\bullet,\catf{N}^\catf{r} P\otimes X^{\vee\vee})$ by going in counterclockwise direction in the hexagon on the right (this follows from an analysis of the proof of \cite[Theorem~3.18]{fss}).
As a consequence, the hexagon on the right commutes.
A direct computation shows that the square and the triangle on the left commute. Therefore, the entire diagram commutes.
After taking the linear dual of the entire diagram and remembering that the isomorphisms `YL' and $(\diamond)$ combine into the isomorphisms from Proposition~\ref{propdualhom}, we see that the diagram
\begin{equation}
\begin{tikzcd}
\ar[rrr,"\text{Proposition~\ref{propdualhom}}"] \ar[dd,swap,"\vee"] \cat{C}(P\otimes X,Y_\bullet) &&& \cat{C}(Y_\bullet,\catf{N}^\catf{r}( P\otimes X)) ^*\cong \cat{C}(Y_\bullet,\catf{N}^\catf{r} P\otimes X^{\vee\vee}) ^* \ar[dd,"\vee"] \\ \\
\cat{C}(P,Y_\bullet\otimes X^\vee) \ar[rrr,swap,"\text{Proposition~\ref{propdualhom}}"] &&& \cat{C}(Y_\bullet\otimes X^\vee,\catf{N}^\catf{r} P) ^* \\
\end{tikzcd}
\end{equation}
commutes.
Since $P$ is projective (and hence $\catf{N}^\catf{r} P$ injective --- in fact, the projective objects in $\cat{C}$ even coincide with the injective ones), this reduces to the commutative diagram
\begin{equation}
\begin{tikzcd}
\ar[rrr,"\text{Corollary~\ref{cortheiso}}"] \ar[dd,swap,"\vee"] \cat{C}(P\otimes X,Y) &&& \cat{C}(Y,\catf{N}^\catf{r}( P\otimes X)) ^*\cong \cat{C}(Y,\catf{N}^\catf{r} P\otimes X^{\vee\vee}) ^* \ar[dd,"\vee"] \\ \\
\cat{C}(P,Y\otimes X^\vee) \ar[rrr,swap,"\text{Corollary~\ref{cortheiso}}"] &&& \cat{C}(Y\otimes X^\vee,\catf{N}^\catf{r} P) ^* \\
\end{tikzcd}
\end{equation}
in which the horizontal maps have specialized to the ones from Corollary~\ref{cortheiso}.
If we spell out the commutativity of this diagram
in equations for morphisms $g:P\otimes X\longrightarrow Y$ and $h:Y\otimes X^\vee \longrightarrow \catf{N}^\catf{r} P$,
we obtain with the bracket notation from Definition~\ref{deftracefc}
(we use here additionally the graphical calculus for morphisms in a monoidal category --- to be read from bottom to top; we refer to \cite{kassel} for a textbook treatment)
\begin{equation}{\footnotesize\tikzfig{trace}} \label{eqntrace}
\end{equation}
As another preparation, recall that the double dual functor $-^{\vee \vee}:\cat{C}\longrightarrow\cat{C}$ is monoidal, hence it preserves the duality pairing $d _{{^\vee X}} : X \otimes {^\vee X} \longrightarrow I$ (we use here the canonical identification ${^\vee (X^\vee)}\cong X$) and therefore sends it to $d_{X^\vee} :X^{\vee \vee} \otimes X^\vee \longrightarrow I$. Using Theorem~\ref{thmnakamodule} we find
the equality
of morphisms
\begin{align} \catf{N}^\catf{r} (P\otimes d _{{^\vee X}})=\catf{N}^\catf{r} P\otimes d_{X^\vee} : \catf{N}^\catf{r} (P) \otimes X^{\vee\vee} \otimes X^\vee \longrightarrow \catf{N}^\catf{r} P \ , \end{align}
which implies for a morphism $f:P\otimes X\longrightarrow \catf{N}^\catf{r} (P\otimes X)$
\begin{equation}{\footnotesize\tikzfig{trace2}}\label{trace2}
\end{equation}
\normalsize
The desired equality~\eqref{eqnpartialtrace} now follows from:
\footnotesize
\begin{equation}\tikzfig{trace3}
\end{equation}
\normalsize\end{proof}
\begin{theorem}\label{thmmtrace}
For any finite tensor category $\cat{C}$,
the twisted trace
$ (\catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \longrightarrow k)_{P\in \operatorname{\catf{Proj}}\cat{C}}$
from Definition~\ref{deftracefc}
is cyclic, non-degenerate and satisfies the partial trace property
in the sense of Proposition~\ref{proppartialtrace}.
