text:
Need help with Mysqli query

Hi Experts
I have written a mysqli query for a chart (chartjs).
http://www.chartjs.org/docs/

I can't seem to figure out where I have made an error
Please can someone look over the code and see if I have built query correct.

Basically I am making a query with 2 parameteres.
I am looking to output the results into a json format to populate the chart.
If there is a better way of doing this I would appreciate the input.

I get the following errors
==================================
errors
==================================
Notice: Undefined variable: data1_layer in C:\Program Files (x86)\Zend\Apache2\htdocs\trident\inc_widget_orv_performace.php on line 77
Warning: join(): Invalid arguments passed in C:\Program Files (x86)\Zend\Apache2\htdocs\trident\inc_widget_orv_performace.php on line 77
Notice: Undefined variable: data2_layer in C:\Program Files (x86)\Zend\Apache2\htdocs\trident\inc_widget_orv_performace.php on line 78
Warning: join(): Invalid arguments passed in C:\Program Files (x86)\Zend\Apache2\htdocs\trident\inc_widget_orv_performace.php on line 78
Notice: Undefined variable: data3_layer in C:\Program Files (x86)\Zend\Apache2\htdocs\trident\inc_widget_orv_performace.php on line 79
Warning: join(): Invalid arguments passed in C:\Program Files (x86)\Zend\Apache2\htdocs\trident\inc_widget_orv_performace.php on line 79
=================================

I have attached the code plus data to support the query
any help would be great

thanks

Matt

data_ranked_competitors.xlsx
test.php
data_ranked_competitors_detail.xlsx
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Author Commented:
test.php = inc_widget_orv_performace.php in errors above
0
Commented:
You are doing a fetch_row() on the $row and accessing the$row as if it were assoc - change to fetch_assoc();

Then have a look at this code
//define variables
$labels = 'null';$data1_layer = 0;
$data2_layer = 0;$data3_layer = 0;

// bind variables
$labels[] =$row["ORV_keyword"];
$data1_layer[] =$row["MAX_yearly_orv"];
$data2_layer[] =$row["Missed_Opportunity_ORV"];
$data3_layer[] =$row["Business_yearly_orv"];

You define $data1_layer as an int by assigning it to 0 and then try to use it as an array... and you overwrite the definition on each iteration of the loop Thats good for starters - will post more as I find them 0 Commented: A more detailed description. HINT: spend some time formatting the layout of your code - makes it easier to debug both for you and those trying to assist. $stmt = $mysqli->stmt_init(); // CHANGED TO HEREDOC STRING TO IMPROVE READABILITY$query = <<< QUERY
SELECT
data_ranked_competitors_detail.keyword AS ORV_keyword,
data_ranked_competitors_detail.ryorv_max AS MAX_yearly_orv,
(data_ranked_competitors_detail.ryorv_max - data_ranked_competitors_detail.yorv_real) AS Missed_Opportunity_ORV,
FROM
data_ranked_competitors
LEFT JOIN data_ranked_competitors_detail ON data_ranked_competitors_detail.comp_id = data_ranked_competitors.comp_id
WHERE
data_ranked_competitors.dfy_id = ?
GROUP BY ORV_keyword
ORDER BY
data_ranked_competitors_detail.ryorv_max DESC
LIMIT ?
QUERY;
if ($stmt->prepare($query)){
$stmt->bind_param("is",$dfy_id, $range);$stmt->execute();

// get results
$result =$stmt->get_result();

// build chart string
if ($result) { // OK UP TO HERE BUT NOW THINGS GO PEAR SHAPED // YOU DEFINE$labels AND $data - BUT YOU NEVER USE THEM$labels = array();
$data = array(); // NOT REALLY REQUIRED BUT OK$result->data_seek(0);
while ($row =$result->fetch_assoc()) {
// YOU ARE DEFINING, IN THE LOOP, VARIABLES YOU INTEND TO MODIFY
// INSIDE THE LOOP.
// ON EACH ITERATION YOU WIPE OUT WHAT YOU DID ON THE PREVIOUS ITERATION
//define variables

// THESE VARAIBABLES ARE BEING ASSIGNED SCALARS - BUT LOWER DOWN
// YOU ARE USING THEM AS ARRAYS
$labels = 'null';$data1_layer = 0;
$data2_layer = 0;$data3_layer = 0;

// bind variables
// APART FROM $labels WHICH IS DEFINED AS A SCALAR AND IS NOW // BEING USED AS AN ARRAY - THE OTHER 3 ARRAYS HAVE // NEVER BEEN DEFINED ANYWHERE$labels[] = $row["ORV_keyword"];$data1_layer[]   = $row["MAX_yearly_orv"];$data2_layer[]   = $row["Missed_Opportunity_ORV"];$data3_layer[]   = $row["Business_yearly_orv"]; } // NOT REALLY NECESSARY AS THE STATEMENTS BELOW OVERWRITE - BUT OK //define variables$data_string = 0;
$data_string2 = 0;$data_string3 = 0;
$labels_string = 0; // Now you can aggregate all the data into one string$data_string = "[" . join(", ", $data1_layer) . "]";$data_string2 = "[" . join(", ", $data2_layer) . "]";$data_string3 = "[" . join(", ", $data3_layer) . "]";$labels_string = "['" . join("', '", $labels) . "']"; } else { print('MySQL query failed with error: ' .$mysqli->error);
}
$result->close(); }  0 Commented: Last point - binding the$range to the LIMIT appears to break the query so that it does not return rows. This is because you are binding it as a string.

If you set $range = 20 and then $stmt->bind_param("ii", $dfy_id,$range);

It resolves the problem.

Maybe one of the other experts can comment on how to put a max,min limit in with a bind - personally I would just include it directly in the query as a string variable.
0

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Author Commented:
HI Julian
Thanks for the help.
Got it working. Coding late at night does not work!

Here is what I ended up with
<?php
// Create the PHP MySQLi object
$mysqli =$framework_builder_mysqli;

// Check for any connection errors
if ($mysqli->connect_errno) { echo "Failed to connect to MySQL: (" .$mysqli->connect_errno . ") " . $mysqli->connect_error; } // set parameters$dfy_id = $_SESSION['dfy_id']; // set filter range$range = '0,20';

if(isset($_POST["orv_range"])==''){$_POST["orv_range"] = '0,20';
} else{
$range =$_POST["orv_range"];
}
//prepare statement
$stmt =$mysqli->stmt_init();
if ($stmt->prepare(" SELECT data_ranked_competitors_detail.keyword AS ORV_keyword, data_ranked_competitors_detail.ryorv_max AS MAX_yearly_orv, (data_ranked_competitors_detail.ryorv_max - data_ranked_competitors_detail.yorv_real) AS Missed_Opportunity_ORV, data_ranked_competitors_detail.yorv_real AS Business_yearly_orv FROM data_ranked_competitors LEFT JOIN data_ranked_competitors_detail ON data_ranked_competitors_detail.comp_id = data_ranked_competitors.comp_id WHERE data_ranked_competitors.dfy_id = ? GROUP BY ORV_keyword ORDER BY data_ranked_competitors_detail.ryorv_max DESC LIMIT$range
")){
$stmt->bind_param("i",$dfy_id);
$stmt->execute(); // get results$result = $stmt->get_result(); // build chart string if ($result)
{
// define arrays
$labels = array();$data1_layer	= array();
$data2_layer = array();$data3_layer	= array();

//loop through data
while ($row =$result->fetch_assoc())
{

$labels[] =$row["ORV_keyword"];
$data1_layer[] =$row["MAX_yearly_orv"];
$data2_layer[] =$row["Missed_Opportunity_ORV"];
$data3_layer[] =$row["Business_yearly_orv"];
}

// Now you can aggregate all the data into one string
$data_string = "[" . join(", ",$data1_layer) . "]";
$data_string2 = "[" . join(", ",$data2_layer) . "]";
$data_string3 = "[" . join(", ",$data3_layer) . "]";
$labels_string = "['" . join("', '",$labels) . "']";
} else {
print('MySQL query failed with error: ' . $mysqli->error); } //$result->close();
}
?>

0
Commented:
You are welcome.
0
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text:
Browse Questions

# Figure shows the top view of two horizontal forces pulling a box along the floor the box is displaced 70cm along the broken line?What is the total work done by the two forces in pulling the box this distance?

$\begin{array}{1 1}81.2J\\51.5J\\29.7J\\71.2J\end{array}$

In each case take the component of the force in the direction of the displacement :
$(85\cos 30^{\large\circ}N)(0.70m)=51.5$J
$(60\cos 45^{\large\circ}N) (0.70m)=29.7$J
Work is a scalar,so add the work done by each force to give
$51.5J+29.7J=81.2J$

format: plain


text:
# How weak/strong is this hand cipher? (updated) [closed]

In simple steps here the system:

first write text.

hand cipher

then divide into groups of three letters:

han dci phe r

then choose a key consisting of three letters, for example car and encrypt first three letters with the key han + car => kbf.

Then take the result kbf and encrypt the second three letters dci + kbf and the result to encrypt the next third three letters and so on until the end.

How strong is this system?

oct22 -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- oct26 (updated)

Hello again , I read a little and work a much and come with this, if the old system above is weak. What about that;

will divide the plain text into groubs of three letters, then choose three pass keys each three letters ex. ( are, you, mad )

ill sec ure cip her = plain text divide into sectors

are jdq cit xay ajo = use key no.1 'are'

jdq cit xay ajo iog = cipher text round.1

you isl lbf jcb kmq = use key no.2 'you'

isl lbf jcb kmq tbx = cipher text round.2

vtp hvv ryx clo wnm = cipher text round.3

ill secure cipher = plain text

vtp hvvryx clownm = cipher text

S0................? so embarrassing, really ill secure cipher! just leave the post open a couple of days, no more answers needed today

• Is it absolutely trivial too .? Oct 26 '12 at 15:17
• The attack from before still works, except you need to run it three times. Oct 26 '12 at 15:23
• I know that but its like 3DES now u need to test every key three times is that possible any way, and note that if the txt longer it could be 5,7 or ten letters in each sector. How can u try every key ? and any way I've something hidden will maximum the security. just for this any possible attack. Oct 26 '12 at 15:45
• The attack does not require any key guessing. The key is simply the ciphertext of the previous block. Any data after 3*n chars where n is the length of each key can be trivially decrypted without knowing any of the keys. Oct 26 '12 at 16:02
• How I miss that the only block that encrypted is the first, well... give me more time to think. Oct 26 '12 at 16:22

$c_0 = m_0 + k$

$c_i = m_i + c_{i-1}$ for $i>0$

It's absolutely trivial to break this, since the attacker knows the ciphertext, and thus knows both $c_i$ and $c_{i-1}$.

