text:
407 493 6601

# Algebra 2 - Adding and Subtracting Rational Expressions

## Introduction

• Addition and subtraction of rational expressions can only be performed if they have common denominators.
• If the two rational expressions to be added don't have a common denominator, we need to modify the rational expressions so that equivalent rational expressions have the same denominators.

## Adding & Subtracting Rational Expressions

For simplicity's sake, we classify rational expressions' addition and subtraction into two cases.

1. When rational expressions have a common (same) denominator:

• Keep the denominator as it is, and add or subtract the numerators.
•  Simplify (cancel out common factors) the remaining expression if possible.

2. When rational expressions have different denominators:

• Find the LCM (least common multiple) of all the denominators.
• Modify the rational expression into an equivalent rational expression with the same denominator (as we do while adding or subtracting fractions with unlike denominators).
• Simplify the resulting rational expression if possible.

## Solved Examples

Example 1: Add $\frac{2x}{3-5x}$ and $\frac{9}{3-5x}$.

Solution: Here, we see that the denominators of the rational expressions to be added are the same. Hence, we can directly add the numerator terms.

Thus, $\frac{2x}{3-5x}+$$\frac{9}{3-5x}$$=\frac{2x+9}{3-5x}$

Example 2: Subtract $\frac{7x+1}{{x}^{2}-4}$ from $\frac{1-8x}{{x}^{2}-4}$.

Solution: In this problem, the denominators of the rational expressions to be subtracted are the same. Hence, we can directly perform subtraction operations.

Thus, $\frac{1-8x}{{x}^{2}-4}-\frac{7x+1}{{x}^{2}-4}=$$\frac{\left(1-8x\right)-\left(7x+1\right)}{{x}^{2}-4}=$$\frac{1-8x-7x-1}{{x}^{2}-4}=$$\frac{-15x}{{x}^{2}-4}$

Example 3: Simplify $\frac{12}{{x}^{2}-9}+\frac{2}{x+3}$.

Solution: $\frac{12}{{x}^{2}-9}+\frac{2}{x+3}$$=\frac{12}{\left(x+3\right)\left(x-3\right)}+\frac{2}{x+3}$$=\frac{12+2·\left(x-3\right)}{\left(x+3\right)\left(x-3\right)}$$=\frac{12+2x-6}{\left(x+3\right)\left(x-3\right)}$$=\frac{2x+6}{\left(x+3\right)\left(x-3\right)}$$=\frac{2\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}$$=\frac{2}{\left(x-3\right)}$

Example 4: Simplify $\frac{x-1}{x}-\frac{y}{y-1}$.

Solution: $\frac{x-1}{x}-\frac{y}{y-1}$

Example 5: Simplify $\frac{5x}{\left(x+3\right)}+\frac{x+1}{{x}^{2}+2x-3}-\frac{x}{\left(x-1\right)}$

Solution: $\frac{5x}{\left(x+3\right)}+\frac{x+1}{{x}^{2}+2x-3}-\frac{x}{\left(x-1\right)}$$=\frac{5x}{\left(x+3\right)}+\frac{x+1}{\left(x+3\right)\left(x-1\right)}-\frac{x}{\left(x-1\right)}$

$=\frac{5x·\left(x-1\right)+\left(x+1\right)-x·\left(x+3\right)}{\left(x+3\right)\left(x-1\right)}$

$=\frac{5{x}^{2}-5x+x+1-{x}^{2}-3x}{\left(x+3\right)\left(x-1\right)}$$=\frac{4{x}^{2}-7x+1}{\left(x+3\right)\left(x-1\right)}$

## Cheat Sheet

• To add or subtract rational expressions, they must have common denominators.
• If the denominators of rational expressions to be added are different, then we must first express them into an equivalent rational expression with common denominators.

## Blunder Areas

• Subtracting $\frac{{f}_{1}\left(x\right)}{{g}_{1}\left(x\right)}$ from $\frac{{f}_{2}\left(x\right)}{{g}_{2}\left(x\right)}$ means $\frac{{f}_{2}\left(x\right)}{{g}_{2}\left(x\right)}-\frac{{f}_{1}\left(x\right)}{{g}_{1}\left(x\right)}$.


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# How to determine if a number is odd or even in JavaScript?

JavascriptWeb DevelopmentFront End Technology

Use the modulus % operator to determine if a number if odd or even in JavaScript.

## Example

You can try to run the following code to check if a number is odd or even −

Live Demo

<!DOCTYPE html>
<html>
<body>
<script>
var num = 5;
document.write("Number = "+num+"<br>");
if(num % 2 == 0) {
document.write('Number is even!');
} else {
document.write('Number is odd!');
}
</script>
</body>
</html>

## Output

Number = 5
Number is odd!
Published on 03-Feb-2018 17:11:28

format: plain


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Semantics for minimal logic

Minimal logic is a fragment of intuitionistic logic that rejects not only the classical law of excluded middle (as intuitionistic logic does), but also the principle of explosion (ex falso quodlibet). Essentially, from proof-theoretical viewpoint, this means that in minimal logic the bottom $\bot$ is considered as a propositional variable, without any special inference rule involving it.

Question: Is there a semantics $\mathcal{S}$ for minimal logic such that a completeness theorem holds? By completeness theorem I mean a statement of the form (for both propositional and first-order languages):

For every formula $A$, $A$ is provable in minimal logic if and only if $A$ is valid in all $\mathcal{S}$-structures.

I guess such a semantics could be a generalization of intuitionistic Kripke models.

I would like also to have some references about completeness theorem in (propositional and/or first-order) minimal logic.

• I find it boggling that one would throw in a gratuitous propositional constant without any axioms relating to it. I find it especially boggling that one would name the constant $\bot$ when it is not a bottom element.
– user14972
Jan 28, 2018 at 22:52
• Can I ask the reason of the recent downvote? Apr 5, 2018 at 19:04


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Finite-dimensional subalgebras of $C^\star$-algebras

Let $A$ be a unital $C^\star$-algebra and let $a_1,\dots,a_n$ be a finite list of normal elements in $A$ which (together with their adjoints) generate a norm-dense $\star$-subalgebra $B \subset A$. Clearly, if $A$ is finite-dimensional, then every element in $A$ (and hence $B$) has finite spectrum. I am asking for the converse.

Question: Assume that each element of $B$ has finite spectrum. Is it true that $A$ has to be finite-dimensional?

The existence of finitely-generated infinite torsion groups shows that this might be a highly non-trivial problem. In this case one would consider the reduced group $C^\star$-algebra and note that all monomials in the generators of the group and their inverses (which are equal to the adjoints of the generators) would have finite spectrum. However, the generated algebra would still be infinite-dimensional. I do not know of any simpler way to come up with such an example. In this case it is conceivable that the random-walk operator associated with the generating set (which is an element in the real group ring) has infinite spectrum, even though I did not prove this.

Maybe there is also need to consider the spectra of elements in matrices over $B$ (which of course follow to be finite if $A$ is finite-dimensional.) In view of this, I am not only asking for an answer to the question but also for the right question (or a better one) if the answer to the original question is negative.

A stronger assumption would be to assume that $A$ itself consists only of elements with finite spectrum. This case seems much easier to approach and the answer seems to be positive. In fact every infinite-dimensional $C^\star$-algebra should contain an element with infinite spectrum.

Just to get started, a more concrete instance of the question above is:

Question: Let $p_1, p_2$ and $p_3$ be three projections in a $C^*$-algebra with the property that every (non-commutative) polynomial in $p_1, p_2$ and $p_3$ has finite spectrum. Is it true that the projections generate a finite-dimensional algebra?

-
I have just learned that Choi-Elliott proved in (ams.org/mathscinet-getitem?mr=1081290) that the set of self-adjoint elements with finite spectrum in certain rotation algebras are dense in the set of all self-adjoint elements. (Interestingly, this result is not explicit, in the sense that there is no concrete $\theta$ for which it happens.) However, from what I understand, it seems unclear whether there is a finitely generated dense sub-algebra such that each element has finite spectrum. –  Andreas Thom Aug 17 '10 at 15:05
Suppose you have a C$^*$ algebra generated by $g_1$, $g_2$ and $g_3$. Consider the elements $x_1g_1+x_2g_2+x_3g_3$, where the $x_i$ are variables. If the spectra of these elements are the solutions of a polynomial equation in $x_1$, $x_2$ and $x_3$, is the algebra finite dimensional? –  Peter Shor Aug 19 '10 at 23:13
I do not know. Beautiful question! Do you want to (or may I) ask this as a separate question? –  Andreas Thom Aug 20 '10 at 9:07
@Andreas, Why don't you go ahead and ask this as a separate question. I was thinking about this as an approach to your question. One next step (which would be much harder) would to say if the spectra of these elements isn't the solution of a polynomial equation, then one of these elements has an infinite spectrum. I have no idea how to prove this, or even if it's true, but it seems to work for a few examples. –  Peter Shor Aug 20 '10 at 12:54
Could be; I do not know. I was just talking with Joachim Cuntz and George Elliott about the problem, but they did not mention Richard Kadison. There is a related problem to find an element with infinite spectrum in an infinite dimensional $C^\ast$-algebra, which is a lot easier. Where did you hear that Kadison asked the question? –  Andreas Thom Sep 4 '10 at 21:28


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# Need set notation help.

1. Nov 9, 2011

### ektrules

Ok, I'm not very familiar with set notation. I was just wondering if the following is correct notation and means what I think:

{x$\in$S : $\exists$y$\in$x, y$\in$W}

S is a set of pairs of symbols (tuples of length 2 is the technical term I believe). W is a set of symbols.

What I want is the set of pairs in S that contain at least 1 symbol from set W.