Under the additional assumption that on the finite tensor category $\cat{C}$ a pivotal structure has been chosen,
the twisted trace
$( \catf{t}_P : \cat{C}(P,\catf{N}^\catf{r} P) \longrightarrow k)_{P\in\operatorname{\catf{Proj}}\cat{C}}$
from Definition~\ref{deftracefc}
can be naturally identified with a right modified $D$-trace,
where $D\in\cat{C}$ is the distinguished invertible object of $\cat{C}$.
\end{theorem}
\begin{remark}
More precisely, the twisted trace from Definition~\ref{deftracefc}
yields a \emph{canonical} right modified $D$-trace and thereby
trivializes the $k^\times$-torsor of right modified $D$-traces in a canonical way.
\end{remark}
\begin{proof}[{\slshape Proof of Theorem~\ref{thmmtrace}}]
We use the pivotal structure $\omega:-^{\vee\vee}\cong \mathrm{id}_\cat{C}$ to obtain isomorphisms \begin{align} \cat{C}(P,\catf{N}^\catf{r} P)\stackrel{\text{Theorem~\ref{thmnakamodule}}}{\cong} \cat{C}(P,D^{-1}\otimes P^{\vee \vee}) \stackrel{\omega \ \text{and duality}}{\cong} \cat{C}(D\otimes P,P) \quad \text{for}\quad P\in\operatorname{\catf{Proj}} \cat{C} \ . \label{eqnthemaps}
\end{align}
As a consequence, the twisted trace from Definition~\ref{deftracefc} gives us maps
$\cat{C}(D\otimes P,P)\longrightarrow k$ which are cyclic and non-degenerate (Lemma~\ref{lemmatrace}).
Moreover, Proposition~\ref{proppartialtrace} combined with Remark~\ref{rempartialtrace} gives us the usual partial trace property in presence of a pivotal structure.
Note that one needs really a \emph{monoidal}
isomorphism $-^{\vee\vee}\cong \mathrm{id}_\cat{C}$
to get the desired maps $\cat{C}(D\otimes P,P)\longrightarrow k$. If $\omega$ is just linear, one would get maps $\cat{C}(D\otimes P,P)\longrightarrow k$, but they would not necessarily satisfy the partial trace property:
The proof of the partial trace property in its $\catf{N}^\catf{r}$-twisted version (Proposition~\ref{proppartialtrace}) relies on the monoidal structure of $-^{\vee\vee}$. The partial trace property only transfers along the isomorphisms $\cat{C}(P,D^{-1}\otimes P^{\vee \vee})\cong \cat{C}(P,D^{-1}\otimes P)$ if $-^{\vee\vee}$ is replaced by $\mathrm{id}_\cat{C}$ \emph{as monoidal functor}.
\end{proof}
\needspace{5\baselineskip}
\section{The trace field theory\label{sectracefieldtheory}}
We now introduce the topological conformal field theory induced by the Calabi-Yau structure appearing the previous section.
To this end, let us recall from \cite{costellotcft} the definition of the (differential graded) open-closed two-dimensional cobordism category, see \cite{egas,wahlwesterland} for models of this symmetric monoidal differential graded category in terms of fat graphs.
An \emph{open-closed Riemann surface}
is a Riemann surface with the following data:
\begin{itemize}
\item A subset of its boundary components, the so-called \emph{closed boundary components}. They are para\-metrized and labeled as incoming or outgoing.
\item A finite number of embedded intervals in the remaining boundary components,
the so-called \emph{open boundary intervals}.
They are also parametrized and labeled as incoming or outgoing.
\end{itemize}
The free boundary components are defined as the complement (in the boundary) of the closed boundary components and the open boundary intervals.
It will be required that each connected component of the Riemann surface has at least one free boundary component or at least one incoming closed boundary.
An example (that additionally contains certain labels that will be discussed in a moment) is depicted in Figure~\ref{figoc}.
One can now define
the symmetric monoidal differential graded category $\catf{OC}$ of \emph{open-closed cobordisms} for a set $\Lambdait$ of labels (that we will fix later and that will be suppressed in the notation; the set of labels is sometimes referred to as set of `D-branes'):
The objects are pairs of finite sets $O$ and $C$ (that in a moment will play the role of open boundary intervals and closed boundary components of Riemann surfaces) and two maps $s,t : O\longrightarrow \Lambdait$ (that attach a `start' and an `end' label
to any
open boundary).
The chain complex of morphisms from $(O,C,s,t)$ to $(O',C',s',t')$ is given by the $k$-chains on the moduli space of Riemann surfaces $\Sigmait$
with \begin{itemize}
\item an identification of its set of incoming open and incoming closed boundary components with $(O,C)$, an identification of its set of outgoing open and outgoing closed boundary components with $(O',C')$,
\item a label in the set $\Lambdait$ of D-branes for each free boundary component
\end{itemize}
subject to the following requirement:
First observe that any incoming open boundary interval $o\in O$ inherits a label for its start point and its end point, namely the label of the free boundary component that it is bounded by. We require that this label agrees with $(s(o),t(o))$; the analogous requirement is imposed for outgoing open boundary intervals.