Decrypting the first group is hard. But to decrypt any group but the first, simply subtract the previous group from the current group $m_i = c_i - c_{i-1}$.

If you use the same key twice, you can also attack the first block by calculating $m^1_0-m^2_0 = c^1_0 - c^2_0$ which for typical texts allows some kind of frequency analysis on the messages. This is a generic attack stream ciphers which are used multiple times with the same key/IV.

Your updates scheme is secure for the first three blocks, and can be broken in exactly the same way as the old one. Just apply the above method three times giving you everything but the first 9 chars.

So once again it's secure if the key is used only once and the message is shorter than the key and insecure otherwise. You're still inferior to vignere with the same key-size, and that's already a pretty bad cipher.

• ok 1.if u dont know how many letters in each block will it be easy too, 2.And what if you dont know anything about the system if you had cipher text only , 3.And last thing what if I zigzag(cipher) the cipher text will that make it more secure. Oct 22 '12 at 10:08
• 1. An unknown block-size doesn't help much either. The attacker simply tries different block-sizes until one fits. 2. The standard assumption in cryptography is that the algorithm is public, and only the key is secret. It's hard to say how much security is gained by hiding the algorithm, so we simply assume that no security is gained. 3. Zig-zag and other obfuscations don't increase security. The attacker can simply undo the transformation, since its unkeyed. Oct 22 '12 at 10:11
• one more question is there any way I can make hard yet easy to apply cipher using a key or there is no way for that? cuz I almost giving up Oct 22 '12 at 10:23
• Oct 22 '12 at 10:24
• would you look at the updated Oct 26 '12 at 15:16

The system looks like Cipher Block Chaining mode with a secret initialization vector (your key) and the identical transformation as the (key-less) block cipher, using character-wise addition instead of bit-wise one.

The initialization vector in CBC is not the thing which provides secrecy (it is usually not secret), and as shown in CodesInChaos' answer, with a keyless cipher it is not secure at all.

For the first block, you essentially have the one-time pad, which means that it is secure, if and only if you only use each "key" once.

So if you make your block size larger than your message size, generate your key truly randomly (dice-throwing or similar) and never ever use a key twice, then your scheme is secure (as it is the one-time-pad). (But then we cut away the core of the scheme, which is the "use the ciphertext as the next key").

• I understand now and study for adding more steps to make it more secure, will post it when done to know if it still had any weakness. Oct 23 '12 at 18:57


text:
# Distinct degrees and homogeneous sets

Eoin Long, Laurentiu Ploscaru

Research output: Working paper/PreprintPreprint

## Abstract

In this paper we investigate the extremal relationship between two well-studied graph parameters: the order of the largest homogeneous set in a graph $G$ and the maximal number of distinct degrees appearing in an induced subgraph of $G$, denoted respectively by $\hom (G)$ and $f(G)$. Our main theorem improves estimates due to several earlier researchers and shows that if $G$ is an $n$-vertex graph with $\hom (G) \geq n^{1/2}$ then $f(G) \geq \big ( {n}/{\hom (G)} \big )^{1 - o(1)}$. The bound here is sharp up to the $o(1)$-term, and asymptotically solves a conjecture of Narayanan and Tomon. In particular, this implies that $\max \{ \hom (G), f(G) \} \geq n^{1/2 -o(1)}$ for any $n$-vertex graph $G$,which is also sharp. The above relationship between $\hom (G)$ and $f(G)$ breaks down in the regime where $\hom (G) < n^{1/2}$. Our second result provides a sharp bound for distinct degrees in biased random graphs, i.e. on $f\big (G(n,p) \big )$. We believe that the behaviour here determines the extremal relationship between $\hom (G)$ and $f(G)$ in this second regime. Our approach to lower bounding $f(G)$ proceeds via a translation into an (almost) equivalent probabilistic problem, and it can be shown to be effective for arbitrary graphs. It may be of independent interest.
Original language English Submitted - 12 Apr 2022

### Bibliographical note

38 pages, 2 figures

## Keywords

• math.CO
• 05D40 05C07 (Primary) 05D10 05C69 05C80 (Secondary)

## Fingerprint

Dive into the research topics of 'Distinct degrees and homogeneous sets'. Together they form a unique fingerprint.


text:
A finite distribution assigns probabilities to a finite collection of values. This includes categorical distributions.

dst_finite(y, probs, data, ...)

## Arguments

y

<data-masking> Outcomes to comprise the distribution. Should either evaluate to an (atomic) vector, or be a name in the specified data.

probs

<data-masking> Probabilities corresponding to the outcomes in y. Must not be negative, but must sum to 1 (unlike dst_empirical()). Should either evaluate to a vector, or be a name in the specified data.

data

Data frame containing the outcomes y and/or probabilities probs. Optional.

...

## Value

An object of class c("finite", "dst").

## Note

This distribution is called "finite" and not "discrete", because a discrete distribution could have an infinite amount of possible outcomes, as in the Poisson distribution.

dst_empirical()

## Examples

dst_finite(1:5, probs = rep(0.2, 5))
#> finite dst
#>
#>  probabilities :
#> # A tibble: 5 × 2
#>   location  size
#>      <int> <dbl>
#> 1        1   0.2
#> 2        2   0.2
#> 3        3   0.2
#> 4        4   0.2
#> 5        5   0.2


text:
## Does anyone want to collaborate?

General discussion about LÖVE, Lua, game development, puns, and unicorns.
Yozzaxia
Prole
Posts: 6
Joined: Fri Jul 16, 2021 3:36 pm

### Re: Does anyone want to collaborate?

togFox wrote: Thu Sep 09, 2021 7:34 am I suggested a 2D tiled map editor above. I'm thinking a grid would be presented for ppl to drag and drop tiles onto it. There can be an easy way to change the tiles too (images) so you could have desert/grass/hills etc or change the tiles to be house/road/sky-scraper or whatever.

The point of the tool is to export into a 2D table/list/array for use in any game/application you like. Is that close to what you're thinking?
Mostly what I thought of.

Thinking about it now, I'm not sure how useful a tilemap editor is, when Tiled and Ldtk editors already have incredible support for game frameworks/engines. To achieve the tile-swapping effect you describe, I would just change the tileset in code.
togFox
Party member
Posts: 392
Joined: Sat Jan 30, 2021 9:46 am

### Re: Does anyone want to collaborate?

I personally think those editors are hugely complex and overkill for a small project but you are right in that it's been done before.

What about a tool that lets you create, graphically, a binary tree? You can drag nodes onto the screen and connect child to parent. You can have multiple childs and attach data/attributes to nodes. You can then export for use in a project.

You can import a tree for inspection and debugging.

Lots of projects need trees for searching or AI etc.
Yozzaxia
Prole
Posts: 6
Joined: Fri Jul 16, 2021 3:36 pm

### Re: Does anyone want to collaborate?

togFox wrote: Thu Sep 09, 2021 9:23 pm I personally think those editors are hugely complex and overkill for a small project but you are right in that it's been done before.

What about a tool that lets you create, graphically, a binary tree? You can drag nodes onto the screen and connect child to parent. You can have multiple childs and attach data/attributes to nodes. You can then export for use in a project.

You can import a tree for inspection and debugging.

Lots of projects need trees for searching or AI etc.
Kind of like Godot's SceneTree or Blender's node editor? I've always seen advance editors do their own node systems, but I've never seen a standalone tool. That would be super useful!
togFox
Party member
Posts: 392
Joined: Sat Jan 30, 2021 9:46 am

### Re: Does anyone want to collaborate?

NOT this but something like this:

In other words - a graphic representation of a tree with a root, parents and children and data associated with each node.
togFox
Party member
Posts: 392
Joined: Sat Jan 30, 2021 9:46 am

### Re: Does anyone want to collaborate?

Are we also happy to experiment with crowd source.io? I'd like to see how that works with collaborating, structure, organisation and tracking individual effort.
togFox
Party member
Posts: 392
Joined: Sat Jan 30, 2021 9:46 am

### Re: Does anyone want to collaborate?

Need a project name. I'll start with "LÖVE Trees" and see if anyone can better that.

Does someone want to host a github? I don't know how the clone/pull/merge thingy works but I'm happy to try.

What about an official project goal to keep us on track?

"To create a graphic editor for developers to create, edit and visualise a tree structure. This tree structure will support parents, at least two children per parent (nodes) and at least one attribute associated with each node".

Good starting point? If we get really enthusiastic we can add libraries for search and sorting etc.
togFox
Party member
Posts: 392
Joined: Sat Jan 30, 2021 9:46 am

### Re: Does anyone want to collaborate?

Project is live. Some short same easy tasks have been suggested.

https://crowdsourcer.io/project/778
ddabrahim
Prole
Posts: 32
Joined: Mon May 17, 2021 8:05 pm
Contact:

### Re: Does anyone want to collaborate?

It essentially lets anyone join the project and then choose from a list of tasks that need to happen. Those tasks have 'contribution points' against one. The more tasks you do, especially high work-effort tasks, the more contribution points you earn. You can can drop in and drop out. Take a break. Choose what tasks interest you etc.
This is sounds pretty good and flexible, no commitment..etc However, on all project page I see roles like Programmer, 2D Artist..etc that I can apply for by sending a private message and no tasks. It is sounds more like a permanent position to fill than something I can simply jump in, get it done and then opt out.

Personally I would be more interested in contributing with tasks I choose from than filling a permanent position.
If I apply for a position, can I accept tasks after my application get accepted? I mean tasks listed on the website and not something we have a Zoom call to discuss.... I am not personally interested at the moment to join a team and have video calls about what to do. But I would be interested to contribute with tasks as described in the OP.
togFox
Party member
Posts: 392
Joined: Sat Jan 30, 2021 9:46 am

### Re: Does anyone want to collaborate?

I have set 9 vacancies for all roles. I can add more and even more role types. Keen to see if you can join more than one role.