Does my set builder notation correctly describe what I'm looking for? I don't necessarily have to use set builder notation; I just can't think of a way to describe it with unions, intersections, and quantifiers.

2. Nov 9, 2011

### CompuChip

I think it is correct, however it may become slightly clearer if you explicitly write down the pairs:

$$\{ (x, y) \in S \mid x \in W \vee y \in W \}$$

If you would like to omit the set builder notation, you could consider something like
$$(W \times X) \cup (X \times W)$$
where X is the set of all symbols (and $W \subseteq X$).


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## Number Theory Seminar: Distribution of Farey fractions

Seminar | April 10 | 3:40-4:40 p.m. | 748 Evans Hall

Jeffrey C. Lagarias, University of Michigan

Department of Mathematics

The Farey fractions of level $n$ are the set of rationals in $[0,1]$ in lowest terms having denominator at most $n$. It is known that a measure of equally weighted point masses (of total mass 1) on the points of the Farey sequence $F_n$ converges to the uniform distribution on $[0,1]$ as $n$ goes to infinity. The Riemann hypothesis is equivalent to suitably fast rates of convergence (to zero) of certain statistics measuring distance to uniform distribution, given by theorems of Franel (1924) and Landau (1924) . This talk addresses a toy model consisting of unreduced Farey fractions (allowing fractions not in lowest terms) and studies similar statistics.

ribet@berkeley.edu


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# Nokia 8110 4G firmware dumping

## Step 0: Intro

Earlier this year I bought a Nokia 8110 4G phone. Hoping this was a return to glory for the brand, I quickly realized that the new KaiOS (fork of Mozilla’s Firefox OS) is laggy, slow and kind of shitty. Being a developer I started digging in the docs and source code of KaiOS. I found out that it is loosely based on Android, but running a gecko browser where the entire UI is HTML5 and JavaScript. That sounds hackable indeed. After some websearching I found a guide to get to the recovery and enable developer mode using an update zip called smith.zip. First you need an SD card with 1-4GB capacity depending on what you want to dump. You will need to run a Linux of some sort to open the dumps.

If you just want to dump your firmware and skip the technicals, dowload the dumper-signed.zip, copy it to SD card (or use ADB sideload), reboot to recovery and apply the update.

## Step 1: Running code from recovery

First of all you need to boot the recovery. Start by powering off the phone completely. Press and hold the DPAD UP button and power it on. Keep DPAD UP pressed. After some seconds you should see the glorious Android recovery in the eyeblistering resolution of 320x400.

An update.zip file for Android is basically just a signed script. The folder structure inside the zip is as follows:

[your files go in the root here]

META-INF/CERT.RSA  # this is generated
META-INF/CERT.SF   # so is this
META-INF/com/google/android/update-binary  # this is the same for all
META-INF/com/google/android/updater-script # <-- this script copies your files and starts them

You can find a lot more detailed descriptions of the format over on XDA Developers forum.

### Step 1.1: Locating the suff we want to dump

We need to print the partition structure on the phone. Let’s start with a small script that prints the directory structure. First we need to create the updater-script:

ui_print("Running script...");

package_extract_file("recon.sh", "/tmp/recon.sh");
package_extract_file("busybox", "/tmp/busybox");

set_perm(0, 0, 0777, "/tmp/busybox");
run_program("/tmp/busybox", "sh", "/tmp/recon.sh");

wipe_cache();
ui_print("Done.");

The above script assumes two files, busybox and recon.sh. We need busybox to execute sh scripts and we write our dump script in recon.sh. The contents of recon.sh is the following:

#!/tmp/busybox sh

/tmp/busybox mount -t vfat -o rw /dev/block/mmcblk1p1 /sdcard
/tmp/busybox ls -R /dev/block > /sdcard/dirlist.txt
/tmp/busybox umount /sdcard

The above mounts the SD card, lists everything under /dev/block recursively and dumps it to a file on the SD card called dirlist.txt. We need to sign the file before uploading. Generate a certificate (just search for Android signing). I used the signjar tool from the Ubuntu repositories with some random cert and it worked great. After it’s been signed, copy the zip to the SD card and boot the Nokia in recovery. Use the DPAD to navigate and the POWER button to select. If everything works as intended your SD card will have a file called dirlist.txt on it.

Notice under /dev/block/platform/soc.0/7824900.sdhci/by-name there are a bunch of devices (uninteresting ones omitted for brevity):

boot
config
recovery
...
splash
...
system
...
userdata

As you can see it resembles the layout of your basic android phone. We want to poke around the OS so let’s take a look at what’s under system.

### Step 1.2: Dumping /system

We utilize the same procedure as above, albeit we alter the shell script a little. We make a new script dump.sh and alter our updater script:

ui_print("Running script...");

package_extract_file("dump.sh", "/tmp/dump.sh");
package_extract_file("busybox", "/tmp/busybox");

set_perm(0, 0, 0777, "/tmp/busybox");
run_program("/tmp/busybox", "sh", "/tmp/dump.sh");

wipe_cache();
ui_print("Done.");

The contents of dump.sh:

/tmp/busybox mount -t vfat -o rw /dev/block/mmcblk1p1 /sdcard

partitions="boot config recoverysplash ssd system userdata"

for part in $partitions do echo "Dumping$part" >> /sdcard/run.log
echo $part >> /sdcard/run.log /tmp/busybox sync /tmp/busybox dd if=/dev/block/platform/soc.0/7824900.sdhci/by-name/$part of=/sdcard/$part.bin /tmp/busybox sync done /tmp/busybox umount /sdcard echo "Done" Same procedure as before, sign, copy to SD and run from recovery. If everything is working you now have a bunch of .bin files on your SD card. ### Step 1.3: Mounting the system image The system image is a simple ext4 partition. Just mount it as rw: mount -t ext4 -o rw,loop system.bin /mnt Voila! You have your Nokia firmware mounted. If you want to play a game of Snake, navigate to: /mnt/b2g/webapps/snake.gaiamobile.org and extract the application.zip. Run the index.html file in Firefox and enjoy Snake in actual 60fps. ## Step 2: Changing the SMS app Apps are, as the Snake app demonstrate, just zip files with HTML in them. Let’s take our dump of the SMS app and try changing something. The SMS app is fairly generic. Let’s try chaning a color in it. First locate the app, it’s in the b2g/webapps/sms.gaiamobile.com folder. Extract application.zip. Locate style/message.css. Locate the selector .outgoing .bubble .message-content-body and change the background color. To package it as a KaiOS app you need to zip it into an application.zip and replace the SMS app on the phone. For this you need to create an update.zip with a script that overwrites the app in question. The updater-script: ui_print("Running script..."); package_extract_file("application.zip", "/tmp/smsapp.zip"); package_extract_file("swap.sh", "/tmp/swap.sh"); package_extract_file("busybox", "/tmp/busybox"); set_perm(0, 0, 0777, "/tmp/busybox"); run_program("/tmp/busybox", "sh", "/tmp/swap.sh"); wipe_cache(); ui_print("Done."); This is the swap.sh script: /tmp/busybox mount -t ext4 -o rw /dev/block/platform/soc.0/7824900.sdhci/by-name/system /system SOURCE="/tmp/smsapp.zip" DESTINATION="/system/b2g/webapps/sms.gaiamobile.org/application.zip" /tmp/busybox cp$SOURCE \$DESTINATION
/tmp/busybox umount /system

Now package it, zip it and sign it. Congratulations, you now have a modified SMS app.

## What now?

KaiOS is a nice little platform and the Android base gives it a lot potential. The built-in apps are no way near optimized for the crappy CPU that is in the Nokia 8110, but one can imagine a more optimized set of apps. Let’s see how far we can push this little piece of hardware. Go download the firmware dumper and start poking around.

Until next time, happy hacking.

© 2022 Jesenko Mehmedbasic - this is a footer.


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# The space of polynomials with all real roots

The question stems from an attempt to answer a question of David Speyer. Let $R \subseteq \mathbb{R}^n$ be the space of coefficients of all polynomials of degree $n$ whose all roots are real, i.e.

$R := \lbrace (a_0,\ldots, a_{n-1}): x^n + a_{n-1}x^{n-1} + \cdots + a_0$ has only real roots$\rbrace$

Let $R_+$ be the intersection of $R$ with the all-non-negative orthant. If $R_+$ has more than one connected component, the answer to David's question would be negative.

As I describe below, for $n=3$, $R$ and $R_+$ are connected, and it 'looks' from the plot that David's question has affirmative answer (which, in this special case, can surely be proved in a more straightforward way). However, I would think the spaces $R$ and $R_+$ are interesting in their own right, and would like to know if they have been studied. The answers to even the simplest questions, e.g. if they are connected, seem not very obvious.

Edit: As Richard shows below, $R_+$ is connected. It follows by an even easier version of the same argument that $R$ is connected as well.

Below follows a description of $R$ for the case $n=3$, following the excellent answer to this question.