Explicitly, for the objects $X=(O,C,s,t)$ and
$X'=(O',C',s',t')$,
the morphism complexes are given, up to equivalence, by
\begin{align}
\catf{OC} \left( X,X' \right) \simeq \bigoplus_{S : X \longrightarrow X'} C_*(B \catf{Map}(S);k) \ ,
\end{align}
where the direct sum is running over all topological types of compact oriented open-closed bordisms $S$ with incoming and outgoing boundary described by $X$ and $X'$, respectively, and $\catf{Map}(S)$ is the mapping class group of $S$; we refer to \cite{egas,wahlwesterland} for a description of these morphism complexes by means of classifying spaces of categories of fat graphs.
Composition in $\catf{OC}$ is by gluing. Disjoint union provides a symmetric monoidal structure.
\begin{figure}[h]\centering
\tikzfig{oc}
\caption{An open-closed surface with D-brane labels. As a morphism in $\catf{OC}$, we read the surface from left to right, i.e.\ with the source object (constituted by the incoming boundary components) on the left and the target object (constituted by the outgoing boundary components) on the right.
We will, however, deviate from the left-to-right drawing convention at times if it simplifies the surface; for this reason, we also indicate by `in' and `out' whether a boundary component is incoming or outgoing. The source object is given by $\{c,o_1,o_2\}$
(where the identification with boundary components is through the dotted arrows in the picture) plus the assignment $s(o_1)=s(o_2)=P$ and $t(o_1)=t(o_2)=Q$. The target object is $\{o'\}$ plus the assignments $s(o')=t(o')= R$. }
\label{figoc}
\end{figure}
\begin{definition}[Costello \cite{costellotcft} following Getzler~\cite{getzler} and Segal~\cite{segal}]For a fixed set $\Lambdait$
of $D$-branes,
an \emph{open-closed topological conformal field theory} is a symmetric monoidal functor $\Phiit : \catf{OC} \longrightarrow \catf{Ch}_k$.
An \emph{open topological conformal field theory}
is a symmetric monoidal functor $\catf{O}\longrightarrow\catf{Ch}_k$ defined only on the subcategory $\catf{O}\subset \catf{OC}$ of open bordisms.
\end{definition}
Open-closed topological conformal field theories
are a differential graded generalization
of ordinary vector space-valued
two-dimensional (open-closed) topological field theories.
The latter can be constructed and
classified in terms of symmetric and
commutative Frobenius algebras,
see \cite{kock,laudapfeiffer} for the precise statements.
In \cite{costellotcft},
Costello proves that one may construct an \emph{open} topological conformal field theory from a (linear) Calabi-Yau category (Costello actually considers differential graded Calabi-Yau categories, but we just need the linear case).
By homotopy left Kan extension, one obtains an open-closed topological conformal field theory:
\needspace{5\baselineskip}
\begin{theorem}[Costello \cite{costellotcft}]\label{thmcostello}\begin{pnum}\item
Any linear Calabi-Yau category $\cat{A}$ gives rise to
an open-closed topological conformal field theory $\catf{OC} \longrightarrow \catf{Ch}_k$ with the object set of $\cat{A}$ as the set of $D$-branes.
\item The value of this field theory on the circle is equivalent to the Hochschild complex of $\cat{A}$. \end{pnum}
\end{theorem}
In particular, if $\cat{M}_{p,q}$ is the moduli space of Riemann surfaces with $p$ incoming closed $(p\ge 1)$, $q$ outgoing closed and no open and no free boundary components, there are maps
\begin{align} C_*(\cat{M}_{p,q};k) \otimes \left( \int_\mathbb{L}^{a \in \cat{A}}\cat{A}(a,a)\right)^{\otimes p} \longrightarrow \left( \int_\mathbb{L}^{a \in \cat{A}}\cat{A}(a,a)\right)^{\otimes q} \ .
\end{align}
We refer to \cite{dva} for details on the homotopy coends appearing here.
In the present text, this is only needed to a very limited extent.
It suffices to know that in degree zero the complex $\int_\mathbb{L}^{a \in \cat{A}} \cat{A}(a,a)$ is given by
$\bigoplus_{a\in\cat{A}} \cat{A}(a,a)$.
\begin{remark}
In \cite{costellotcft}, it is actually required that $k$ has characteristic zero, but an extension to fields of arbitrary characteristic is given in \cite{egas,wahlwesterland}.