When you are in a role you will see tasks you can take or leave. I can make more tasks if you have suggestions (and will need suggestions!)

Crowd source is very role / task focused but I don't care about the roles and nigyht collapse them all into a single role so everyone is in the same role and everyone sees all the tasks.

Zoom etc is not what this is about. I'm hoping we can collaborate by accepting a small 1 hour task and submitting that and then get on with your life until you're ready to take on another 1 hour task.
togFox
Party member
Posts: 392
Joined: Sat Jan 30, 2021 9:46 am

### Re: Does anyone want to collaborate?

PROGRAMMER now has 3 tasks. Join that first.

The other two roles are empty but keen to see if you can join them as well.

### Who is online

Users browsing this forum: No registered users and 29 guests


text:
# Weight, Index, and Congruence Subgroup of Classical Jacobi Theta Functions

On the very first page in the Introduction of Eichler and Zagier's text on Jacobi forms, they mention that the theta function

$$\Theta_{x_{0}}(\tau, z) = \sum_{x \in \mathbb{Z}^{N}} q^{Q(x)} y^{B(x, x_{0})}$$

is a holomorphic Jacobi form of weight $N/2$ and index $Q(x_{0})$ for some congruence subgroup of $SL_{2}(\mathbb{Z})$. In this formula, $Q: \mathbb{Z}^{N} \to \mathbb{Z}$ is a positive-definite quadratic form, $B(x, x_{0}) = \frac{1}{2}(Q(x + x_{0}) - Q(x) - Q(x_{0}))$ is the associated bilinear form, and $x_{0}$ is some lattice vector. Finally, $q = e^{2 \pi i \tau}$ and $y = e^{2 \pi i z}$, and these are conventions I'd like to stay with for my purposes.

I would like to apply this to the example of the four classical Jacobi theta functions, but I haven't found any conclusive references, and those references I have found, all seem to use different conventions. I believe the Jacobi theta functions are given for my definition above of $q$ and $y$ by

$$\vartheta_{1}(\tau, z) = - \sum_{n \in \mathbb{Z}} q^{\frac{1}{2}(n + \frac{1}{2})^{2}}(-y)^{n+\frac{1}{2}}$$

$$\vartheta_{2}(\tau, z) = \sum_{n \in \mathbb{Z}} q^{\frac{1}{2}(n + \frac{1}{2})^{2}} y^{n + \frac{1}{2}}$$

$$\vartheta_{3}(\tau, z) = \sum_{n \in \mathbb{Z}} q^{n^{2}/2} y^{n}$$

$$\vartheta_{4}(\tau, z) = \sum_{n \in \mathbb{Z}} q^{n^{2}/2} (-y)^{n}$$

My questions simply is, what is the weight and index of each of these theta functions, and with respect to what congruence subgroup? Is there a nice reference?

I'm struggling to even reconcile these functions with the general theta series above from Eichler and Zagier. For example, $n^{2}/2$ is not an integer-valued quadratic form, and even if it were, $n$ has the wrong coefficient to be the corresponding bilinear form. This leads me to worry that I'm using the wrong conventions for $q$ and $y$. Moreover, don't the factors of $(-1)^{n+1/2}$ and $(-1)^{n}$ in $\vartheta_{1}$ and $\vartheta_{4}$ respectively, prevent us from putting these in the general form of $\Theta_{x_{0}}(\tau, z)$ above?

• Why do you think that the Jacobi theta functions are Jacobi forms? Apr 18 '18 at 0:46
• Well because they look schematically like that general form in the first equation of my OP in the case of N=1. And in fact, just after Eichler and Zagier write that formula they mention that it's a generalization of what Jacobi studied for N=1. I figured they were referring to these theta functions. Apr 18 '18 at 1:05
• That is a good answer, but even for N=1, specializing a Jacobi form doesn't mean you can get exactly a Jacobi theta function. Some adjustments have to be made. Eichler and Zagier should have given explicit details of how, for example, $\vartheta_3$ is a specialization of a Jacobi form. Did they? Apr 18 '18 at 1:50
• Eichler and Zagier were certainly inspired by Jacobi theta functions, but the definition they give of Jacobi forms requires that the weight and index are integers. The Jacobi theta functions you give are weight one-half and so are not Jacobi forms by the definition given in Eichler-Zagier. Apr 18 '18 at 2:13
• Theta functions with half integral weight are best thought of as vector-valued forms which transform under the Weil representation of the metaplectic group which is a double cover of $SL(2, \mathbb{Z})$. Apr 18 '18 at 14:08

## 1 Answer

To expand on the comments. All four of the functions in the question are somehow versions of the classical theta function \begin{align*} \vartheta(\tau,z) &= \sum_{n \in \mathbb{Z}} (-1)^n q^{\frac{1}{2}(n+1/2)^2} \zeta^{n + 1/2} \\ &= \zeta^{1/2} q^{1/8} \prod_{n=1}^{\infty} (1 - q^n) (1 - q^n \zeta) (1 - q^{n-1} \zeta^{-1}), \end{align*} where $q = e^{2\pi i \tau}$ and $\zeta = e^{2\pi i z}$. This is indeed a Jacobi form, of half-integer weight and half-integer index and it is not covered explicitly by Eichler-Zagier. To make sense of it you should interpret it as having a multiplier system under $SL_2(\mathbb{Z})$ (or transforming under a character of the metaplectic group) as well as a character of the Heisenberg group.

If you let $\chi$ denote the character of Dedekind's eta function, so $$\eta((M,\sqrt{c \tau + d}) \cdot \tau) = \chi(M,\sqrt{c \tau + d}) \sqrt{c \tau + d} \eta(\tau)$$ then I believe the behavior of $\vartheta$ is $$\vartheta\Big( (M,\sqrt{c \tau + d}) \cdot \tau, \frac{z}{c \tau + d} \Big) = \chi(M,\sqrt{c \tau + d})^3 \sqrt{c \tau + d} e^{\pi i \frac{c z^2}{c \tau + d}} \vartheta(\tau,z)$$ and $$\vartheta(\tau,z+\lambda \tau + \mu) = (-1)^{\lambda +\mu} q^{-\lambda^2 / 2} \zeta^{-\lambda} \vartheta(\tau,z).$$


text:
cases:case027

# Infinite 35 Degree Swept Wing

This study was undertaken by the authors in order to investigate three-dimensional turbulent boundary layers. A two dimensional boundary layer initially developed on a test plate is subjected to adverse pressure gradient leading to three dimensional separation near the trailing edge of the plate.

Reynolds number $Re_{\theta} = 3290$.

#### Geometry

In order to obtain a test set-up suitable for checking three-dimensional boundary-layer calculation methods, the following requirements were made of the experimental model.

1. A flow similar to that which occurs on an infinite swept wing should be simulated.
2. The pressure should be constant along the most forward part of the model, leading to the development of a two-dimensional constant pressure boundary layer. This means the initial conditions for the three-dimensional boundary layer are then well know.
3. The measurements should be carried out in a boundary layer on a flat surface, to eliminate the effects of surface curvature on the structure of turbulence.

A sketch of the resulting model can be seen in figure 1.

Fig. 1: Experimental setup

The model consisted of a smooth flat plate with a pressure distribution induced on it by a large airfoil-like body mounted near the plate. Both the flat test plate and the pressure-inducing body had an angle of sweep of 35o.

In order to achieve the requirement of a constant pressure region along the forward part of the flat plate, a variable number of bars were placed at the channel outlet, allowing the correct amount of air to pass through the channel. Another feature of the model is the presence of guiding vanes mounted between the pressure-inducing body and the test plate, thus providing a reasonably close approximation to the infinite swept-wing condition.

The model was installed in the NLR 3m $\times$ 2m low speed wind tunnel.

#### Flow Conditions/Parameters

The flow through the model consisted of three regimes. The first of these is a constant pressure, two-dimensional boundary layer which develops on the furthest upstream part of the model, (right hand edge of the diagram. The flow in front of the diverging region should be parallel to obtain the constant pressure region along the most forward part of the plate. The parallel flow was achieved by the use of a blockage in the channel.

To achieve the three dimensional separation near the model trailing edge, the pressure is required to increase in the direction of flow. This is achieved by creating a diverging channel flow. The use of the pressure inducing body, a large airfoil like body, mounted near to the plate, also swept at 35o (see diagram) provides the pressure increase in the direction of flow. The pressure rise should not be too steep as this reduces the value of such an experiment as a check on semi-empirical calculation methods. To diminish the influence of the end walls on the flow, curved guiding vanes were introduced, the shape of which were determined experimentally by trial and error so that the pressure distribution on the test plate approximated the infinite swept-wing condition as accurately as possible.

To determine the pressure distribution on the test plate, the static pressure was measured at about 200 locations through holes in the surface of the plate. Such a large number of static pressure holes were required if the pressure gradients were to be deduced with sufficient accuracy.

The direction and magnitude of the velocity at the boundary layer edge were measured with a cylindrical yaw tube and a Pitot static tube, respectively. The velocity measurements inside the boundary layer were carried out with a rotatable hot wire probe. The probes at the measuring stations were protruded through the plate surface. At the reverse side of the plate, a traversing mechanism was mounted, by which the probe could be translated and rotated. The measurements extended from outside the boundary layer up to a distance of 0.15mm from the surface. The direction of the velocity was found by rotating the hot wire and subsequently determining the line of symmetry of the data.

The hot wire was calibrated at the boundary layer edge, where the magnitude of the velocity is known from Pitot static tube measurements. Calibrations were made before and after each traverse through the boundary layer. The traverse was repeated when the difference between both calibrations was larger than 1% in the mean velocity.

The magnitude of the wall shear stress was determined by wall Pitots of the Stanton type, with slit heights of 0.1mm and 0.2mm. Both wall Pitots were calibrated in a long pipe with fully developed flow. The wall Pitots were rotated to establish the symmetry line, and thus the approximate direction of the wall shear stress. More details concerning the experimental techniques employed are given in van den Berg & Elsenaar (1972).