The computation outlined in the above answer shows that the interior $R^0$ of $R$ is the region of positive definiteness of some symmetric matrix whose entries are polynomials in $a_j$'s. For $n=3$, using the characterization of positive definiteness in terms of minors, it follows that

$R^0 = \lbrace (a_0, a_1, a_2): f_i(a_0, a_1,a_2) > 0,\ 1 \leq i \leq 2\rbrace$

where $f_1 = a_2^2 - 3a_1$, $f_2 = -(27a_0^2 - a_0(18a_1a_2 - 4a_2^2) + 4a_1^3 - a_1^2a_2^2)$. The discriminant of $f_2$ (as a quadratic equation in $a_0$) is $16(a_2^2 -3a_1)^3 = 16f_1^3$ which is positive on $R$. For each $(a_1, a_2)$ such that $f_1(a_1, a_2) > 0$, the admissible $a_0$ values are therefore those belonging to the interval between the minimum $r_m(a_1, a_2)$ and the maximum $r_M(a_1, a_2)$ of the two roots of $f_2(\cdot,a_1,a_2)$. Setting $t := (a_2^2 - 3a_1)^{1/2}$ gives

$r_M(a_1, a_2) = \frac{1}{27}(a_2^3 - 3a_2t^2 + 2t^3)$

$r_m(a_1, a_2) = \frac{1}{27}(a_2^3 - 3a_2t^2 - 2t^3)$

It follows that

$R^0 = \lbrace (a_0, a_1, a_2): a_2^2 - 3a_1 > 0,\ r_m(a_1, a_2) < a_0 < r_M(a_1, a_2) \rbrace$

Here is a plot of the boundary of $R$ ($R$ is the region between the two surfaces):

And here is the boundary of $R_+$:

I must be missing something. A monic polynomial with real roots has nonnegative coefficients if and only if all roots are nonpositive. If we have two vectors $u,v\in \mathbb{R}^n$ with nonpositive entries, then there is a continuous path (e.g., a line segment) from $u$ to $v$ that keeps the entries nonnegative. This induces a continuous path between the polynomials with roots $u$ and $v$ that keeps the roots nonpositive.

• But the roots may no longer be real... Jun 5, 2015 at 0:18
• See for instance the comments to the referenced question: mathoverflow.net/questions/207971/… Jun 5, 2015 at 0:18
• @SamHopkins: $u$ and $v$ are vectors of roots, as are the vectors on the line segment joining them. In the link you gave, there is also the issue of the coefficients increasing monotonically. Jun 5, 2015 at 1:02
• Ah I understand now. (By the way, you want to say that u and v are nonpositive vectors.) Jun 5, 2015 at 1:44

Space $R$ of hyperbolic(i.e. real-rooted) polynomials of one variable, have actually been studied quite a lot, mostly in papers by V.I. Arnold and his students at the end of 80's-beginning of 90's

I would recommend a book by V.P. Kostov Topics in hyperbolic polynomials of one variable. Societe Mathematique de France, 2011, especially it's chapter 2.

It has a lot of references.

Moreover, Section 2. of a recent preprint http://arxiv.org/abs/1512.08645 has some review on this and some other close questions.

In particular, one can prove that $R$ is homeomorphic to ${\mathbb R}\times{\mathbb R_+^{n-1}}.$

This follows from the fact that the correspondence between roots and coefficients of polynomials is morphism of symmetric product of topological spaces (i.e. quotient by $S_n$ acting by permutations of roots -- see http://www.ams.org/journals/tran/1954-077-03/S0002-9947-1954-0065924-2/S0002-9947-1954-0065924-2.pdf P.538 or http://link.springer.com/article/10.1007%2FBF01094483) and $n$-th symmetric product of real line is exactly ${\mathbb R}\times{\mathbb R_+^{n-1}}$ (see e.g. Proposition 3 there: https://ncatlab.org/nlab/show/symmetric+product+of+circles).


text:
### U.S. Could Do Much More To Protect The Environment, Report Finds

In a new report that ranks countries by how well they protect the environment, the U.S. comes in at a disappointing 26th place among 180 nations.

### Sensitivity of freshwaters to browning in response to future climate change

Many boreal waters are currently becoming browner with effects on biodiversity, fish production, biogeochemical processes and drinking water quality. The question arises whether and at which speed this browning will continue under future climate change. To answer the question we predicted the absorbance (a420) in 6347 lakes and streams of the boreal region under future climate change. For the prediction we modified a numerical model for a420 spatial variation which we tested on a temporal scale by simulating a420 inter-annual variation in 48 out of the 6347 Swedish waters.

### Leisure time computer use and overweight development in young adults – a prospective study

The prevalence of overweight among Swedish young adults has nearly doubled since the 1980s. The weight increase has been paralleled by the increased use of computers at work, at school, and at leisure time. The aim was to examine leisure time computer use for gaming, and for emailing/chatting, in relation to overweight development in young adults.

Original Source

### Africa: Sweden's We Effect and FAO to Support Small-Scale Forest and Farm Producers

FAO and the Swedish-based development cooperation organization We Effect have agreed to work together to strengthen small-scale forest and farm producers' organizations in developing countries so t

### Water, forests, people – building resilient landscapes

The Stockholm International Water Institute (SIWI) Swedish Water House (SWH) Cluster Group for Water and Forests started with a mission to highlight the importance of forests and sustainable forest management to securing water resources globally.

### Significant increase of aerosol number concentrations in air masses crossing a densely trafficked sea area

In this study, we evaluated 10 months data (September 2009 to June 2010) of atmospheric aerosol particle number size distribution at three atmospheric observation stations along the Baltic Sea coast: Vavihill (upwind, Sweden), Utö (upwind, Finland), and Preila (downwind, Lithuania).

### Climate impact of beef: an analysis considering multiple time scales and production methods without use of global warming potentials

An analysis of the climate impact of various forms of beef production is carried out, with a particular eye to the comparison between systems relying primarily on grasses grown in pasture ('grass-fed' or 'pastured' beef) and systems involving substantial use of manufactured feed requiring significant external inputs in the form of synthetic fertilizer and mechanized agriculture ('feedlot' beef). The climate impact is evaluated without employing metrics such as ${\mathrm{CO}}_{2}{\rm{e}}$ or global warming potentials. The analysis evaluates the impact at all time scales out to 1000 years.

### Traffic-related air pollution and dementia incidence in northern Sweden: A longitudinal study

Exposure to ambient air pollution is suspected to cause cognitive effects, but a prospective cohort is needed to study exposure to air pollution at the home address and the incidence of dementia. The researchers aimed to assess the association between long-term exposure to traffic-related air pollution and dementia incidence in a major city in northern Sweden.

Original Source

### Zambia: Sweden Resumes Healthcare Aid to Zambia

The Swedish government, one of Zambia's closest development allies has unblocked up to 400 million Swedish Kroners (sek), which amounts to about US \$47,3 million meant for improving the Zambian hea

### Swedish city Kiruna being moved to avoid sinking from iron ore mining

Have you ever just wanted to pick up and leave the foundation you call home? Kiruna knows the feeling.


text:
Deepak Scored 45->99%ile with Bounce Back Crack Course. You can do it too!

# Evaluate the following integrals:

Question:

Evaluate the following integrals:

$\int \tan ^{-1}\left(\frac{\sin 2 x}{1+\cos 2 x}\right) d x$

Solution:

Let I $=\int \tan ^{-1}\left(\frac{\sin 2 x}{1+\cos 2 x}\right) d x$

We know $\cos 2 \theta=2 \cos ^{2} \theta-1$

Hence, in the denominator, we can write $1+\cos 2 x=2 \cos ^{2} x$

In the numerator, we have $\sin 2 x=2 \sin x \cos x$

Therefore, we can write the integral as

$\mathrm{I}=\int \tan ^{-1}\left(\frac{2 \sin \mathrm{x} \cos \mathrm{x}}{2 \cos ^{2} \mathrm{x}}\right) \mathrm{dx}$

$\Rightarrow \mathrm{I}=\int \tan ^{-1}\left(\frac{\sin \mathrm{x}}{\cos \mathrm{x}}\right) \mathrm{dx}$

$\Rightarrow \mathrm{I}=\int \tan ^{-1}(\tan \mathrm{x}) \mathrm{dx}$

$\Rightarrow \mathrm{I}=\int \mathrm{xdx}$

Recall $\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$

$\Rightarrow I=\frac{X^{1+1}}{1+1}+c$

$\therefore \mathrm{I}=\frac{\mathrm{x}^{2}}{2}+\mathrm{c}$

Thus, $\int \tan ^{-1}\left(\frac{\sin 2 \mathrm{x}}{1+\cos 2 \mathrm{x}}\right) \mathrm{dx}=\frac{\mathrm{x}^{2}}{2}+\mathrm{c}$


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Article | Open | Published:

# Centiloid method evaluation for amyloid PET of subcortical vascular dementia

Scientific Reportsvolume 7, Article number: 16322 (2017) | Download Citation

## Abstract

Reference region selection is important for proper amyloid PET analysis, especially in subcortical vascular dementia (SVaD) patients. We investigated reference region differences between SVaD and Alzheimer’s disease (AD) using Centiloid scores. In 57 [C-11] Pittsburgh compound B (PiB) positive (+) AD and 23 PiB (+) SVaD patients, we assessed standardized PiB uptake and Centiloid scores in disease-specific cortical regions, with several reference regions: cerebellar gray (CG), whole cerebellum (WC), WC with brainstem (WC + B), pons, and white matter (WM). We calculated disease group differences from young controls (YC) and YC variance according to reference region. SVaD patients showed large effect sizes (Cohen’s d > 0.8) using all reference regions. WM and pons showed larger YC variances than other regions. Findings were similar for AD patients. CG, WC, and WC + B, but not WM or pons, are reliable reference regions for amyloid imaging analysis in SVaD.

## Introduction

The use of amyloid PET scans for quantitative measurement of amyloid-beta deposition has grown in recent years. As a result, the development and standardization of methods for amyloid-PET data analysis, particularly methods applicable across centers, has increasingly interested clinicians and investigators1,2,3. The Centiloid project, an effort to standardize quantitative amyloid-PET plaque estimation across centers, is one such development. The ratio of target to reference region is an effective method for calculating amyloid load, with sufficient discriminatory power between Alzheimer’s disease (AD) and matched controls2,4. This method is easy, useful, has obvious merit for clinical use as it requires no arterial blood sampling2, and has been adopted widely for semi-quantification of amyloid load. In probable AD, target to reference ratio is usually large and variance of the ratio matters little clinically2. However, there are sometimes subjects with borderline amyloid PET tracer uptake. Aβ deposition occurs on a continuum; at present there is no clear a priori way to separate individuals who have Aβ in the brain from those who do not. Therefore, small differences in quantification result according to the reference region used can present an issue especially in subjects with borderline uptake. Furthermore, when target-to-reference ratios are converted to the Centiloid scale, the values are increased, as the dynamic range of the Centiloid scale is greater than that of the target-to-reference ratio; consequently, choosing an inappropriate reference region may exaggerate errors in the Centiloid procedure especially in patients with borderline amyloid burden.