\end{remark}
From Corollary~\ref{cortheiso} and Costello's result, we immediately obtain:
\begin{corollary}\label{corollarytracefieldtheory}
For a finite category $\cat{C}$, any trivialization of the
Nakayama functor
$
\catf{N}^\catf{r} :\cat{C}\longrightarrow\cat{C}$
yields a Calabi-Yau structure on $\operatorname{\catf{Proj}} \cat{C}$
and hence gives rise to a topological conformal field theory $\Phiit_\cat{C} : \catf{OC} \longrightarrow \catf{Ch}_k$ with set of D-branes given by the set of projective objects of $\cat{C}$.
\end{corollary}
\begin{remark}\label{remtraces}
The evaluation of the field theory $\Phiit_\cat{C}$ on the disk
with one incoming open boundary interval whose complementing free boundary carries the D-brane label $P\in\operatorname{\catf{Proj}}\cat{C}$
\begin{equation} \Phiit_\cat{C} \left( \tikzfig{diskohne} \right)\ : \ \cat{C}(P,P)\longrightarrow k
\end{equation}
is exactly the trace function of the Calabi-Yau structure from Definition~\ref{deftracefc} (this follows directly from Costello's construction), while
for $P,Q\in \operatorname{\catf{Proj}}\cat{C}$,
the map
\begin{equation} \Phiit_\cat{C} \left( \tikzfig{composition} \right)\ : \ \cat{C}(P,Q) \otimes \cat{C}(Q,P) \longrightarrow \cat{C}(P,P)
\end{equation}
is the composition over $Q$.
\end{remark}
The construction of Corollary~\ref{corollarytracefieldtheory} does not do much:
It just translates a trivialization of $\catf{N}^\catf{r}$ to a Calabi-Yau structure and then a topological conformal field theory, and in fact, this construction
will only be of limited use to us since we want to treat finite tensor categories and not just linear categories.
Fortunately, we can give a natural refinement: The construction from Corollary~\ref{corollarytracefieldtheory} becomes more meaningful
in the context of finite tensor categories if $\catf{N}^\catf{r}$ is trivialized not just as a linear functor, \emph{but as a right $\cat{C}$-module functor relative to a pivotal structure}. Let us define what we mean by that:
\begin{definition}\label{defsymfrob}
For any finite tensor category $\cat{C}$ and a pivotal structure $\omega: -^{\vee \vee} \cong \mathrm{id}_\cat{C}$,
denote by $(\mathrm{id}_\cat{C},\omega)$ the identity functor endowed with the structure of a right $\cat{C}$-module functor $\cat{C}_\cat{C}\longrightarrow\cat{C}^{\vee\vee}$ by means of $\omega$.
We refer to an isomorphism $\catf{N}^\catf{r} \cong (\mathrm{id}_\cat{C},\omega)$
of right $\cat{C}$-module functors as a trivialization of $\catf{N}^\catf{r}$ as a right $\cat{C}$-module functor relative to $\omega$.
We define a \emph{symmetric Frobenius structure} on a finite tensor category $\cat{C}$
as a trivialization of $\catf{N}^\catf{r}$ as right $\cat{C}$-module functor relative to a pivotal structure, where the pivotal structure is part of the data.
\end{definition}
\begin{remark}\label{remunpacksymFrob}
Thanks to Theorem~\ref{thmnakamodule},
a symmetric Frobenius structure on a finite tensor category is a pivotal structure plus a trivialization of $D$. We use the term symmetric Frobenius structure not only as convenient shorthand for the rather clumsy description of `pivotal unimodular finite tensor category with a trivialization of the distinguished invertible object as part of the data', but also for a deeper reason: The symmetric Frobenius algebra structure on a finite tensor category allows us to write $\cat{C}$, as a linear category, as modules over a symmetric Frobenius algebra. This can be
read off from the Corollary~\ref{cordualhom}
because it provides
canonical natural isomorphisms
\begin{align} \cat{C}(X,Y_\bullet)^* \cong \cat{C}(Y_\bullet, X ) \ , \end{align}
where $X\in\cat{C}$
and $Y_\bullet$ is a projective resolution of $Y\in\cat{C}$.
However,
a finite tensor category with symmetric Frobenius structure requires
a compatibility of the monoidal structure with
the trivialization of $\catf{N}^\catf{r}$. It is not just an identification of $\cat{C}$, as \emph{linear category}, with modules over a symmetric Frobenius algebra. In the latter sense, the notion is used in \cite{shimizucoend}.
\end{remark}
\begin{definition}\label{deftracefieldtheory}
Let $\cat{C}$ be a finite tensor category with symmetric Frobenius structure.
For the trivialization of the right Nakayama functor $\catf{N}^\catf{r}$ (that $\cat{C}$
by Definition~\ref{defsymfrob}
comes equipped with), we refer to the topological conformal field theory
$\Phiit_\cat{C} : \catf{OC} \longrightarrow \catf{Ch}_k$ built from \emph{this particular} trivialization in the sense of Corollary~\ref{corollarytracefieldtheory}
as the \emph{trace field theory of $\cat{C}$}.