#### Numerical Techniques

For a summary of the numerical techniques employed in order to provide a comparison with experimental results, refer to van den Berg et al (1975).

The available data consists of:

• Pressure coefficient distribution along the wall, and at a distance from it of $y=75$ mm.
• Mean velocity and Reynolds stress profiles across the boundary layer at a number of streamwise locations.

Sample plots of selected quantities are available.

f0251a-pt1.dat Contains mean velocity profiles at nine locations: $x= 0.52$, $0.62$, $0.72$, $0.82$, $0.92$, $1.02$, $1.12$, $1.22$, $1.32$ and $1.42$ m.

f0251a-pt2.dat Contains Reynolds stress profiles at eight locations: $x= 0.495$, $0.795$, $0.895$, $0.995$, $1.095$, $1.195$, $1.295$ and $1.395$ m.

f0251a-pt3.dat Contains the distribution of static pressure coefficient along the flat plate wall, and along a line at $y=75$ mm.

1. Van Den Berg, B., Elsenaar, A. Lindhout, J.P.F., Wesseling, P. (1975). Measurements in an incompressible three dimensional turbulent boundary layer, under infinite swept-wing conditions, and comparison with theory. J. Fluid Mech., Vol. 70, p. 127.
2. Van Den Berg, B., Elsenaar, A. (1972). NLR Tech. Rep. No. 72092 U.

Indexed data:

case027 (dbcase, flow_around_body)
case027
titleInfinite 35 degree swept wing
authorVan den Berg, Elsenaar, Lindhout, Wesseling
year1975
typeEXP
flow_tag3d, streamlined_body


text:
Braid group on 4 strands

Consider the braid group $B_4$ with generators

$a=\$, $b=\$ and $c=\$

Assume $$\alpha, \alpha' \in \langle a,c\rangle{\smallsetminus}(\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle c\rangle),$$ where $\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle c\rangle=\{\,c^k\cdot a^l\cdot c^m\mid k,l,m\in\mathbb{Z}\,\}$ and $$\beta, \beta' \in \langle b,c\rangle{\smallsetminus}(\langle c\rangle{\cdot}\langle b\rangle{\cdot}\langle c\rangle).$$

Is it true that $$\alpha{\cdot}\beta{\cdot}\alpha'{\cdot}\beta'\ne 1?$$

P.S. I am also interested in algorithms for solving this and similar problems. (There are some issues with the suggestion of Lee Mosher.)

• P.S. It is motivated by the second comment in here: mathoverflow.net/questions/139274 – Anton Petrunin Sep 26 '15 at 20:31
• What does the notation $<c>\cdot<a>\cdot<c>$ denote? – Ryan Budney Sep 28 '15 at 16:47
• It should be, but I would suggest not in, rather than setminus. In the current presentation, it is hard for me to tell if you mean set difference or quotient. What you call $\langle c\rangle{\cdot}\langle a\rangle{\cdot}\langle c\rangle$ would be more clearly called "Charlie", in my mind. Declaring the alphas to lie outside of Charlie would be more clear to me. Gerhard "Not Doing The Obvious Joke" Paseman, 2015.09.28 – Gerhard Paseman Sep 28 '15 at 18:38

This may be resolved by identifying $B_4$ with the mapping class group of the 4-punctured disk (fixing the boundary). I.e. $B_4\cong Mod(D^2-\{p_1,p_2,p_3,p_4\})$. I will consider isotopy classes of arcs in $D^2-\{p_1,p_2,p_3,p_4\}$ with endpoints on $\partial D^2$ and how they intersect each other, and the action of $B_4$ on the isotopy classes of arcs. In your pictures defining the generators of $B_4$, I'm assuming the points $p_i$ are numbered in order from top to bottom. Choices of isotopy classes of arcs may be made by choosing a complete hyperbolic metric on the punctured disk so that the boundary is totally geodesic, and making the arcs geodesics. Then the intersection number between pairs of arcs will be minimized by these representatives. Given an arc $w$ and $\sigma \in B_4$, I'll abuse notation $\sigma(w)$ to mean the arc made geodesic after applying $\sigma$ to the arc $w$.

Consider the arcs $x, y, z$:

Then we have that $stab(y)=\langle a,c\rangle$, $stab(x)=\langle b,c\rangle$, $stab(x\cup y)=\langle c\rangle$, which may be seen again by regarding the braid group as a mapping class group.

What you would like to know is the intersection of the double cosets $$U=( \langle a,c\rangle\cdot\langle b,c\rangle ) \cap ( \langle b,c\rangle\cdot\langle a,c\rangle ).$$

Suppose that we have $\gamma \in U$. Then one may see that $\gamma(x)\cap y =\emptyset$ and $\gamma(y)\cap x = \emptyset$. Moreover $\gamma(x)\cap \gamma(y)=\emptyset$ since $x\cap y=\emptyset$. Write $\gamma=\alpha \beta, \alpha\in \langle a,c\rangle, \beta \in \langle b,c\rangle$. Then $\beta(x)=x$ and $\alpha(y)=y$, so we see that $\gamma(x)\cap y =\alpha\beta(x)\cap \alpha(y) =\alpha(x)\cap \alpha(y)=\emptyset$, and similar for the other intersection.

Consider the subsurface $P\subset D^2$ obtained by cutting $D^2$ along $x \cup y$, and keeping the middle piece containing $p_2, p_3$. Then $P'=P-\{p_2,p_3\}$ is a twice-punctured disk, and there is only one isotopy class of essential arc with endpoints on $\partial P'$. Then each component of $\gamma(x)\cap P', \gamma(y)\cap P'$ must be isotopic to this arc (the case when one of the arcs is boundary parallel may be dealt with in a similar fashion). However, all but one component of $\gamma(x)\cap P'$ must have endpoints on $x\subset \partial P'$, and all but one component of $\gamma(y) \cap P'$ must have endpoints on $y\subset P'$. Up to composing $\gamma$ with an element of $\langle c\rangle = Mod(P')$, we must see a picture like this:

Thus replace $\gamma$ by $c^{-i}\gamma$ if necessary to get this configuration. We see now that $\gamma(x)\cap z=\emptyset$, $\gamma(y)\cap z=\emptyset$. Now focus on the subsurface $L\subset D^2$ obtained by cutting along $z$ and taking the left piece containing $p_1, p_2$. There is only one isotopy class of arc $x$ and $\gamma(x)$ in $L'=L-\{p_1,p_2\}$. But the endpoints of these arcs lie in $\partial L' -z$, and hence we see a picture like this:

Thus we see that up to composing with a power of $a$ (which generates $Mod(L')$), we may assume that $\gamma(x)=x$. Similarly, up to composing with a power of $b$, we may assume that $\gamma(y)=y$.

But now $stab(x \cup y) =\langle c\rangle$ in $B_4$. So we've shown that $\gamma = c^i a^j b^k c^l$.

Now we have $$\gamma = c^ia^jb^kc^l = \alpha \beta, \alpha\in\langle a,c\rangle, \beta\in \langle b,c\rangle.$$

Then $$\alpha^{-1} c^i a^j = \beta c^{-l} b^{-k} \in \langle a,c\rangle\cap \langle b,c\rangle = stab(x\cup y)=\langle c\rangle.$$

Thus $\alpha = c^i a^j c^{-m}, \beta =c^m b^k c^l$, as desired.

The fact that braid groups are automatic, proved by Thurston and recorded in the book "Word Processing in Groups" by Epstein, Cannon, Holt, Levy, Patterson, and Thurston, leads, I believe, to an algorithm for solving this problem. I don't know what the result of the algorithm will be --- yes or no --- and there is one step I'm a little unsure about. And, I admit, this is not the desired nontechnical solution to your question.

First, using the automatic structure, one writes down a regular expression representing a language $L$ in the alphabet $\mathcal{A} = \{a,A,b,B,c,C\}$ having exactly one word for each element of the braid group $B_4$.

Next, one has to work out regular expressions for the sublanguages of $L$ representing the following subsets of the braid group:

• $L\langle a,c\rangle$ representing each element of the subgroup $\langle a,c\rangle$
• $L\langle b,c\rangle$ representing each element of the subgroup $\langle b,c\rangle$
• $L_{cac}$ representing each element of the subset $\langle c \rangle \cdot \langle a \rangle \cdot \langle c \rangle$
• $L_{cbc}$ representing each element of the subset $\langle c \rangle \cdot \langle b \rangle \cdot \langle c \rangle$

The fact that these four sublanguages are regular needs to be demonstrated, however that should be straightforward, and the proof should yield the desired regular expressions.

Next, there is a theorem that if $L_1 \subset L_2$ are regular languages over the same alphabet then $L_2 - L_1$ is a regular language. The proof gives an algorithm for computing a regular expression for $L_2-L_1$ given regular expressions for $L_1$ and $L_2$. There exists standard software for doing these computations. One can therefore compute regular expressions for the two languages $L'_{ac} = L\langle a,c\rangle - L_{cac}$ and $L'_{bc} = L\langle b,c\rangle - L_{cbc}$.

Next, from the automatic structure one obtains a multiplier automaton. Using this, one has an automaton which inputs a pair of elements of the language $L$ and output the element of $L$ representing their product in the braid group. Here's the part I am a little unsure of: given regular expressions for the two sublanguages $L'_{ac}$ and $L'_{bc}$, one should be able to use the multiplier automaton to compute a regular expression for the sublanguage $L'' \subset L$ representing all group elements of the form $\alpha \cdot \beta \cdot \alpha' \cdot \beta'$ where $\alpha,\alpha' \in L'_{ac} = L\langle a,c\rangle - L_{cac}$ and $\beta,\beta' \in L'_{bc} = L\langle b,c\rangle - L_{cbc}$.

Finally your question is answered by using the regular expression for $L''$ and seeing whether or not it accepts the empty word (which represents the identity element of the braid group).

• Thank you, looks very interesting, but I have to learn few thing to understand it completely. So, you are saying that the problem can be solved on computer. If possible, could suggest where to start. (This can lead to an algorithm to check the more general conjecture for small $n$ --- maybe it will produce a counterexample and I will not have to thing about it any longer.) – Anton Petrunin Oct 2 '15 at 15:07
• I believe that @DerekHolt oversees the best automatic group software, perhaps he could weigh in. – Lee Mosher Oct 3 '15 at 15:15

I don't know the answer. However, this is what I would try.