Cerebrovascular disease (CVD) and amyloid burden are the most frequent pathologies in cognitively impaired subjects5, and these two distinct pathologies present concurrently at a high rate. Mixed pathology is present in approximately half of all clinically diagnosed AD cases6,7,8,9, even in clinical trials with participants extensively screened for pure AD10. Conditions such as subcortical vascular dementia (SVaD), which exhibit both CVD and amyloid pathology, also require amyloid quantification; however, reference region selection in SVaD presents additional difficulties because pathology may occur within and impact measurement of certain reference regions.

According to Thal amyloid phase, pons and cerebellum are among the latest regions presenting amyloid pathology in AD11. The cerebellum has been used as a reference region in most previous studies using [C-11] Pittsburgh compound B (PiB) PET scans2,4,12. However, there is concern that cerebellum may not be suitable as a reference region for analyses of amyloid burden in conditions other than late-onset AD, as cerebellar amyloid deposits may be present in cerebral amyloid angiopathy (CAA)13, prion diseases14, and genetic AD15,16. Therefore, alternative reference regions such as pons have been evaluated for reliability4. Edison and colleagues reported that the pons is a reliable reference region for analysis in [C-11] PiB studies where cerebellum is not an appropriate reference region4. A recent study also suggested that WM as a reference region improved discrimination between clinically-defined groups17. However, the appropriateness of different reference regions for assessing amyloid uptake in patients with SVaD has not yet been investigated.

In this study, we investigated the use of different reference regions in analyzing amyloid uptake in SVaD patients. Given that SVaD patients affect more severe white matter (WM), brainstem and cerebellum than AD patients18,19,20, we hypothesized that there might be differences between SVaD and AD in the appropriateness of different reference regions. For testing the hypothesis, we used five reference regions cerebellar gray matter (CG), whole cerebellum (WC), WC with brainstem (WC + B), pons, and WM which have been frequently used in previous studies.

## Results

### Participant demographics

Table 1 shows demographic information of participants in this study. There were no differences in gender proportion between PiB (−) old control (OC), PiB (+) AD, and PiB (+) SVaD groups. However, ANOVA (F = 14.2, p < 0.001) and post-hoc tests indicated the PiB (+) AD group was younger than the PiB (−) OC (t = 3.4 and p = 0.001) and PiB (+) SVaD (t = 4.3 and p < 0.001) groups.

### Global cortical target (CTX) ROIs in PiB (+) AD and PiB (+) SVaD

Figure 1 shows disease-specific CTX ROIs representing large differences in group-averaged PiB images between OC and patients (see the section of “Disease-specific CTX ROI”). SUVr of every voxel in the ROIs is significantly higher in patients (see Supplementary Fig. S1). Both CTXAD and CTXSVaD were mostly located in frontal, temporal and parietal regions. When evaluating proportions by lobe, CTXSVaD showed relatively larger proportions in parietal and lower proportions in temporal regions than CTXAD (Table 2).

### Effect size of Centiloid (CL) values between disease groups and young control (YC) according to reference ROIs

The differences of CL between disease groups and YC are represented as effect sizes (Table 3). The effect sizes were large enough (Cohen’s d > 0.8) to show consistent group differences in all reference ROIs. The highest effect size of differences between PiB (+) AD and YC was generated using WM ROI (5.543) followed by pons (4.363), WC + B (4.200), WC (4.043) and CG (3.698). The highest effect size of differences between PiB (+) SVaD and YC was generated using WM ROI (4.721) followed by WC + B (3.957), WC (3.928), CG (3.851) and pons (3.615).

### Variance of CL values according to reference ROIs in YC group

The results of assessing variance of CL values are shown in Table 3. The Bartlett’s statistics for equal variance indicated that there was within-group variability for all groups except SVaD patients; the results of post hoc tests are shown in Figs 2 and 3. The largest variances of CL values in YC were found using WM as reference region, followed in order by pons, WC + B, WC and CG (Table 3). Compared to other ROIs, WM and pons ROIs showed greater variance of CL values.

## Discussion

In this study, we translated our PiB PET data into the Centiloid scale as described previously1, and tested multiple reference regions for evaluating amyloid-beta deposits in cortical target regions, with a focus on patients with PiB (+) SVaD. We found that there were large magnitudes of effect sizes for differences between SVaD and YC regardless of reference ROI. However, using WM or pons ROI as a reference region showed larger variances in YC than using other ROI regions. In PiB (+) AD patients, the results were similar. Taken together, our findings show that CG, WC, and WC + B could be used as reference regions for amyloid imaging analysis in patients with PiB (+) SVaD, which are the same for AD patients.

We found that CG, WC or WC + B were reliable as reference regions for amyloid PET analyses in SVaD patients. These regions are commonly used as reference for amyloid PET analyses in AD patients because they are among the last regions where amyloid pathology is known to occur2,4,12, and are classified as Thal phase 511. Although some studies have raised the possibility of requiring different reference regions for specific diseases4,14,15,16, our findings demonstrate that appropriate reference regions for SVaD patients are the same as those for AD.

In this study, WM showed higher variability of Centiloid-scaled values in YC group than other reference regions. [C-11] PiB binding to WM is mainly non-saturable and non-specific21, but PiB retention has been found to be reduced in WMH regions compared to normal appearing WM regions22. Some have raised concerns that using a WM-containing reference region for analysis of amyloid imaging data could be problematic in a population with high WMH22. Consistent with this, our findings suggest that the quantification of amyloid deposits based on the ratio of target to WM may be an inappropriate approach in SVaD patients.

We also found that pons showed higher variability than other reference regions of Centiloid-scaled values in YC. This may appear inconsistent with a recent amyloid imaging study demonstrating the value of the pons as a reference region4. The discrepancy may be partly explained by different analysis methods, as well as by different study subjects. Unlike the previous study, patients with genetic AD and prion disease were excluded in our study. However, our findings are consistent with a previous Centiloid study showing pons with higher variability and lower effect size than other reference regions1. In addition, the inherent variability of normalization processes could affect the results of our study. For example, poor performance of normalization process in the pons can occur frequently, due to its small size and the fact that the SPM normalization algorithm seems to handle brainstem structures less well than cortical structures1, and could influence the [C-11] PiB processing and results.

One of the important findings of our study is that CG, WC, and WC + B, but not pons nor WM, are appropriate reference regions in both AD and SVaD. However, several limitations must be noted. First, our conclusions that WM and pons are not reliable reference regions, while CG, WC, and WC + B are suitable for analysis in the scaled data set, is not based on neuropathological studies. The precise performance of these reference regions will require confirmation by further neuropathological studies. Second, other pathologies such as prion disease, tau pathology, or hippocampal sclerosis are not considered because we did not perform a pathologic study. Third, due to the small number of SVaD patients used in this study, it will be necessary to replicate our results with larger samples in further study. Fourth, selecting a disease-specific CTX ROI as well as an appropriate reference region is important for calculating amyloid uptake. We therefore chose the disease-specific CTX ROI in the SVaD and AD groups, separately. Finally, we did not perform correction of PiB retention for brain atrophy, which might have effects on quantification.

Nevertheless, this study demonstrates that CG, WC, and WC + B are reliable reference regions for amyloid PET analysis in patients with SVaD, suggesting that amyloid burden in SVaD patients might be analyzed using the same reference regions as AD.

## Methods

### Participants

Patients were evaluated by clinical interview and neurological and neuropsychological examinations as previously described27. All patients underwent laboratory tests including complete blood count, blood chemistry, vitamin B12/folate, syphilis serology, and thyroid function tests. Brain MRI confirmed the absence of structural lesions including territorial cerebral infarction, brain tumors, hippocampal sclerosis, and vascular malformation.

### Ethics statement

This study protocol was approved by the Institutional Review Board of Samsung Medical Center. We obtained written consent from each participant and all methods were carried out in accordance with the approved guidelines.

### MR and [C-11] PiB-PET imaging techniques

We acquired standardized three-dimensional T1 turbo field echo images from all participants at Samsung Medical Center, using the same 3.0 T MRI scanner (Philips Achieva; Philips Healthcare, Andover, MA), using the following parameters: sagittal slice thickness, 1.0 mm, over contiguous slices with 50% overlap; no gap; repetition time (TR) of 9.9 msec; echo time (TE) of 4.6 msec; flip angle of 8°, and matrix size of 240 × 240 pixels, reconstructed to 480 × 480 over a field of view (FOV) of 240 mm.

All patients underwent [C-11] PiB-PET imaging at Samsung Medical Center or Asan Medical Center (Seoul, Korea) with identical settings using a Discovery STe PET/CT scanner (GE Medical Systems, Milwaukee, WI, USA). [C-11] PiB-PET scanning was performed in 3-dimensional scanning mode examining 35 slices of 4.25-mm thickness spanning the entire brain. [C-11] PiB was injected into an antecubital vein as a bolus with a mean dose of 420 MBq (range 259–550 MBq). CT scans were performed for attenuation correction 60 minutes after injection. A 30-minute emission static PET scan was then initiated. The specific radioactivity of [C-11] PiB at time of administration was more than 1,500 Ci/mmol for patients and the radiochemical yield was more than 35%. The radiochemical purity of the tracer was more than 95% in all PET studies.