\end{definition}
The name is chosen because $\Phiit_\cat{C}$
does not only recover the trace of the Calabi-Yau structure by Remark~\ref{remtraces}, but can also be recovered from the trace itself.
\begin{theorem}\label{thmtracefieldtheory}
Let $\cat{C}$ be a finite tensor category with symmetric Frobenius structure and $\Phiit_\cat{C}:\catf{OC}\longrightarrow \catf{Ch}_k$ its trace field theory.
The evaluation of $\Phiit_\cat{C}$ on the disk
with one incoming open boundary interval whose complementing free boundary carries the label $P\in\operatorname{\catf{Proj}}\cat{C}$
\begin{equation} \Phiit_\cat{C} \left( \tikzfig{diskohne} \right)\ : \ \cat{C}(P,P)\longrightarrow k
\end{equation}
is a right modified trace, while the evaluation of $\Phiit_\cat{C}$ on the cylinder with one incoming open boundary interval with complementing free boundary label $P\in\operatorname{\catf{Proj}}\cat{C}$ and one outgoing closed boundary circle
\begin{align}
\Phiit_\cat{C} \left( \tikzfig{ht} \right)\ : \ \cat{C}(P,P)\longrightarrow \int_\mathbb{L}^{P\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(P,P) \label{eqnht}
\end{align}
agrees, after taking zeroth homology, with the Hattori-Stallings trace of $\cat{C}$.
\end{theorem}
\begin{proof}
The statement about the modified trace can be extracted from Theorem~\ref{thmmtrace}
and Remark~\ref{remtraces}.
For the second statement,
we first observe that by the construction of $\Phiit_\cat{C}$
the map~\eqref{eqnht} is just the inclusion of $\cat{C}(P,P)$ into the direct sum $\bigoplus_{P\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(P,P)$, which is the degree zero term of the Hochschild complex. After taking homology, we get the natural map
$\cat{C}(P,P)\longrightarrow HH_0(\cat{C})$, i.e.\ the quotient map projecting to zeroth Hochschild homology, and hence the Hattori-Stallings trace for $\cat{C}$.
The connection to the traditional Hattori-Stallings trace \cite{hattori,stallings}
uses that by writing $\cat{C}$, as a linear category,
as finite-dimensional modules over a finite-dimensional algebra $A$ (which we can always do), the zeroth homology
$HH_0(\cat{C})$ is isomorphic to the zeroth Hochschild homology $HH_0(A)=A/[A,A]$
of $A$. This is a consequence of the Agreement Principle
of McCarthy \cite{mcarthy} and Keller \cite{keller}, see also \cite[Section~3.2]{dva} for this principle in the context of finite tensor categories.
\end{proof}
We formulate the result in Theorem~\ref{thmtracefieldtheory}
topologically (although the Theorem~\ref{thmmtrace} that it relies on is purely algebraic) because, instead of traces, we will in Section~\ref{sectrivE2} use the trace field theory as an efficient tool
for computations.
\begin{remark}
The reader should appreciate that the trace field $\Phiit_\cat{C}$ is defined through a specific trivialization of the Nakayama functor and not by choosing a modified trace (although Theorem~\ref{thmtracefieldtheory} tells us that we could have done that).
This has the advantage that, through the closed formula for $\catf{N}^\catf{r}$, the trace field theory $\Phiit_\cat{C}$ becomes very accessible.
In fact, we will rely on the particular definition of $\Phiit_\cat{C}$ given above in future work.
\end{remark}
\needspace{5\baselineskip}
\section{The block diagonal product on Hochschild chains\label{sectrivE2}}
We will now see how we can profit from the topological description of traces. First we define a multiplication by evaluation on the pair of pants:
\begin{definition}\label{defstarprod}
Let $\cat{C}$ be a finite tensor category with symmetric Frobenius structure and $\Phiit_\cat{C}:\catf{OC}\longrightarrow \catf{Ch}_k$ the associated trace field theory.
Then we define the \emph{block diagonal $\star$-product} on the Hochschild complex $\int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X)$ by
\begin{align}
\star := \Phiit_\cat{C}\left(\tikzfig{pop}\right)\ : \ \int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X) \otimes \int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X)\longrightarrow \int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X) \, . \label{eqnstarproduct}
\end{align}
\end{definition}
The sense in which the product $\star$ is block diagonal will be discussed in Proposition~\ref{propdiag}.
The results of
Wahl and Westerland on the product obtained from a topological conformal field theory in \cite[Section~6]{wahlwesterland} imply
that, up to homotopy, the multiplication is concentrated in degree zero (they prove it for symmetric Frobenius algebras, but their proof carries over to our situation).