$\alpha$ is constrained to have as a subword $ac^ma$, for nonzero $m$ and where I abuse notation and use $a$ to represent itself or its inverse. So get a red bead out to start threading. After doing the first part of $\alpha$, after the first series of a twists, slip the bead on to the a-c string (second from top) so that when you do the c twists, the bead indicates a demarcation between the a and the next (possibly first) set of c-twists. In particular, after you have done the c twists (and in particular after you have twisted $\alpha$, there is a red bead in the middle trapped between a twists and c twists.

It may be more useful to use more beads to track your progress, but I use the bead to make the following conjecture: After any other sequence of twists $\beta$, $\alpha'$, $\beta'$ of the claimed form, you can't slip the bead off the string without passing it through a twist. This may just be reexpressing your intuition. It may also suggest how to show that there will always be a twist after the bead that can't be undone in the desired fashion.

Gerhard "Proof By Exhaustive Infinite Twisting" Paseman, 2015.09.28


text: Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Lie Theory 27 (2017), No. 3, 745--769Copyright Heldermann Verlag 2017 Topological Frobenius Reciprocity for Representations of Nilpotent Groups and Motion Groups Robert J. Archbold Institute of Mathematics, University of Aberdeen, King's College, Aberdeen AB24 3UE, Scotland, England r.archbold@abdn.ac.uk Eberhard Kaniuth Institut für Mathematik, Universität Paderborn, 33095 Paderborn, Germany kaniuth@math.uni-paderborn.de [Abstract-pdf] Let $G$ be a locally compact group and $H$ a closed subgroup of $G$, and let $\pi$ and $\tau$ be irreducible representations of $G$ and $H$, respectively. If $G$ is compact then, by the classical Frobenius reciprocity theorem, $\pi$ is contained in the induced representation ${\rm ind}_H^G \tau$ if and only if $\pi|_H$ contains $\tau$. Topological Frobenius properties, which a general locally compact group may or may not satisfy, are obtained by replacing containment by weak containment of representations. We investigate the if' and the only if' assertions for nilpotent locally compact groups and for motion groups. Keywords: Locally compact group, nilpotent group, motion group, SIN-group, unitary representation, induced representation, weak containment, topological Frobenius reciprocity, tensor product. MSC: 22D10, 22D30 [ Fulltext-pdf  (340  KB)] for subscribers only.


text:
# pVACtools¶

pVACtools is a cancer immunotherapy tools suite consisting of the following tools:

pVACseq
A cancer immunotherapy pipeline for identifying and prioritizing neoantigens from a list of tumor mutations.
pVACfuse
A tool for detecting neoantigens resulting from gene fusions.
pVACvector
A tool designed to aid specifically in the construction of DNA-based cancer vaccines.
pVACviz
A browser-based user interface that assists users in launching, managing, reviewing, and visualizing the results of pVACtools processes.

## New in version 1.1.1¶

This is a hotfix release. It fixes the following issue(s):

• In version 1.1 we updated VAFs to be fractions, rather than percentages. A bug in this code change resulted in an error when using custom VAF cutoff values instead of the default. This has now been fixed.

## New in version 1.1¶

This version adds a host of new features to pVACtools:

• pVACseq is now able to parse VAF, depth, and expression information directly from the VCF. This makes the --additional-input-file-list option obsolete. The --additional-input-file-list option is now deprecated and will be removed in an upcoming release. For more information on how to annotate your VCF with readcount and expression information, see the Input File Preparation page.
• pVACseq is now able to handle proximal germline and somatic variants. In order to incorporate those into the epitope predictions, you will need to provide a phased variants VCF to your pVACseq run using the --phased-proximal-variants-vcf option. For more information on how to create this file, see the Input File Preparation page.
• We added support to pVACseq for filtering on transcript support levels. This requires the input VCF to be annotated with the TSL field by VEP. Be default, any transcripts with a TSL above 1 will be filtered out.
• The binding filter of pVACseq and pVACfuse can now be run with flexible, allele-specific binding-thresholds. This feature can be enabled using the --allele-specific-binding-thresholds flag. The thresholds used are taken from the IEDB recommendations.
• pVACseq now supports a --pass-only flag that will result in any VCF entries with a FILTER to be skipped. Using this flag, only VCF entries with a FILTER of PASS or . will be processed.
• We added support for the MHCflurry and MHCnuggets prediction algorithms. These can be used by listing MHCflurry, MHCnuggetsI (for MHC Class I alleles), and/or MHCnuggetsII (for MHC Class II alleles) as the prediction algorithms in your run commands.
• The default --tdna-vaf and --trna-vaf cutoff values have been updated from 0.4 to 0.25. This is the minimum VAF threshold that an epitope candidate must meet in order to pass the coverage filter.
• We now offer a graphical user interface, pVACviz, to run pVACseq as an alernative to using the command line. pVACviz, can also be used to plot and filter your pVACseq results.


text:
# Chapter 10 - Exponents and Radicals - 10.1 Radical Expressions and Functions - 10.1 Exercise Set: 109

$(-\infty,0)\cup(0,\infty)$

#### Work Step by Step

The denominator of the function, $g(x)=\dfrac{1}{x},$ cannot be zero. Hence, the domain is the interval $(-\infty,0)\cup(0,\infty) .$

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.


text:
## sexta-feira, 19 de agosto de 2011

Artigo aceito para publicação no MNRAS Letters.

http://arxiv.org/abs/1108.3745

In this Letter, we present a model connecting the cosmic star formation rate (CSFR) to the growth of supermassive black holes. Considering that the evolution of the massive black hole is dominated by accretion (Soltan's argument) and that the accretion process can be described by a probabilistic function directly regulated by the CSFR, we obtain the evolution of the black hole mass density. Then using the quasar luminosity function, we determine both the functional form of the radiative efficiency and the evolution of the quasar duty-cycle as functions of the redshift. We analyze four different CSFRs showing that the quasar duty-cycle, $\delta(z)$, peaks at $z\sim 8.5-11$ and so within the window associated with the reionization of the Universe. In particular, $\delta_{\rm max}\sim 0.09-0.22$ depending on the CSFR. The mean radiative efficiency, $\bar\eta(z)$, peaks at $z\sim 0.6-3.0$ with $\bar\eta_{\rm max}\sim 0.10-0.46$ depending on the specific CSFR used. Our results also show that is not necessary a supercritical Eddington accretion regime to produce the growth of the black hole seeds. The present scenario is consistent with the formation of black hole seeds $\sim 10^3{\rm M}_\odot$ at $z\sim 20$.


text:
Article | Open | Published:

# Self-decoupled radiofrequency coils for magnetic resonance imaging

## Abstract

Arrays of radiofrequency coils are widely used in magnetic resonance imaging to achieve high signal-to-noise ratios and flexible volume coverage, to accelerate scans using parallel reception, and to mitigate field non-uniformity using parallel transmission. However, conventional coil arrays require complex decoupling technologies to reduce electromagnetic coupling between coil elements, which would otherwise amplify noise and limit transmitted power. Here we report a novel self-decoupled RF coil design with a simple structure that requires only an intentional redistribution of electrical impedances around the length of the coil loop. We show that self-decoupled coils achieve high inter-coil isolation between adjacent and non-adjacent elements of loop arrays and mixed arrays of loops and dipoles. Self-decoupled coils are also robust to coil separation, making them attractive for size-adjustable and flexible coil arrays.

## Introduction

Magnetic resonance imaging (MRI) is a widely used non-invasive medical imaging modality that provides a variety of high-resolution images of the human body, with versatile soft-tissue and functional contrast1. The radiofrequency (RF) coils used in an MRI scanner to transmit RF energy into the body and receive signals from it play a critical role in determining image quality in terms of signal-to-noise ratio (SNR) and image uniformity, as well as other technical constraints on scanner performance. For three decades, arrays of receive coils have been used in MRI to achieve flexible coverage of large imaging volumes with high SNR2,3, and to accelerate scans using parallel imaging techniques4,5,6,7,8. Advances in RF array performance have usually come through increasing the number of coils in the array6,9,10. For example, receive arrays with large numbers of coils (≥32) have been fundamental to recent advances in the speed, sensitivity and resolution of neuroimaging8,11,12. 3 Tesla MRI scanners used for clinical applications are today equipped with 32 receiver channels as standard, and array coils are used exclusively for signal reception. Array coils are also used for RF transmission at ultra-high magnetic field strengths (7 Tesla and higher), as a means to achieve spatially homogeneous transmit RF fields in a patient-adaptive manner, while controlling tissue heating (quantified as global and local specific absorption rate (SAR)13,14,15,16,17). In that application, increasing the number of coils in the array improves power efficiency and image uniformity. Current 7 Tesla scanners are typically equipped with 8 transmit channels, and there have been reports of systems developed with as many as 3218 channels. Even for scanners with a smaller number of transmit channels, the benefits of many-coil transmit arrays can be realized using array-compression networks19.

The most widely used RF coil array element is a resonant loop coil comprising a circle or rectangle of copper wire or tape, which is uniformly segmented along its length by identical impedances (usually capacitors) to achieve a uniform current distribution and avoid antenna effects. When assembled into an array, these loops electromagnetically couple to each other due to magnetic flux linkage, which is referred to as loop-mode coupling. Minimizing this coupling is a central challenge in building RF coil arrays with many coil elements. When receiving RF signals from the body, strong coupling can lead to noise amplification and may limit the degree to which image acquisition can be accelerated using parallel imaging. When transmitting RF energy into the body, coupled power is absorbed by protection circuits and wasted. These and other power losses necessitate the use of very large RF amplifiers, with high cost and siting requirements. Transmit coil coupling also limits the ability to control the shape of the RF fields produced within the subject being scanned, because each coil’s transmit field profile becomes less distinct from those of its neighbours.