### Image processing

We employed image processing described in the Centiloid paper1, and replicated the Centiloid procedure to validate methodological consistency (see Supplementary information).

### Reference ROIs

The four data-driven ROIs (CG, WC, WC + B and pons) in Klunk et al.1 have been used as reference ROIs for quantifying of amyloid retention in numerous previous studies28,29,30,31,32,33, because no significant binding differences between AD and OC are observed. WM has also been evaluated as a reference ROI in recent [C-11] PiB-PET studies34, as those studies reveal comparable WM amyloid burden between diagnostic groups and low inter-subject variability. Therefore, we included WM as a candidate reference region. To generate the WM ROI, the WM probability map on template of SPM8 was thresholded at 0.7. This threshold provided a WM region that avoided contributions from CSF and amyloid retention in gray matter.

### Statistical analysis

Individual SUVr values were calculated for normalized PiB-PET using five different reference ROIs (including WM), and CTXAD and CTXSVaD. SUVr values were obtained by ratio with each of the reference ROIs. CL values for individual subjects in AD group were computed by comparing to YC group which are considered not to have any brain amyloid pathology, and it defined as follows:

$${\rm{CL}}=100\times (SUV{r}_{IND\ast }-SUV{r}_{YC-0\ast })/(SUV{r}_{AD-100\ast }-SUV{r}_{YC-0\ast })\,$$
(1)

Where $$SUV{r}_{IND\ast }$$ represents the individual SUVr values of all YC-0 and AD-100 subjects, and $$SUV{r}_{YC-0\ast }$$ and $$SUV{r}_{AD-100\ast }$$ represent each group’s mean values. For SVaD group, the term of $$SUV{r}_{AD-100\ast }$$ was replaced as $$SUV{r}_{SVaD-100\ast }$$.

To describe the dissimilar amyloid binding patterns between CTXAD and CTXSVaD ROIs, we measured regional volumetric proportion of CTX by dividing into frontal, parietal, temporal, occipital lobe and other regions.

Selection of the standard reference was based on the effect size of the group differences between patients and YC and on the variance of reference ROIs. Effect size for each group was evaluated with Cohen’s d with pooled standard deviation,

$${\rm{Effect}}\,{\rm{Size}}=({\mu }_{p}-{\mu }_{n})/\sqrt{({N}_{p}{{\sigma }_{p}}^{2}+{N}_{n}{{\sigma }_{n}}^{2})/({N}_{p}+{N}_{n}-2)}\,$$
(2)

where:

μ p and μ n mean the average of SUVr in each patient and YC group.

$${\sigma }_{p}^{2}$$ and $${\sigma }_{n}^{2}$$ are the variance in each patient and YC group.

N p and N n are the number of subjects in each patient and YC group.

For variance tests, Bartlett’s statistics were employed for validating equal variances of reference ROIs in each clinical group35. Post hoc variance tests were also performed to investigate regional specificities for reference ROIs.

For quantitative analysis of lobar SUVr values among all groups, mean values of SUVr (using WC as reference) in each lobe were calculated.

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## Acknowledgements

This research was supported by the Brain Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2016M3C7A1913844), the Korea government (MSIP) (NRF-2016R1A2B3016609 and NRF-2017R1A2B2005081) and Research of Korea Centers for Disease Control and Prevention (2016-ER6203-00), and by National Institutes of Health (F32-AG050389).

## Author information

### Author notes

1. Hyuk Jin Yun and Seung Hwan Moon contributed equally to this work.

### Affiliations

1. #### Department of Biomedical Engineering, Hanyang University, Seoul, 04763, Korea

• Hyuk Jin Yun
•  & Jong-Min Lee
2. #### Fetal Neonatal Neuroimaging and Developmental Science Center, Division of Newborn Medicine, Boston Children’s Hospital, Harvard Medical School, Boston, 02115, MA, USA

• Hyuk Jin Yun
3. #### Department of Nuclear Medicine, Sungkyunkwan University School of Medicine, Seoul, 06351, Korea

• Seung Hwan Moon
• , Yearn Seong Choe
•  & Kyung Han Lee
4. #### Department of Neurology, Samsung Medical Center, Sungkyunkwan University School of Medicine, Seoul, 06351, Korea

• Hee Jin Kim
• , Duk L. Na
•  & Sang Won Seo
5. #### Neuroscience Center, Samsung Medical Center, Seoul, 06351, Korea

• Hee Jin Kim
• , Duk L. Na
•  & Sang Won Seo
6. #### Helen Wills Neuroscience Institute, University of California, Berkeley, 94720, CA, USA

• Samuel N. Lockhart
7. #### Department of Internal Medicine, Division of Gerontology and Geriatric Medicine, Wake Forest School of Medicine, Winston-Salem, 27157, NC, USA

• Samuel N. Lockhart
8. #### Department of Health Sciences and Technology, Sungkyunkwan University, Seoul, 06351, Korea

• Duk L. Na
•  & Sang Won Seo
9. #### Department of Clinical Research Design and Evaluation, SAIHST, Sungkyunkwan University, Seoul, 06351, Korea

• Sang Won Seo

### Contributions

Designed the experiments (H.J.Y., S.H.M., J.M.L. and S.W.S.), performed the experiments (H.J.Y.), analyzed the data (H.J.Y. and S.W.S.), contributed reagents/materials (S.H.M., H.J.K., Y.S.C., K.H.L., D.L.N., J.M.L. and S.W.S.) and wrote the paper (H.J.Y., S.H.M., S.N.L. and S.W.S.).

### Competing Interests

The authors declare that they have no competing interests.

### Corresponding authors

Correspondence to Jong-Min Lee or Sang Won Seo.

## Electronic supplementary material

### DOI

https://doi.org/10.1038/s41598-017-16236-1


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# Chapter 1 Atomic Structure¶

## Example 1_1 pgno:6¶

In [12]:
print'The Atomic weight of 12.0111 for Natural Carbon shows that the 12C nuclide must be present to a larger extent.'
print'\nLet 100 atoms of natural carbon contain x atoms of 12C nuclide.\n'
X=(13.0034-12.0111)*100/(13.0034-12.0000);#percentage of 12C in natural carbon#
print'Percentage of 12C in Natural carbon=X=',round(X,3)
Y=100-X;#percentage of 13C in natural carbon#
print'\nPercentage of 13C in Natural carbon=',round(Y,3)

The Atomic weight of 12.0111 for Natural Carbon shows that the 12C nuclide must be present to a larger extent.

Let 100 atoms of natural carbon contain x atoms of 12C nuclide.

Percentage of 12C in Natural carbon=X= 98.894

Percentage of 13C in Natural carbon= 1.106

## Example 1_2 pgno:9¶

In [13]:
C=3*10**2;#velocity of light in megametre/sec#
v=C*10**-6/L;#frequency of radiation in Teracycles per sec#
v1=1/L;#wave number of radiation in per meter#

Frequency of radiation=v=Teracycles per sec=1.0*10**15Hz 1000.0

## Example 1_3 pgno:12¶

In [14]:
V=0.85;#external voltage in volts#
e=1.6*10**-19;#electron charge in coloumbs#
m=9.1*10**-28;#electron mass in grams#
v=(2*V*e*10/m)**0.5;#velocity of electron in motion in Kilocm per sec#
print'velocity of electron in motion=v= Kilocm per sec=5.47*10**7cm per sec',round(v,3)
W=(3.198*10**-12)/(1.6*10**-12);#Threshold energy in eV#
print'\nThreshold energy of electron=W=eV',round(W,3)
v0=(3.198*10**-12)/(6.625*10**-15);#Threshold frequency in tera per sec#
print'\nThreshold frequency=v0=Tera per sec=4.83*10**14per sec',round(v0,3)

velocity of electron in motion=v= Kilocm per sec=5.47*10**7cm per sec 54671.848

Threshold energy of electron=W=eV 1.999

Threshold frequency=v0=Tera per sec=4.83*10**14per sec 482.717

## Example 1_4 pgno:13¶

In [15]:
E=118.5*10**3*4.2*10**7;#energy of ions in ergs#
C=3*10**10;#velocity of light in cm/sec#
h=6.625*10**-27;#plank's constant#
l=(L*h*C*10**8)/E;
print'wavelength required to cause ionization=l=Angstrums',round(l,3)

wavelength required to cause ionization=l=Angstrums 2405.206

## Example 1_5 pgno:17¶

In [16]:
n1=2.;
n2=4.;
dE=21.7*(10**-12)*((1/n1**2)-(1/n2**2));
h=6.625*10**-27;#plank's constant#
C=3*10**10;#velocity of light in cm/sec#
l=h*C*10**8/dE;#Wavelength of second line in balmer series in Angstrums#
print'wavelength of the second line in balmer series=l=Angstrums',round(l,3)#here the answer given in textbook is slightly wrong the original answer should be the one comes through execution#

wavelength of the second line in balmer series=l=Angstrums 4884.793

## Example 1_6 pgno:18¶

In [17]:
n1=1;
dE=21.7*(10**-12)/(1.6*10**-12*n1**2);#energy required to promote an electron from ground to infinity in eV#
print'Ionisation potential for an electron=dE=eV',round(dE,3)

Ionisation potential for an electron=dE=eV 13.563

## Example 1_7 pgno:20¶

In [18]:
h=6.625*10**-27;#plank's constant#
V=2*10**3;#velocity of Cricket Ball in cm/sec#
m=170;#weight of Cricket Ball in grams#
l=h/(m*V);#DeBroglie Wavelength of CricketBall in Angstrums#
print'DeBroglie Wavelength of CricketBall=l==1.95*10**-24Angstrums',l