They also give a formula for the degree zero part of the homotopy commutative multiplication. We will below give a slightly different formula which, when working with a Calabi-Yau category instead of a symmetric Frobenius algebra, is a little more convenient.
\begin{lemma}\label{lemmadeligneinH0}
The degree zero part
\begin{align}
\star\ : \bigoplus_{P,Q \in \operatorname{\catf{Proj}} \cat{C}} \cat{C}(P,P)\otimes\cat{C}(Q,Q) \longrightarrow \bigoplus_{P\in \operatorname{\catf{Proj}} \cat{C}} \cat{C}(P,P)
\end{align}
of the product from Definition~\ref{eqnstarproduct} is given on the summand $\cat{C}(P,P)\otimes\cat{C}(Q,Q)$, up to boundary, by the linear map
\begin{equation}
\Phiit_\cat{C}\left(\tikzfig{multchain}\right) \ : \ \cat{C}(P,P)\otimes\cat{C}(Q,Q) \longrightarrow \cat{C}(P,P) \ . \label{multviaPhieqn}
\end{equation}
\end{lemma}
\begin{proof} Let $P$ and $Q$ be projective objects in $\cat{C}$.
The following morphisms in $\catf{OC}$ can be deformed into each other and hence represent homologous zero chains:
\begin{align}{\footnotesize \tikzfig{pairofpants}}
\end{align}
But this means that the square in $\catf{OC}$
\begin{align} \tikzfig{multsquare}
\end{align}
commutes up to boundary. If we apply $\Phiit_\cat{C}:\catf{OC}\longrightarrow\catf{Ch}_k$ to the square, we see that the square
\begin{equation}
\begin{tikzcd}
\ar[rrr,"\eqref{multviaPhieqn}"] \ar[dd,swap] \cat{C}(P,P)\otimes\cat{C}(Q,Q) &&& \cat{C}(P,P) \ar[dd] \\ \\
\int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X) \otimes \int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X) \ar[rrr,swap,"\star"] &&& \int_\mathbb{L}^{X\in\operatorname{\catf{Proj}}\cat{C}} \cat{C}(X,X) \\
\end{tikzcd}
\end{equation}
commutes up to chain homotopy, where the vertical maps are just the usual embeddings of endomorphism spaces as summands in the Hochschild complex. Since we now recover the linear map \eqref{multviaPhieqn} as the upper horizontal arrow, the assertion follows.
\end{proof}
In the case of one object,
using the Sweedler notation for the coproduct of a symmetric Frobenius algebra, we recover the formula of Wahl and Westerland \cite[Section~6, page~41]{wahlwesterland} up to boundary.
In the sequel, it will always be implicit that the $\star$-product is applied in degree zero (because of the fact that it only contains information in that particular degree).
\begin{proposition
\label{propdiag}
For any finite tensor category $\cat{C}$
with symmetric Frobenius structure, the product $\star$ is block diagonal in the sense that it vanishes on two elements in components indexed by projective objects $P$ and $Q$ with vanishing morphism space $\cat{C}(P,Q)$;
\begin{align} f\star g= 0 \quad
\text{for}\quad f \in \cat{C}(P,P)\ , \quad g \in \cat{C}(Q,Q) \quad \text{if}\quad \cat{C}(P,Q)=0 \quad \text{(or equivalently $\cat{C}(Q,P)=0$)} \ .
\end{align}
\end{proposition}
\begin{proof}
From the formula for the $\star$-product given in
Lemma~\ref{lemmadeligneinH0} one can see that the map describing $\star$ on $\cat{C}(P,P)\otimes\cat{C}(Q,Q)$
factors through $\cat{C}(P,Q)$ or $\cat{C}(Q,P)$.
\end{proof}
We now prove a formula for the $\star$-product of identity endomorphisms of two projective objects.
As a preparation, we make the following Definition:
\begin{definition}\label{handleelements}
Let $\cat{C}$ be a finite tensor category with symmetric Frobenius structure.
For $P,Q \in \cat{C}$,
we define the \emph{handle element of $P$ and $Q$} as the endomorphism
$\xi_{P,Q} \in \cat{C}(P,P)$ obtained by evaluation of the trace field theory on the annulus:
\begin{equation}
\xi_{P,Q} := \Phiit_\cat{C}\left( \tikzfig{handleelement} \right)\in\cat{C}(P,P) \ .
\end{equation}
\end{definition}
\begin{remark} The name of the element $\xi_{P,Q}$
is chosen
for the following reason:
If we were in the situation $P=Q$, the element $\xi_{P,P}$ would be the composition `$\text{multiplication}\circ \text{comultiplication} \circ \text{unit}$' in the symmetric Frobenius algebra $\cat{C}(P,P)$.
In \cite[page~128]{kock}, this element is called the \emph{handle element} of the symmetric Frobenius algebra.