Here we propose a simple and practical RF coil design for arrays called the “self-decoupled” coil, which retains the loop structure of conventional coils and does not constrain array geometry or require additional circuitry. In this design, instead of uniformly segmenting the coil’s conductor with capacitors, a relatively large impedance (Xmode) is positioned opposite the coil’s feed port so that the coil behaves as a combination of a loop and a folded dipole superimposed on each other40, as illustrated in Fig. 1a. Proximal loop coils couple magnetically to each other, and proximal dipole antennas couple electrically to each other. Thus, coupling between a self-decoupled coil and a second coil is a mixture of magnetic (loop-mode) and electric (dipole-mode) coupling, as illustrated in Fig. 1b. If the magnetic (Km) and electric (Ke) coupling coefficients have the same magnitude and opposite signs, they will cancel and the coil will be self-decoupled from other coils. The self-decoupled concept can be used to decouple adjacent or non-adjacent elements, and applies to both transmit and receive arrays. It can also be used for arrays of mixed types such as loop plus dipole arrays. The unique structure of a self-decoupled coil makes its decoupling performance much more robust as a function of the distance between coil elements compared to conventional coils. With this feature, the self-decoupled design can improve the performance of size-adjustable and flexible coils used to match individual patients and improve patient comfort.

## Results

### Balancing magnetic and electric coupling

We simulated and built self-decoupled coils with a capacitor Cmode that was positioned opposite to the feed port (illustrated in Fig. 2a), to adjust the impedance of the bottom coil conductor and balance the magnetic and electric coupling. When changing Cmode, the coils’ resonance frequencies (298 MHz for 7 Tesla MRI) and input impedance were maintained by tuning the impedances Xarm (which were capacitors or inductors) and the capacitor Cmatch. Figure 2b shows an electromagnetic simulation model of a pair of 10 × 10 cm2 loops separated by 1 cm. Figure 2c plots the magnetic coupling coefficients (Km) and the electric coupling coefficients (Ke) as a function of Cmode, which were calculated using the methods of Hong et al.41 and Chu et al.42; further details are provided in Supplementary Table 1 and Equations 1-3. Note that the Ke definition used here represents electric coupling via free space, and unlike the definition used in Roemer et al.3, it is primarily a reactive coupling and does not include resistive coupling through the conductive sample. Figure 2c, d, e shows that setting Cmode to a large value such as 8 pF in both coils (corresponding to low impedance and thus a relatively large current in the bottom rung) causes the magnetic coupling to dominate, while setting Cmode to a small value such as 0.1 pF (corresponding to high impedance and thus a relatively small current in the bottom rung) causes the electric coupling to dominate. The currents in the two loops are of the same magnitude and opposite directions in magnetic- and electric-coupling-dominated cases, which validates that the two coupling coefficients have opposite signs. The negative sign of the magnetic coupling coefficients is consistent with Lenz’s law, which dictates that the current induced in the passive coil will generate magnetic flux to cancel out the active magnetic flux through it. As Cmode decreases, the two self-decoupled coils increasingly behave as a pair of co-linear folded dipoles that couple electrically, and the positive sign of the electric coupling coefficients is consistent with prior results on the mutual coupling between co-linear dipoles separated by less than 1/10 of a wavelength43. Figure 2f shows that the mutual coupling is minimized and the coils are self-decoupled when Cmode ~ 0.44 pF so that Ke = −Km. Figure 2g, h, i further shows current distributions and scattering (S−) parameter plots when the coils are self-decoupled.

### Constructed self-decoupled coils

Although the required Cmode can be obtained through a simulation analysis such as in Fig. 2, it can also be tuned on the benchtop based on S21 measurements. Specifically, to achieve the self-decoupled condition, Cmode should be adjusted so that the frequency with minimum S21 (denoted fm) is equal to the Larmor frequency (f0). In the Supplementary Information we show that $$f_{\mathrm{m}} = \sqrt {\left| {K_{\mathrm{e}}} \right|/\left| {K_{\mathrm{m}}} \right|} f_0$$, so when fm < f0, magnetic coupling dominates and Cmode should be decreased, and when fm > f0, electric coupling dominates and Cmode should be increased. This process is illustrated in Supplementary Figure 1 for two initial values of Cmode. In an array with more than two elements, the optimal Cmode for a two-coil array can be used as an initial value for all coils. As each new self-decoupled coil is added to the array, the Cmode and Xarm of its neighbours (in practice, mainly Xarm) may then be re-tuned.

A pair of constructed self-decoupled coils (loop–loop configuration) with the same sizes as the simulated coils are shown in Fig. 3a, b, along with a pair of conventional non-decoupled coils. Figure 3c, d plots the measured S-parameters of the two arrays. As expected, the non-decoupled coils are strongly coupled to each other, with a transmission coefficient S21 of −3.6 dB (i.e., power cross-talk of 44%). The strong coupling also makes the reflection coefficient (S11) plot asymmetrical and splits the resonance peaks. However, the coupling is only −29.3 dB (i.e., power cross-talk of 0.1%) between the two self-decoupled coils. Figure 3e, f shows that the RF transmit field strength (B1+) maps for each coil element match the ideal maps (where only one coil is present) for the self-decoupled coils, and that they have ~70 and ~47% greater power efficiency than the conventional non-decoupled coils. Note that the measured self-decoupled S21 is better than the simulated one in Fig. 2f, because the Cmode capacitors could be tuned continuously in the constructed coils but only discretely in the simulation. Simulated and measured multi-slice B1+ maps in multiple axial planes across the coil lengths are shown in Supplementary Figures 2 and 3, respectively. Although the current of the self-decoupled coils is non-uniform along the conductor length (stronger near the feed port and weaker near the Cmode capacitor), the slice-by-slice B1+ measurements show negligible decay along the z-direction. This can be understood by considering that B1+ = (Bx + iBy) / 244 is mainly produced by the current on the vertical conductor segments arms for these square coils, where the current distribution is relatively uniform. The maximum 10-gram local SARs of the ideal single coils and the self-decoupled coils were 1.35 W Kg-1 and 1.66 W Kg-1, respectively. As described later, the SAR of the self-decoupled coil can be reduced to 1.44 W Kg-1 by using multiple larger Cmode capacitors placed in series. Overall, the self-decoupled coils were nearly identical to ideal conventional coils in isolation, in terms of RF transmit field and SAR efficiency.

A pair of transformer-decoupled (also referred to as inductively decoupled) coils20 was also built for comparison, as shown in Supplementary Figure 4. Like the self-decoupled coils, the transformer-decoupled coils had much lower (−19.1 dB) coupling than the non-decoupled coils and similar B1+ maps as the ideal single conventional coils. However, transformer decoupling is extremely sensitive to coil separation and is limited to loop arrays, while (as will be shown later) self-decoupling is insensitive to changes in coil separation and can be applied to non-adjacent elements and mixed dipole and loop arrays.

### Mixed element-type self-decoupled arrays

The self-decoupled design was applied to a loop in a two-element loop–dipole array, shown in Fig. 4a, b. Although a loop coil is intrinsically decoupled from a radiative antenna such as a dipole or monopole when that antenna is placed across the loop’s center, in an array there is strong coupling between loops and adjacent dipoles/monopoles, especially when the loop coils are closely spaced or overlapped31,32. Figure 4c, d shows that, as in the loop–loop case, the currents induced in the loop coil have opposite directions for Cmode capacitors of very large versus very small values, which means the coupling coefficients Km and Ke have opposite signs. Thus, Cmode can again be tuned to balance the two kinds of coupling and minimize total coupling (Fig. 4e, f, g). The experimental results in Fig. 5c, d show that the isolation between the loop and dipole was improved from −5.1 to −14.8 dB using the self-decoupled design. Compared to the non-decoupled case, the dipole and self-decoupled loop also have ~39% and ~24% greater power efficiency, respectively (Fig. 5e, f).

Compared to a conventional loop coil, the matching and tuning of a dipole antenna are more sensitive to differences in subject positioning and size due to its higher sensitivity to the parasitic capacitance induced by a load45. A self-decoupled coil has a dipole-mode contribution, so a similar concern about robustness in different loading conditions arises. To investigate this, the robustness of self-decoupled coils to different coil separations and loading conditions were evaluated in benchtop experiments and compared to conventional loop coils, using the setup illustrated in Fig. 6a, b.

Figure 6c, d shows the measured S11 and normalized S21 of two conventional coils and two self-decoupled coils with different distances between them. S-parameter plots vs. frequency (center 298 MHz, bandwidth 60 MHz) are shown in Supplementary Figure 5. When the two coils were far away from each other, e.g., 7 cm, they were individually well-matched to 50 Ohms with S11 better than −30 dB (<0.1% power reflection). Although the conventional coils’ coupling was relatively stronger than the self-decoupled coils’ coupling (−17 vs. −29 dB), it was still acceptable with a power cross-talk of ~2%. As the coil-to-coil distance decreased, the coupling increased significantly and was as strong as −4.9 dB for 1 cm separation, leading to resonance peak splitting and an impedance mismatch in the conventional coils (Supplementary Figure 5). For the self-decoupled coils, however, good matching (<−22 dB) and decoupling performance (<−20 dB) were maintained across coil separations. This can be understood by considering that both the electric coupling and magnetic coupling vary similarly as the coil-to-coil distance varies, so they always cancel each other out.

Figure 6e, f shows the measured S11 and normalized S21 of two overlapped conventional coils and two self-decoupled coils with different coil-to-phantom distances; loading was higher for shorter distances. S-parameter plots vs. frequency (center 298 MHz, bandwidth 60 MHz) are shown in Supplementary Figure 6. Both the conventional overlapped coils and the self-decoupled coils became mismatched as the coil-to-phantom distance changed. The largest frequency shift and worst impedance matching were 3.6 MHz/−2.8 dB and 7.3 MHz/−3.2 dB for the conventional overlapped and self-decoupled coils, respectively. The normalized S21 of the conventional overlapped coils ranged from −17.5 to −4.1 dB, while that of self-decoupled coils ranged from −23.7 to −8.8 dB. Although they experienced larger frequency shifts with varied loading (7.3 vs. 3.6 MHz when the coil-to-phantom distance changed from 4.5 to 1.5 cm), the self-decoupled coils’ matching robustness was similar to that of the conventional overlapped coils, and their overall decoupling performance was better, especially for light loading.