DeBroglie Wavelength of CricketBall=l==1.95*10**-24Angstrums 1.94852941176e-32

## Example 1_8 pgno:21¶

In [19]:
from math import pi,sqrt
r2=4*r1;#Bohr radius in second state in cm#
h=6.625*10**-27;#plank's constant#
m=9.11*10**-28;#electron mass in grams#
v2=h/(pi*m*r2);#electron velocity in second state in cm per sec#
print'\nElectron velocity in second state=v2=cm per sec',v2
l=(h*10**8)/(m*v2);#De Broglie wavelength of electron in second state in Angstrums#
print'\nDe Broglie wavelength of electron in second state=l=Angstrums',l
e=1.6*10**-12;#electron charge in ergs#
v=sqrt((2*(10**4)*e)/m);#velocity of the moving electron in second state in cm/sec#
print'\nVelocity of moving electron in second state=v=cm per sec',v
l1=(h*10**8)/(v*m);#De Broglie wavelength of moving elctron in Angstrums#
print'\nDe Broglie wavelength of moving electron in secondstate=l1=Angstrums',round(l1,4)

Bohr radius in second state=r2=2.12*10**-8cm

Electron velocity in second state=v2=cm per sec 109189724.953

De Broglie wavelength of electron in second state=l=Angstrums 6.66017642561

Velocity of moving electron in second state=v=cm per sec 5926738977.44

De Broglie wavelength of moving electron in secondstate=l1=Angstrums 0.1227

## Example 1_9 pgno:21¶

In [20]:
from math import pi
m=9.11*10**-28;#electron mass in grams#
v=1.1*10**8;#velocity of electron in cm per sec#
p=m*v;#momentum of electron in gram cm per sec#
print'momentum of electron=p=10.01*10**-20gram cm per sec'
dp=p*10**-2;#Uncertainity in momentum in gram cm per sec#
print'\nUncertainity in momentum=10.01*10**-22gram cm per sec'
h=6.625*10**-27;#plank's constant#
dx=(h*10**8)/(4*pi*dp);#Uncertainity in position in Angstrum#
print'\nUncertainity in position=dx=Angstrum',round(dx,2)

momentum of electron=p=10.01*10**-20gram cm per sec

Uncertainity in momentum=10.01*10**-22gram cm per sec

Uncertainity in position=dx=Angstrum 52.61

## Example 1_10 pgno:24¶

In [21]:
h=6.625*10**-27;#plank's constant#
g=10**3;#particle mass in grams#
l1=1.;#length of one dimensional box in cm#
n1=1.;
n2=2.;
dE1=((n2**2-n1**2)*h**2)/(8*g*l1**2);#Energy difference between two energy levels of particle in eV#
print'Energy difference between two energy levels of particle=dE1=1*10**-44eV'
l2=2*10**-8;#length of one dimensional box in cm#
m=9.11*10**-28;#electron mass in grams#
dE2=((n2**2-n1**2)*h**2)/(8*m*l2**2*1.6*10**-11);#Energy difference between two energy levels of electron in eV#
print'\nEnergy difference between two energy levels of electron=dE2=eV',round(dE2,3)

Energy difference between two energy levels of particle=dE1=1*10**-44eV

Energy difference between two energy levels of electron=dE2=eV 2.823


text:
Chapter 5 - Section 5.2 - Properties of Rational Functions - 5.2 Assess Your Understanding - Page 351: 25

Domain: All real numbers; $x \ne \pm 3$

Work Step by Step

The domain of the function is all real numbers, except for when the denominator is 0. Thus, we let the denominator equal 0: $$4(x^2-9) =0 \\ 4x^2 -36 =0 \\ x^2 =9 \\ x= \pm 3$$ Thus, we know that x cannot equal 3 or -3. Domain: All real numbers; $x \ne \pm 3$

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:
# List of TeX commands for Generic Problems¶

TODO - Work in progress.

Default value is the value that the command will get when it is omitted from the TeX file.

TeX-Command Description
\correctorprecision{<number>} Default value: 2
defines the amount of digits that will be used behind the decimal mark, indicated by <number>, when correcting the final answer of the student.
\displayprecision{<number>} Default value: 2
defines the amount of digits that will be visible behind the decimal mark, indicated by <number>, when displaying the number on the screen. Numbers defined through \randdouble are rounded to the same number of digits behind the decimal mark. Typically the displayprecision $\ge$ correctorprecision. Displayprecision only affects variables of type \randdouble, results of calculated functions will always display the full double value, unless the command below is used.
\function[calculate, <number>]{<var>}{<expression>} Default value: 16
Will calculate the <expression> and round the result according to the value of <number>. The option <number>, should only be used when you want to display the result of the calculated function on the screen. If <number> is not provided calculate will result in a number with an accuracy of up to 16 digits behind the decimal mark.

Add picture from clipboard (Maximum size: 500 MB)


text:
Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
 Already signed up? New to KöMaL?

# Problem A. 659. (January 2016)

A. 659. For which $\displaystyle n$ are there polynomials $\displaystyle g(x)$ and $\displaystyle h(x)$ with real coefficients and degrees smaller than $\displaystyle n$ such that $\displaystyle g\big(h(x)\big)= x^n+x^{n-1}+x^{n-2}+\dots +x^2+x+1$?

Miklós Schweitzer Competition, 2015

(5 pont)

Deadline expired on February 10, 2016.

### Statistics:

 6 students sent a solution. 5 points: Glasznova Maja, Imolay András, Williams Kada. 3 points: 2 students. 0 point: 1 student.

Problems in Mathematics of KöMaL, January 2016


text:
# What is -10g-29=31?

Mar 15, 2018

$- 6$

#### Explanation:

First step is to add 29 to both sides of the equation.
$31 + 29 = 60$
==> $- 10 g = 60$

Next, you want to divide my Negative ten. 60÷-10=-6

Your answer: $g = - 6$


text:
# Standing wave on a circle [closed]

Suppose that we have a standing wave on a circle. I heard that by gradually increasing the radius of the circle, the wavelength will also increase to keep the standing wave. Is it right? If yes, what's the physics behind this?

EDIT: Since this question is marked as "ON HOLD" due to ambiguity, I should make it a bit more clear. Since I've already got my answer, this explanation is not a try to ask the question again but an attempt to demonstrate my intentions for asking this question.

Suppose that we're living in a 1-sphere expanding universe (that is just a simple circle which its radius increases with time). The metric of this space is:

$$ds^2 = a^2(t)\ d\phi^2$$

As you can see, the comoving position of galaxies (just the angle $\phi$) in this universe does not vary by time, however, since the scale factor ($a(t)$) is increasing by time, the universe is blowing up too. So now consider a case in which we have two galaxies, A and B. Due to the expansion of this universe, these two galaxies have a relative velocity which could be greater than the speed of light too. (The expansion obeys the Hubble law since the distance between galaxies is $D = a(t)\ d\phi$ and then $$\dot{D} = V = \dot{a(t)}\ d\phi = \frac{\dot{a(t)}}{a(t)}\ a(t)\ d\phi = H(t)\ D.$$ Now consider the case where an observer on galaxy A is observing galaxy B. The light of galaxy B as seen by the observer is redshifted. We can explain this redshift with two similar but a bit different interpretations. First we can say this redshift happens due to the Doppler effect related to the relative velocity of two galaxies, and second we can say that the light is redshifted due to the expansion of the universe, so it's a continues effect on a photon which is traveling between two galaxies. Ok, you may say these two interpretation are exactly the same, which I believe it's right, but you can also think of the scale factor, not as a factor that expands the universe, but a factor which decrease the speed of light as time goes on. (I may be wrong, in this case please correct me.)

Well, I was watching one of the Susskind lectures and he mentioned that if we have a standing wave on a circle, by slowly increasing the radius of the circle, the wavelength will change in a way to keep the standing wave as it was. So my question was about this claim and I think this could be related to the expansion of the universe (since he mentioned it in a cosmology course). Also Anna's answer about de Broglie wavelength clears thing up for me.

## closed as unclear what you're asking by ACuriousMind♦, Sebastian Riese, Kyle Kanos, Norbert Schuch, gigacyanMar 11 '16 at 13:03

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

• I don't understand the metric. Why isn't the coordinate $r$ used? And why isn't there a time component, ${\rm d}t^2$? – Kyle Kanos Mar 12 '16 at 15:16
• Think of it like this, $a(t)$ is actually $r(t)$. This is the metric of space, not the space-time, here's the space-time metric: $d\tau^2 = - dt^2 + ds^2$ – Saeed Mar 12 '16 at 16:23

Suppose you have a circle of radius $r$. In order to model a standing wave on the circle, you need to keep in mind appropriate boundary conditions. In this case, the positive integer multiple of the wavelength should be equal to the circumference of the circle, a condition for constructive interference. This is because otherwise the wave will interfere destructively with itself on the entire circle, resulting in the vanishing of the wave.

Therefore we have

$$n \lambda = 2 \pi r$$

which implies

$$\lambda = 2 \pi r/n$$

So basically for a fixed $n$, the wavelength required for a standing wave increases with $r$.

• Thanks for the answer, but my question was something else. I thing I couldn't ask it properly, let me try it again. When we increase the radius of the circle, since we're not changing the properties of the environment, the propagation speed of the wave should remain unchanged. So, if the wavelength remains the same too, it means that we couldn't have a standing wave anymore. But, it seems that the nature tend to tune the wavelength to keep the standing wave. Well, my question is why does the nature tune the wavelength? – Saeed Mar 10 '16 at 2:46
• Since you said "Nature tunes the wave", you should provide us with a specific example. I am not aware of any cases in real life where this happens. I mean, that's what the Bohr model is about, but that's just a model and all aspects of your conceptual understanding might not apply. Bruce Lee's answer is good, except maybe you wanted him to say "if the wavelength remained the same, we would no longer have a standing wave"? – levitopher Mar 10 '16 at 3:34

The only standing waves one can find in examples on the net are the ones on membranes and water and those are not the waves you mean, since you assume the frequency on the perimeter of the circle.