\end{remark}
\begin{theorem}\label{thmhandleelement}
Let $\cat{C}$ be a finite tensor category with symmetric Frobenius structure.
\begin{pnum}
\item For $P,Q\in\operatorname{\catf{Proj}}\cat{C}$,
the $\star$-product of $\mathrm{id}_P$ and $\mathrm{id}_Q$ is the handle element $\xi_{P,Q}$ of $P$ and $Q$, up to boundary in the Hochschild complex of $\cat{C}$;
\begin{align} \mathrm{id}_P \star \mathrm{id}_Q \simeq \xi_{P,Q} \ .
\end{align}\label{tracefieldi}
\item All handle elements in the sense of Definition~\ref{handleelements} are central in the endomorphism algebras of $\cat{C}$.\label{tracefieldii}
\item The modified trace of the handle element is given by
\begin{align} \catf{t}_P \xi_{P,Q}=\mathrm{dim} \,\cat{C}(P,Q) \ . \label{eqntraceformula}
\end{align}\label{tracefieldiii}
\end{pnum}
\end{theorem}
Of course, the numbers $\mathrm{dim}\,\cat{C}(P,Q)$ on the right hand side
of \eqref{eqntraceformula} are
the entries of
the Cartan matrix of $\cat{C}$, considered here as elements in $k$.
If $P$ is simple, the handle element is the number
\begin{align}
\xi_{P,Q} = \frac{\mathrm{dim}\, \cat{C}(P,Q)}{d^\text{m} (P)} \in k \ ,
\end{align}
where $d^\text{m} (P):=\catf{t}_P(\mathrm{id}_P)\in k^\times$ is the modified dimension of $P$
(note that $\catf{t}(\mathrm{id}_P)\neq 0$ is a consequence of the non-degeneracy of the trace).
\begin{proof}[{\slshape Proof of Theorem~\ref{thmhandleelement}}]
In order to compute $\mathrm{id}_P \star \mathrm{id}_Q$ for $P,Q\in\operatorname{\catf{Proj}}\cat{C}$, we use
Lemma~\ref{lemmadeligneinH0} and the functoriality of $\Phiit_\cat{C}$: \begin{equation}
\mathrm{id}_P\star \mathrm{id}_Q \simeq \Phiit_\cat{C}\left(\tikzfig{multchain}\right) \circ \Phiit_\cat{C}\left(\tikzfig{twodisks}\right)= \Phiit_\cat{C}\left( \tikzfig{handleelement} \right) =\xi_{P,Q}
\end{equation}
This proves~\ref{tracefieldi}.
For the proof of \ref{tracefieldii}, recall that in
the setting of symmetric Frobenius algebras, it is shown in \cite[page~128]{kock} that the handle element is central.
A straightforward computation
by means of the trace field theory $\Phiit_\cat{C}$ shows that this holds still true in our more general situation: In fact, one can directly see that both the map $\xi_{P,Q}\circ - : \cat{C}(P,P)\longrightarrow\cat{C}(P,P)$
and the map $-\circ \xi_{P,Q}: \cat{C}(P,P)\longrightarrow\cat{C}(P,P)$ are given by
\begin{align}
\Phiit_\cat{C}\left(\tikzfig{central2}\right)
\end{align}
in terms of the trace field theory.
For the proof of~\ref{tracefieldiii}, first observe
\begin{equation}
\Phiit_\cat{C} \left( \tikzfig{diskohne} \right) \circ \Phiit_\cat{C}\left( \tikzfig{handleelement} \right) = \Phiit_\cat{C}\left( \tikzfig{dimension} \right)=\mathrm{dim}\,\cat{C}(P,Q) \ .
\end{equation}
(This is the generalization of the fact that the counit evaluated on the handle element on a symmetric Frobenius algebra
is the linear dimension \cite[page~129]{kock} to Calabi-Yau categories.)
Now we use Theorem~\ref{thmtracefieldtheory} which
asserts that the evaluation of $\Phiit_\cat{C}$ on the disk with one incoming open boundary interval is actually the modified trace. This proves \eqref{eqntraceformula}.
\end{proof}
Let us also formulate this on the level of homology and denote for an endomorphism $f: P\longrightarrow P$ of $P\in\operatorname{\catf{Proj}}\cat{C}$
by $\catf{HS}(f)\in HH_0(\cat{C})$ the Hattori-Stallings trace. Then Theorem~\ref{thmtracefieldtheory} and Theorem~\ref{thmhandleelement} imply:
\begin{corollary}\label{corhs}
For any finite tensor category $\cat{C}$ with symmetric Frobenius structure,
\begin{align}
\catf{t} (\catf{HS} (\mathrm{id}_P) \star \catf{HS} (\mathrm{id}_Q) ) = \mathrm{dim}\, \cat{C}(P,Q) \quad \text{for}\quad P,Q \in \operatorname{\catf{Proj}} \cat{C} \ .
\end{align}
\end{corollary}
Here, by slight abuse of notation,
we denote the map on $HH_0(\cat{C})$
induced by the modified trace
again by $\catf{t}$.