### 1×3 and 2×2 self-decoupled arrays

Cross-talk between non-adjacent (primarily next-adjacent) elements is a challenging issue in decoupling RF arrays, and coupling may exist in multiple dimensions. To evaluate the potential benefits of self-decoupled arrays in terms of coupling between next-adjacent elements, a three-coil self-decoupled array was built (Fig. 7c). For comparison, we also built conventional loop arrays using next-adjacent coil distances corresponding to a gapped design and an overlapped design (Fig. 7a, b). Only two elements were built in the conventional loop arrays since the aim was to evaluate next-adjacent coupling. Figure 7d, e, f plots the arrays’ S-parameters. Even without the influence of the middle coil, the couplings between the next-adjacent coils in the gapped and overlapped designs were −15.3 and −11.1 dB, respectively. The coupling was relatively stronger in the overlapped design because of the slightly larger coil dimensions and smaller coil separations. For the 1 × 3 self-decoupled array the mutual coupling between next-adjacent elements was −19.4 dB, which was −4.1 and −8.3 dB better than the gapped and overlapped designs, respectively.

The self-decoupled design was further applied to a multi-row array in which three kinds of coupling exist: coupling between neighbouring elements of the same row, coupling between neighbouring elements of different rows and coupling between diagonal elements. A 2 × 2 array was built on a cylindrical former (25 cm diameter), as shown in Fig. 8a. To enable simultaneous decoupling of elements in the same row and different rows (Fig. 8b–d), the Cmode capacitors were positioned in the bottom corners of each coil, and the coils were fed in the opposite top corner. Coils in the same row were partly face-to-face, while elements in different rows were not face-to-face. Therefore, magnetic coupling was slightly weaker between elements in different rows, and the optimal Cmode value to decouple Coil 1 from Coil 3 was slightly larger (0.48 pF) than the optimal Cmode to decouple Coil 1 from Coil 2 (0.37 pF). In the constructed array, the Cmode values were chosen to obtain acceptable decoupling between elements in both the same row and in different rows, and Coil 1’s balanced Cmode value was 0.44 pF. Diagonal coupling was not considered when tuning Cmode. With this construction, Fig. 8f, g, h show that the coupling between all elements was better than −15 dB.

### Reducing SAR using multiple Cmode capacitors in series and in vivo MR images

Up to this point, only one Cmode capacitor was used in self-decoupled coils for simplicity. But this leads to a large electric field across the capacitor’s dielectric and possibly a high local SAR in the subject. Figure 9 shows simulated electric field amplitudes around 10 × 10 cm2 conventional and self-decoupled coils with the same input power. Near the Cmode capacitor, the electric field of the self-decoupled coil is about 4 times higher than for the conventional coil (7.4 × 104 vs. 1.8 × 104 V m−1). This could lead to local temperature rises in the coil housing, but can be addressed by breaking up the Cmode capacitor into a capacitance-equivalent set of larger series capacitors40,46, or using an equivalent transmission line. With five Cmode capacitors, the electric field amplitude was reduced by 72% (2.1 × 104 vs. 7.4 × 104 V m−1). This also reduced the 10-gram local SAR by 15%, from 1.66 W Kg−1 to 1.44 W Kg−1. Therefore, multiple Cmode capacitors were used in the self-decoupled coils constructed for in vivo human imaging, described below.

Figure 10a, b shows a 2-coil self-decoupled array for human imaging (top view and bottom view). Cable-trap circuits and RF cables were used but are not shown. The coils were initially tuned, matched and decoupled when loaded with a human head, with S11 and S21 both better than -30 dB (Fig. 10c). We also note that when the coil was loaded with a human calf, the matching and decoupling remained acceptable without retuning (S11 and S21 both better than −18 dB). Figure 10d, e shows measured axial B1+ maps and low-flip-angle gradient recalled echo (GRE) images of the individual coils and both coils driven together. When imaging using both coils, the array was operated in quadrature mode. Each coil exhibited a distinct B1+ map (linear scale) and GRE image, indicating good decoupling performance, which is also consistent with the S-parameter results in Fig. 10c and the phantom results in Fig. 3. Figure 10f shows the turbo spin echo image, which contains no artifacts such as signal voids due to transmit or receive field nulls that could have resulted from coupling between the coils.

## Discussion

We have proposed and validated a novel but simple and effective approach to decoupling loop coils in RF coil arrays. The method requires only intentional tuning of each coils’ capacitance distribution to balance magnetic coupling and electric coupling so that they cancel. It requires no other decoupling treatments or specific coil geometry (overlapping or orthogonality). Even for two very closely spaced loops whose coupling was as strong as −3.6 dB when conventional equally distributed capacitors were used, we showed that excellent isolation could be achieved (−29.3 dB) using self-decoupling while maintaining nearly ideal RF transmit field distributions and power efficiency. We demonstrated that self-decoupled coils are similarly sensitive to loading but have improved robustness to element separation compared to conventional loop coils. Because the electric coupling and magnetic coupling of a self-decoupled coil vary similarly as the coil-to-coil distance varies, they always cancel each other out, which could make self-decoupled coils advantageous in flexible, screen-printed, and size-adjustable arrays47,48,49. In flexible coil designs, the single Cmode could instead be replaced by a transmission line to maintain coil flexibility, which may also lower local SAR. Self-decoupling was also effective in decoupling loops from other elements in a mixed loop–dipole array where it is challenging to decouple adjacent (non-overlapping) loop and dipole elements. Another concern for mixed arrays is coupling between adjacent dipole/monopole elements. However, several methods have been described to reduce coupling between closely spaced dipoles/monopoles, such as passive antennas50,51 and metamaterials52. This report focused on the characterization and validation of self-decoupling itself, and further work is needed to investigate combinations with these methods. Self-decoupling was also effective in decoupling next-adjacent loops and loops in multi-row loop arrays, and could be a key technique in the development of many-element parallel transmission array coils. We expect that decoupling would be improved in a flat self-decoupled array compared to the cylindrical array we built, because the optimal Cmode values for decoupling elements in the same row and different rows would be almost the same due to more similar coupling geometries.

In this report we have emphasized the broad applicability and simplicity of self-decoupled designs by building arrays that used only self-decoupling. Although two-dimensional (2-D) and cylindrical arrays are suitable for most applications, close-fitting receive-only three-dimensional (3-D) helmet arrays with 32 or more elements may be preferred to capture maximum SNR in brain imaging. For these arrays, the small size, close spacing and varying neighbour geometries may limit the degree of self-decoupling that can be achieved since coils must be decoupled from several neighbours simultaneously. Further optimization of self-decoupled coils may be needed to address this more complex coupling problem. Alternatively, self-decoupling could be paired with conventional decoupling methods such as overlapping, low-impedance preamplifiers, shielding, and transformers, to address coupling in arrays with complex geometries. In particular, for 3-D receive arrays self-decoupling could combined with preamplifier decoupling to achieve high isolation between all neighbours.

Self-decoupled loop coils are built with unequal capacitance and inductance distributions to intentionally generate electric coupling that cancels magnetic coupling. They are distinct from “loopole” coils40, in which vertical conductors are built with unequal capacitance distributions to generate unequal current distributions that approximate transmit- or receive-optimal current patterns at 7 Tesla. However, as shown in Supplementary Figure 7, if a self-decoupled coil is rotated by 90 degrees or fed at its corner, the currents on its two vertical conductors become different so its B1 field becomes more similar to that of a “loopole” coil40,46, which may be helpful to improve transmit or receive efficiency. It should be noted that this may increase coupling with dipole or monopole antennas in the same array. More broadly, self-decoupled coils should enable greater flexibility in RF array design, since they alleviate constraints on element overlap and spacing. As an example, Supplementary Figure 8 shows how decoupling can be made independent of overlapping area using the self-decoupled approach, which provides a new degree of freedom for RF array design to optimize parallel imaging performance and imaging coverage.

Although this work demonstrated the self-decoupled coil concept for 10 × 10 cm2 coils at 7 Tesla, it can be applied to other coil sizes and field strengths so long as the magnetic and electric field coupling can be balanced to cancel each other. Like a conventional loop, the resonant frequency of a self-decoupled coil is dominated by its inductance (self-inductance and Xarm) and capacitance. At lower frequencies and for smaller coils, the Xarm impedance needed to tune a self-decoupled coil’s resonant frequency may be an inductor. In particular, if the Cmode capacitor required for self-decoupling remained fixed as the coil size decreased, Xarm may have a large inductance which could lead to significant power loss. To give some insight on this matter, we simulated a series of square self-decoupled coils with lengths/widths from 10 cm down to 5 cm in 1 cm steps. We found that the Cmode increased approximately linearly as the coil size decreased (Supplementary Table 2), and that, for a 5 × 5 cm2 self-decoupled coil, the Cmode was around 0.8 pF so the required total inductance Xarm remained a relatively small 80 nH, which alleviates this concern. We also built a pair of 5 × 5 cm2 self-decoupled coils and a pair of 5 × 5 cm2 conventional coils. Similar to the 10 × 10 cm2 coil, the smaller self-decoupled coil had better decoupling and efficiency than the non-decoupled coils (Supplementary Figure 9). 5 × 5 cm2 is smaller than the 9.5 and 6.5 cm coils that have been used in state-of-the-art 32- and 64-coil head arrays, respectively8. Supplementary Figure 10 shows a study of self-decoupled coils at 3 Tesla and 1.5 Tesla, where it was found that the ideal B1- fields can be maintained at 3 Tesla for both 10 × 10 cm2 and 20 × 20 cm2 coils, but there was an SNR decrease (~20%) for the 10 × 10 cm2 coil at 1.5 Tesla due to inductor loss. This loss could be avoided using dielectric materials with high permittivity or meander lines, at the cost of increased manufacturing complexity. The radiation losses of the self-decoupled coils were similar to that of conventional coils, even though they behave more like antennas than conventional coils. This means that most of the coils’ power was directed into the high-permittivity and conductivity samples, which is consistent with results using straight dipole antennas53.