You have to go to quantum mechanics of elementary particles and the de Broglie wavelength. There one can have an electron in a magnetic field in a circle and its wavelength could be considered a standing wave on the circle. The radius is given by using the relation of the magnetic induced centripetal force equal to the centrifugal,

$$Bqv=mv^2/r .$$

A bigger radius means a larger energy so a larger de Broglie frequency. This is consistent with the other answer.


text:
# Proving that a statistic is not sufficient (uniform case).

Let $X=(X_1,...,X_n)$ be i.i.d. $U(0,\theta)$. How to show that $$\frac{2}{n}\sum_{i=1}^{n}X_i$$ is not a sufficient statistic?

I have already proven that $\max_{i=1,...,n}X_i$ is a sufficient statistic. Also the PDF of the distribution function of the density can be described as $$P_{\theta}(X_n<x)=\left(\frac{x}{\theta_0}\right)^n\cdot I\left(\max_{i=1,...,n}x_i<\theta\right).$$ Can I just use the factorisation theorem and state that $$P_{\theta}(X_n<x)=\left(\frac{x}{\theta_0}\right)^n\cdot I\left(\max_{i=1,...,n}x_i<\theta\right)$$ cannot be written in terms of $\frac{2}{n}\sum_{i=1}^{n}X_i$?

• What is $\theta_0$ in $P_{\theta}(X_n<x)=\left(\frac{x}{\theta_0}\right)^n\cdot I\left(\max_{i=1,...,n}x_i<\theta\right).$ I think the PDF calculation is not valid! Jul 1 '20 at 15:07

A rigourous way to do this is to first show that $\max X_i$ is a minimal sufficient statistic by a corollary of the factorisation theorem, and then it follows immediately that $\frac{2}{n} \sum_{i=1}^n X_i$ is not sufficient, as the minimal sufficient statistic is not a function of it.

• Could you tell me more about that? Feb 23 '16 at 19:46
• The minimal sufficiency criterion states that a statistic $T = t(X)$ is minimal sufficient if and only if $f(x|\theta)/f(y|\theta)$ is independent of $\theta$ $\iff t(x) = t(y)$, where $f$ is the likelihood function. The wikipedia page en.wikipedia.org/wiki/Sufficient_statistic#Minimal_sufficiency) explains the concept pretty well - feel free to post here if you have more questions.
– user93238
Feb 23 '16 at 19:53

If you want to show a statistic is not a sufficient statistic , you can compare it with minimal sufficient statistic. Use the fact that a minimal sufficient statistic is a function of any sufficient statistic.

Define $$T=\max (X_1,\cdots ,X_n)$$ and $$U=\frac{2}{n}\sum_{i=1}^{n} X_i$$. We want to prove $$U$$ is not a sufficient statistics.

Since $$T$$ is minimal sufficient statistic, so it is a function of any sufficient statistic. It is enough to show that $$T$$ is not a function of $$U$$. So conclude $$U$$ is not a sufficient statistic, that is, the approach is:

1. Find a minimal sufficient $$T$$

2. Show that the minimal sufficient is not a function of $$U$$

3. Compare with the fact that a minimal sufficient statistic is a function of any sufficient statistic. So conclude $$U$$ is not a sufficient statistic.

Obviously $$T=\max (X_1,\cdots ,X_n)$$ is not a function of $$U=\frac{2}{n}\sum_{i=1}^{n} X_i$$ but we prove it. $$T$$ is a function of $$U$$ if $$U(a_1)=U(a_2) \Longrightarrow T(a_1)=T(a_2)$$.

So it is enough to find two points that $$U(a_1)=U(a_2)$$ but $$T(a_1)\neq T(a_2)$$ , and hence $$T$$ is not a function of $$U$$ and hence $$U$$ is not a sufficient statistic.

Lets $$a_1=(.2,.2,\cdots,.2,.1,.3)$$ and $$a_2=(.2,.2,\cdots,.2,.15,.25)$$ so $$U(a_1)=U(a_2)$$ but $$T(a_1)=.3\neq T(a_2)=.25$$


text:
## Introduction

Wearable and flexible electronics, such as ion-sensor arrays, biosensor chips, and vital sensor patches1,2,3,4, have attracted considerable attention due to their potential applications in healthcare. Vital sensors must be enabled constantly for continuous healthcare monitoring. This can be achieved by fabricating wearable sensors. Among the various types of wearable and flexible electronics available for vital sensors, piezoelectric pressure sensors have been extensively studied5,6. Generally, these sensors are used for human motion analysis in applications including pulse wave sensors and motion sensors for motion capture7,8,9,10,11,12. The monitoring and analysis of any information on human physiology is significant in the fields of healthcare and academic science. Moreover, in recent years, printing process technologies have attracted considerable attention as new methods for sensor fabrication. When compared with conventional photolithography and vacuum deposition processes, fabrication processes based on printing technology can drastically improve material utilisation and reduce waste while minimising the processing cost13,14,15.

For the realisation of flexible printed pressure sensors. Poly(vinylidene fluoride) (PVDF) and its copolymer, poly(vinylidene fluoride-co-trifluoroethylene) [P(VDF-TrFE)], are some of the most common ferroelectric materials used in piezoelectric sensors9,16,17,18,19,20. For example, ferroelectric/piezoelectric materials are used in ultrasonic transducers21,22, microelectromechanical devices, and actuators23,24,25, as well as in pressure and strain sensors. Owing to their ferroelectricity, these polymers have high dielectric permittivity and significant remnant polarization26,27,28. In particular, the P(VDF-TrFE) copolymer is advantageous in printing processes because it is soluble in polar solvents and can adapt to the solution/printing processes. A sensor fabricated using this copolymer can monitor various vital signs unerringly for 24 h because of the mechanical and chemical stabilities of the copolymer22,29,30,31. Printed, vital P(VDF-TrFE) sensors can be utilised in healthcare applications because the acoustic impedance of P(VDF-TrFE) is considerably close to that of a living organism. However, heretofore, there have been no reports on the monitoring of signals using wearable printed vital sensors. In addition, the wireless monitoring of vital signals has not been studied. For now, PVDF homopolymer precursor has been used to monitor vital signs32,33,34,35,36,37,38. On the other hand, very few studies have reported the detection of vital signals with sensor devices based on printing technologies. Furthermore, recent works on the effects of polar solvents used for the insulating layers in spin casting have demonstrated that a large dipole moment enhances the ferroelectric characteristics; however, there are few studies on the ferroelectricity of the printed copolymer layers for monitoring the human vital signs39. In addition, the detailed effects of the dipole moments of the polar solvents on the printing process have not been reported.

In this paper, we present a fully printed, wearable vital sensor made of ferroelectric polymer for monitoring the human pulse wave/rate on the skin. This sensor is compact (~4 mm2), thin (~3 μm), and sufficiently flexible, and conforms to the skin while providing high pressure sensitivity (~0.025 MPa), fast response time (~0.2 s), superior operational stability, and excellent mechanical fatigue properties. The surface morphology of the printed P(VDF-TrFE) layer, as a pressure-detecting layer, was improved by using an optimum polar solvent; the optimisation realised satisfactory ferroelectricity, which was approximately 7.0 μC cm−2 of polarization, and high pressure sensitivity. The pressure sensor was connected to a specifically designed wireless sensing system for wireless monitoring of the pulse wave/rate. This sensor system can realise the development of novel healthcare devices for continuous health/wellness monitoring applications.

## Results

### Device design and fabrication

The fully printed vital sensor for wearable devices studied in this work is based on poly(3,4-ethylenedioxythiophene):poly(4-styrenesulfonate) PEDOT:PSS thin electrodes and a P(VDF-TrFE) layer. Figure 1a shows the fabrication process of the proposed printed vital sensor. This sensor was fabricated entirely by screen printing processes and can be fabricated in various sizes, as shown in Fig. 1b. The minimum size of the pressure detection area of the fabricated sensor was 4 mm2. A photograph of the vital sensor device laminated on a forefinger is displayed in Fig. 1c. Figure 1d shows the magnified view of the sensor. The cross-sectional scanning electron microscope (SEM) image of the sensor is shown in Fig. 1e; it shows PEDOT:PSS electrodes and a P(VDF-TrFE) layer with average thicknesses of 500 nm and 2 μm, respectively.

### Homogeneity of the printed piezoelectric polymer layer

Figure 2a,b show the chemical structures of the P(VDF-TrFE) and the polar solvents, respectively, which are used in the fabrication of the sensor. In Table 1, the dipole moments (Debye; D) calculated using the Gaussian density functional theory (calculation conditions of the quantum chemistry is as follows: B3LYP/6-31 G(d), Approximate methods/Basis functions) are depicted. The dotted lines are only for guidance. Figure 3a shows the surface profiles of the printed P(VDF-TrFE) layers for each solvent; the profiles of the printed layer depend on the polar solvents used. In particular, the P(VDF-TrFE) layer for which Dimethyl sulfoxide (DMSO), N,N-dimethylformamide (DMF), and Trimethyl phosphate (TMP) were used does not tend to form flat layers during screen printing. On the other hand, Methyl ethyl ketone (MEK) and Cyclohexanone (CHN) can form ideally flat ferroelectric layers on printing. The above results demonstrate that a polar solvent with a dipole moment less than approximately 3 D can realise a flat ferroelectric layer when used with P(VDF-TrFE) during screen printing. The detailed printability of the copolymer solution is depicted in Supplementary Fig. S1. This information is crucial for the fabrication of vital sensors. Surface atomic force microscope (AFM) images of the screen-printed P(VDF-TrFE) layer are displayed in Fig. 3b. The dependence of the morphology and crystallinity of the P(VDF-TrFE) layer on the polar solvents could be explicitly observed. These results clearly indicate that the solvents can improve the microscale surface morphology also, as shown in the inset images of Fig. 3b. Moreover, when the dipole moment increased beyond 3 D, the trend of the Root-Mean-Square (RMS) value of the height direction, which was calculated from the AFM images, worsened (see Fig. 3c). Therefore, it was determined that a polar solvent with a relatively low dipole moment is appropriate for the fabrication of highly uniform printed P(VDF-TrFE) layers. In this work, we suggest a polar solvent most suited to P(VDF-TrFE) based on the surface homogeneity and crystallinity for avoiding leakage paths between the printed electrodes and the P(VDF-TrFE) layer.