\needspace{5\baselineskip}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,374 |
POSTPONED: The Drag
« Toastmasters
Palm Springs Sunshine Sisters – Tahquitz Canyon Hike »
Sadly, due to another cast member coming down with Covid, we are postponing the run of "The Drag" by Mae West. "The Drag" will now run alongside 'SEX' in a Mae West double feature running May 25th through June 5th. In place of "The Drag" we are extending our run of the acclaimed show "Electricity".
What to do If you already have tickets for "The Drag"?
Current Ticket holders of "The Drag" have the option of seeing "Electricity" on their original date, choose a new date for "Electricity", exchange their tickets for ANY other main Stage production at the Desert Rose, apply the cost of the ticket towards another DRP Special Event, or receive a refund. We ask that you please consider an exchange rather than a refund if at all possible. Current DRAG ticket holders should Please respond to [email protected] to let us know how you would like to proceed and again, THANK YOU for your patience and understanding during this time.
The Drag is a dramatic play written in 1927 by Mae West under her pen name Jane Mast. The play revolves around a complex cast of characters navigating the closeted world of homosexual relationships and culture in 1920's New York. Show times are 7 p.m. on Wednesday and Thursday.
https://thepalmspringspost.com/events/community/add
Desert Rose Playhouse
611 S Palm Canyon Drive Suite 16
Palm Springs, CA 92264 United States + Google Map | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,337 |
class Thread;
class CLREventStatic;
class RuntimeInstance;
class Array;
typedef DPTR(RuntimeInstance) PTR_RuntimeInstance;
class ThreadStore
{
SList<Thread> m_ThreadList;
PTR_RuntimeInstance m_pRuntimeInstance;
CLREventStatic m_SuspendCompleteEvent;
ReaderWriterLock m_Lock;
private:
ThreadStore();
void LockThreadStore();
void UnlockThreadStore();
public:
class Iterator
{
ReaderWriterLock::ReadHolder m_readHolder;
PTR_Thread m_pCurrentPosition;
public:
Iterator();
~Iterator();
PTR_Thread GetNext();
};
~ThreadStore();
static ThreadStore * Create(RuntimeInstance * pRuntimeInstance);
static Thread * RawGetCurrentThread();
static Thread * GetCurrentThread();
static Thread * GetCurrentThreadIfAvailable();
static PTR_Thread GetSuspendingThread();
static void AttachCurrentThread();
static void AttachCurrentThread(bool fAcquireThreadStoreLock);
static void DetachCurrentThread();
#ifndef DACCESS_COMPILE
static void SaveCurrentThreadOffsetForDAC();
#else
static PTR_Thread GetThreadFromTEB(TADDR pvTEB);
#endif
Boolean GetExceptionsForCurrentThread(Array* pOutputArray, Int32* pWrittenCountOut);
void Destroy();
void SuspendAllThreads(CLREventStatic* pCompletionEvent);
void ResumeAllThreads(CLREventStatic* pCompletionEvent);
static bool IsTrapThreadsRequested();
void WaitForSuspendComplete();
};
typedef DPTR(ThreadStore) PTR_ThreadStore;
ThreadStore * GetThreadStore();
#define FOREACH_THREAD(p_thread_name) \
{ \
ThreadStore::Iterator __threads; \
Thread * p_thread_name; \
while ((p_thread_name = __threads.GetNext()) != NULL) \
{ \
#define END_FOREACH_THREAD \
} \
} \
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,552 |
Q: How many DTO objects will be created in a Web based Application On to the Servlets Application , i know that there is only one Servlet created , which perofrms all requests for the Actions
If we have a DTO Object which we use for Setting the Data inside the Servlet , for example
public class Servlet extends HttpServlet
{
public void doGet()
{
EmployeeDTO edto = new EmployeeDTO();
edto.setName("Test");
}
}
Now if there are 100 reuests , how many DTO objects created here ??
A: 100 of course. You don't want to share request-specific data between individual endusers, do you?
On a related note, it may be helpful to read this to learn more about how exactly servlets work behind the scenes: How do servlets work? Instantiation, sessions, shared variables and multithreading.
A: Each time a GET request comes to your servlet, the doGet method is called, and the new EmployeeDTO() statement is executed.
So if 100 requests are done, 100 instances of EmployeeDTO are created. When the request ends, unless you have stored the DTO somewhere where it can still be reached, the DTO is eligible to garbage collection.
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Data processing is the transfer of data from its raw form in to a database format. There are numerous software packages which can be used to do this, and further packages which can be used to assist researchers in drawing conclusions from the data once it has been compiled. Data processing most commonly results in data being tabulated, to make it easier for conclusions to be drawn and insights indentified and understood.
For updated Data Processing information please follow us on @djsresearch. | {
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