## Methods

### Numerical simulation setup

Electromagnetic (EM) simulations were performed using commercially available software (ANSYS Electromagnetics, Canonsburg, PA, USA). As in the experiments, the coils were tuned to 298 MHz (proton Larmor frequency at 7 Tesla) and matched to 50 Ohms (the characteristic impedance of MRI scanner RF chains). Values of all lumped elements were obtained using the EM and RF circuit co-simulation method54. During RF circuit optimization, tuning and matching performance were optimized by varying Xarm and Xmatch with a built-in module in the RF circuit software (ANSYS Designer), while the decoupling performance was optimized by varying Cmode manually. All conductors and lumped elements (capacitors and inductors) were set to be ideal without ohmic loss. The transmit RF fields (B1+) were extracted from the simulations using the relationship: $$B_1^ + = (B_{\mathrm{x}} + {\mathrm{i}}B_{\mathrm{y}})/2$$44, where Bx and By are the x- and y- components of the magnetic field. Local SAR averaged across 10 grams was calculated using the built-in Fields Calculator module in ANSYS HFSS.

For the loop–loop configuration, the feed port was positioned at one end with one Cmode positioned at the opposite end. Two loops (each sized 10 × 10 cm2) were mounted on a 25-cm-diameter former and positioned with a 1 cm gap. A cylindrical phantom (conductivity б = 0.6 S m-1 and relative permittivity ξr = 78) with 15 cm diameter and 20 cm length was placed 4.5 cm below the loops. For the loop–dipole configuration, the loop had one Cmode positioned at each end. The dipole had dimensions 23 × 0.75 cm2 and loop had dimensions 20 × 9.5 cm2; the two were spaced 4 cm apart. A tank phantom with dimensions 35 × 30 × 20 cm3 was placed 4.5 cm below the pair (б = 0.7 S m-1 and ξr = 55). The dipole was shortened using two inductors and was matched using a parallel capacitor55.

### Coil construction and bench test setup

The fabricated self-decoupled coils in both loop–loop and loop–dipole configurations had the same geometry and circuit as those in the simulation. The conventional coils used in the comparisons had the same sizes but equally distributed capacitors along their conductors. All coils were constructed with 7.5-mm-wide copper (3 M, Minneapolis, MN). Fixed capacitors (Passive Plus, 111 C Series, Huntington, NY) were used as distributed capacitors and trimmer capacitors (Johanson Manufacturing, 52 H Series, Boonton, NJ) were used for tuning, matching and decoupling. Handmade inductors using American wire gauge (AWG)-18 copper wires were used for Xarm impedances when needed. In the simulations, six equally distributed Xarm impedances were used for generality. In the constructed coils, only one Xarm was used since the total inductance was only around 35 nH. To avoid confusion with the transformer decoupling method which uses a pair of adjacent windings, the inductors in the two coils were placed far away from each other. Bench tests were performed using a calibrated Agilent 5071 C ENA network analyzer. For the loop–loop configuration, a 3-liter cylindrical phantom (15-cm-diameter, with 0.26 g L−1 NaCl and 0.125 g L−1 NiSO4 × 6H2O) was used for loading. For the loop–dipole configuration, a 20-liter tank phantom (35 × 30 × 20 cm3, with 0.26 g L−1 NaCl and 0.125 g L−1 NiSO4 × 6H2O) was used for loading.

### Phantom transmit (B1+) maps and receive sensitivity (B1-) maps

Phantom B1+ maps were acquired with the constructed loop–loop and dipole-loop arrays on a 7 Tesla Philips Achieva whole-body scanner (Philips Healthcare, Best, Netherlands). In all MR experiments, each coil element was measured individually with the other one terminated with 50 Ohms. B1+ maps were measured using the DREAM method with the same input power56. The DREAM sequence parameters for the loop–loop configuration (10 × 10 cm2) were: field of view (FOV) = 180 × 180 mm2, in-plane resolution = 2 × 2 mm2, slice thickness = 5 mm, TR = 1000 ms. For the multi-slice B1+ of the 10 × 10 cm2 loop coils, the in-plane resolution was set to 1 × 1 mm2. The parameters for the loop–dipole configuration were: FOV = 350 × 200 mm2, in-plane resolution = 1.8 × 1.8 mm2, slice thickness = 5 mm, TR = 1000 ms. To calculate the receive sensitivity (B1-) map of the 5 × 5 cm2 loop coils, low-flip-angle gradient-recalled echo (GRE) images were acquired using the following parameters: FOV = 180 × 180 mm2, nominal flip angle = 10 degree, in-plane resolution = 3 × 3 mm2, slice thickness = 5 mm, TR/TE = 1000/10 ms, bandwidth = 5020 Hz. The B1- was calculated as Signal Intensity/std(noise)/|B1+|, where std(noise) is the standard deviation of noise map acquired with no RF excitation. Then the B1- was normalized to the maximum B1- of the ideal single coils.

### Coil robustness analysis

To evaluate self-decoupled coils’ sensitivity to coil separation, two self-decoupled coils were mounted on a flat acrylic board (thickness 1.2 mm) which had a lift-off of ~4 cm above the large tank phantom. The self-decoupled coils were initially tuned, matched and decoupled when the coils’ distance was 1 cm. Then the distance between the coils (Dcoil) was varied from 1 to 7 cm in steps of 1 cm. No re-tuning or re-matching was applied as Dcoil changed. Two conventional coils with the same Dcoil were measured for comparison. The same bench test was conducted for two conventional coils for comparison. During the test, care was taken to make sure the loading condition did not change.

To evaluate self-decoupled coils’ sensitivity to loading, two self-decoupled coils were mounted on a 25-cm-diameter cylindrical former, and were tuned and matched with the 15-cm-diameter cylindrical phantom positioned 4.5 cm below the coils. The coil-to-phantom distance was then increased to 7.5 cm (lighter loading) and decreased to 1.5 cm (heavier loading) in steps of 1 cm. The same bench test was conducted for two overlapped conventional coils for comparison. The overlapped area was optimized when Dphantom was 4.5 cm, and it was not readjusted when Dphantom changed. In both cases, S11 and S22 were measured to evaluate the matching performance. Note that coils may mismatch when their separation or loading conditions change, so normalized S21 (defined as $$\left| {S_{21}} \right|/\left( {\sqrt {1 - \left| {S_{11}} \right|^2} } \right)$$) was used to evaluate the decoupling performance.

### 1×3 array and 2×2 array

The three-coil self-decoupled array was constructed by adding a third coil to the previously described two-coil array. Adding the third coil had little impact on the decoupling performance of the original two coils but slightly changed the resonance frequency of the middle coil, so the Cmode capacitor and Xarm inductors in the middle coil were retuned manually to achieve isolation and bring its resonant frequency to 298 MHz. We also constructed conventional loop arrays using gapped and overlapping designs as comparisons. Since we focused on evaluating the decoupling between next adjacent elements, only two elements were built in conventional loop arrays. For the gapped array, the two loop elements had the same size (10 × 10 cm2) and the same separation (12 cm) as the self-decoupled coils. For the same coverage, the loop elements using an overlapping design had slightly larger size (12 × 10 cm2) and closer distance (8 cm). Shielded trap circuits were used for all coils to remove the common current along the cables.

The 2 × 2 array was constructed by adding another row to the two-coil array. As explained above, a suitable dipole mode is needed to cancel out the magnetic coupling and realize self-decoupling. Therefore, each coil in the 2 × 2 array was fed at one corner and the Cmode was positioned at the opposite corner to reduce coupling from different rows as well as that from the same row. Due to limited space for the 2 × 2 array, bazooka/sleeve baluns rather than cable traps were used to suppress common-mode current.

### In vivo imaging

The 2-coil self-decoupled array (dimensions 10 × 10 cm2) was mounted on an open cylindrical acyclic former (16-cm diameter). To avoid high electric fields near small Cmode capacitors, five larger capacitors (four fixed 2.7 pF and one variable ~2.5 pF) were connected in series. The coils were initially tuned, matched and decoupled when loaded with a human head. A healthy male volunteer was imaged using the array on the 7 Tesla whole body scanner. B1+ maps and MR images were acquired with individual coils and both coils, respectively. When imaging using both coils, the array was operated in quadrature mode. The imaging sequence was a low-flip-angle GRE sequence with parameters: FOV = 220 × 220 mm2, nominal flip angle = 10 degrees, TR/TE = 1000/1.98 ms, in-plane resolution = 1 × 1 mm2, slice thickness = 5 mm, bandwidth = 887 Hz pixel-1, number of averages = 1. Turbo spin echo (TSE) images were also acquired with the parameters: FOV = 220 × 178 mm2, refocusing flip angle/flip angle = 120/90 degrees, TR/TE = 3000/76 ms, in-plane resolution = 0.5 × 0.5 mm2, slice thickness = 5 mm, bandwidth = 243 Hz pixel-1, TSE factor = 15, number of averages = 1. To achieve the same B1+ in this area, the required input power of the 2-coil self-decoupled array was 73% lower than for a commercial volume transmit array (28 cm diameter, Nova Medical Inc., Wilmington, MA). The human experimental procedures were approved by the local institutional review board of Vanderbilt University, and informed written consent was obtained from the participant.

### Data Availability

The data that support the findings of this study are available from the corresponding author upon request.

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## Acknowledgements

This work was supported by NIH R01 EB016695 and NIH R21 EB 018521. We thank Haoqin Zhu (Quality Electrodynamics, Mayfield Village, OH, USA) for helpful discussion on multiple Cmode capacitors, Saikat Sengupta (Vanderbilt University) for the assistance in coil safety tests and Ping Wang (Vanderbilt University) for assistance with the human scans.

## Author information

### Affiliations

1. #### Vanderbilt University Institute of Imaging Science, Nashville, TN, 37232, USA

• Xinqiang Yan
• , John C. Gore
•  & William A. Grissom
2. #### Department of Radiology and Radiological Sciences, Vanderbilt University, Nashville, TN, 37232, USA

• Xinqiang Yan
• , John C. Gore
•  & William A. Grissom
3. #### Department of Biomedical Engineering, Vanderbilt University, Nashville, TN, 37232, USA

• John C. Gore
•  & William A. Grissom
4. #### Department of Electrical Engineering, Vanderbilt University, Nashville, TN, USA

• William A. Grissom

### Contributions

X.Y. conceptualized the work, performed the coil simulations and carried out the fabrication. Experiments were designed by X.Y., J.G. and W.G. All authors discussed the results and contributed to writing the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Xinqiang Yan.