### Ferroelectricity and piezoelectric responses

Figure 4a shows the electric field (P-E) loops for the ferroelectric polarisation of the printed vital sensor using various polar solvents. As mentioned above, P(VDF-TrFE) exhibits piezoelectric characteristics owing to its ferroelectricity. Therefore, the P(VDF-TrFE) layer has a piezoelectric voltage response26,27,28. We measured the generated voltages in order to quantify the piezoelectricity. The characteristics of PEDOT:PSS as a printed electrode were similarly calculated and depicted in Supplementary Fig. S2. We had previously confirmed that the resistivity of the PEDOT:PSS layer does not affect the piezoelectricity of the fabricated vital sensor. It should be noted that the P-E loop for which TMP was used in the P(VDF-TrFE) layer could not be measured because it did not have fine uniformity (Fig. 3a). The polarization, P (µC cm−2), and the coercive electric field, E (MV m−1), are shown in Fig. 4b. In accordance with the changes in the dipole moment, the polarization changed from 6.5 to 7.5 µC cm−2 and the coercive field changed from 48 to 53 MV m−1. Polarization value of P(VDF-TrFE) typically depends on its crystallinity. The polarization of the printed capacitor fabricated from the PVDF copolymer with MEK as the solvent was slightly lower than that of others due to the dependence on its crystallinity. These results indicate that the polarization of the P(VDF-TrFE) layer in screen printing depends upon the dipole moments of the polar solvents. In Fig. 4c, the crystallite sizes of the P(VDF-TrFE) layer calculated from the X-ray diffraction (XRD) spectra at (110/200) lattice planes are depicted. The average crystallite size, D, was estimated as follows:

$$D\,(\mathring{\rm A} )=\frac{k{\lambda }_{Cu}}{\beta cos\theta }.$$
(1)

Here, k and λ Cu are the Scherrer constant (k = 0.95) and the wavelength of the X-ray (λ = 1.54 Å, Cu source), respectively. β and θ are the full width at half maximum (FWHM) values (calculated from the diffraction widths of effective and full crystals) of the actual survey and Bragg angles, respectively. The out-of-plane XRD spectra of the printed P(VDF-TrFE) layer are shown in Supplementary Fig. S3; the crystallite sizes of the P(VDF-TrFE) were in correlation with the dipole moments of the solvents, which is in accordance with the tendency of the results in previous research40.

For evaluating the pressure sensitivity of the printed vital sensor, the piezoelectric response was measured by applying 0.025–1.5 MPa pressure. As the CHN solvent could realise the smoothest surface for the P(VDF-TrFE) layer, it was selected for the experiment shown in Fig. 5a displays the photographs of the pressure-sensing setup for the sensor, which uses a mechanical compression testing machine. The piezoelectric response of the sensor is shown in Fig. 5b. Voltage peaks were generated in the positive direction when pressure was applied to the sensor (pressure-on). After the pressure was released, the peaks were generated in the negative direction (pressure-off); the generated voltages on the positive-side corresponded to those on the negative-side. The response time of the sensor (approximately 0.2 s) with the applied pressure is depicted in Supplementary Fig. S4. The plots of the peak positive voltages for various applied pressures are shown in Fig. 5c. The generated voltage indicates a linear dependence on the pressure. This is significant because it demonstrates that the fabricated vital sensor is sensitive to low pressure below 0.1 MPa (inset graph in Fig. 5c); the sensor has sufficient sensitivity for monitoring the human pulse rate and exhibits satisfactory pressure sensitivity41. General piezo films such homo- and copolymers generate voltages only as long as the linear regime of the piezoelectric effect does not exist. Our sensor detected the pressure under the piezoelectric effect and monitored the pressure changes. The equation for estimating the piezoelectric constant, d33, of P(VDF-TrFE) is as follows:

$${d}_{33}=\frac{{V}_{gen}{C}_{P(VDF-TrFE)}}{P}.$$
(2)

Here, Vgen, CP(VDF − TrFE), and P are the generated voltage, capacitance of the sensor, and the applied pressure, respectively. In this calculation, Vgen was estimated using the least-squares method from the slope between the pressure and the generated voltage in Fig. 5c. The measured capacitance of the sensor was 56.4 pF (area of the capacitor (S): 1.5 mm2, dielectric constant (ε): 8.5, thickness (t): 2000 nm). The calculated constant d33 was −22.1 pC N−1, which approximately corresponds to the theoretical value22,42,43,44. It can be clearly deduced from equation (2) that the generated voltage from the P(VDF-TrFE) layer changed depending upon the variable parameter P for the applied pressure. Figure 5d shows voltage responsivity characteristics of the sensor with respect to the rate of application of pressure. The generated voltage does not depend upon the rate of the applied pressure. Our sensor generated an output voltage without dependence on the rate of the applied pressure during the experiments performed in this study. Figure 5e depicts the dependence of the frequency-response characteristics on the pressure. The frequencies of the applied pressure (0.75 MPa) were 0.3, 0.5, and 1 Hz. The response time was approximately 0.2 s, which exceeds the requirement for monitoring of pressure in the blood vessels of the human body18. Finally, the results of the mechanical stability are shown in Fig. 5f. The number of pressure cycles was more than 106. Supplementary Fig. S5 shows the additional results of the flexibility and mechanical durability of the vital sensor; a radius of curvature of 1 mm does not affect the ferroelectricity of the sensor. These results establish that the sensor can acquire sustainable and reliable pulse signals due to its high piezoelectric responsivity and mechanical stability.

### Blood pressure wave monitors

One of the promising applications of the vital sensor is the monitoring of the human pulse wave and its rate for use in healthcare4,8,45. Photographs of the fully printed sensor attached to the wrist and neck are displayed in Fig. 6. The demonstrations involved a healthy female volunteer with approval from the institutional review board of the Yamagata University (no. 29-2). In addition, we confirmed that all experiments were performed in accordance with relevant guidelines and regulations of the institutional review board. The sensor was attached to the skin on the wrist and the neck of the volunteer using a skin-compatible adhesive patch, as shown in Fig. 6a,c. We confirmed that there was no allergic reaction, redness, or damage to the skin when the sensor was attached. The changes in the signal owing to the variations in the pressure associated with the pulse wave from the blood flow are depicted in Fig. 6b,d; from the magnified views provided in the insets of the figure, it can be seen that the changes in the amplitude of the signal with the blood flow can be clearly measured. The pulse rate of the volunteer was determined to be 70 min−1 using the sensor. The human pulse wave spectra are divided into P1, a novice wave, and P2, a reflected wave. P1 occurs during cardiac shrinkage and P2 develops when the novice wave is reflected from a peripheral blood artery46,47. The above-mentioned results, particularly in Fig. 6b,d, establish that the sensor can monitor P1 and P2. By comparing the amplitudes of P1 and P2, the radial artery augmentation index (AI) can be calculated as follows:

$${\rm{AI}}\,( \% )=\frac{P2}{P1}\times 100.$$
(3)

The AI value is effective in determining the hardness of human blood vessels48,49. In this study, the experimental AI value in Fig. 6d was 62.5%, which generally indicates a healthy female. This established that the sensor could acquire information regarding arteriosclerosis and heart diseases effectively. These findings can be utilised for realising the development of healthcare applications using printed sensor devices attached to the skin. Figure 6e displays the designed wireless sensing system consisting of the printed vital sensor and wireless communication circuit for the wireless measurement of the pulse rate. Figure 6f displays a photograph of the wireless communication circuit with a tablet PC. The wireless sensing system, including the vital sensor and the wireless communication circuit, can send pulse wave/rate data to an electric display device such as a tablet PC, as shown in Fig. 6f. Figure 6g presents the circuit diagram of the wireless communication circuit, which uses a non-inverting amplifier along with an operational amplifier (op-amp) and high/low-pass filters. The monitored vital signal is sent to a tablet PC through a wireless Bluetooth Low Energy (BLE) module. In the wireless communication circuit, the amplification factor was set to 10 by using 10 and 100 kΩ resistors along with an op-amp. The sampling frequency of this sensor circuit was set to 10 ms. The schematic of the wireless circuit system with the printed sensor and a seat-type cell battery before attachment to the skin is displayed in Supplementary Fig. S6. In Fig. 6h, the monitoring of the pulse wave/rate by the wireless sensor circuit is demonstrated. Despite using wireless communication, data was clearly displayed on the tablet PC. The amplification factor of the monitored signal strength was approximately 10 by comparing with Fig. 6b. The pulse rate of the volunteer was estimated to be 65 min−1 using the sensor circuit. These results establish that the vital sensor and the developed circuit increase potentials of next-generation technologies, such as wearable electronics, for healthcare applications.

## Discussion

The screen-printed ferroelectric polymer layer in the vital sensor was very smooth owing to the optimisation of the polar solvents. This surface morphology and profile are significant in forming each layer of the printed sensor. TMP could not realise a smooth layer; hence, leakage paths between the printed electrodes and P(VDF-TrFE) layer could not be avoided. The differences in the homogeneity of the printed P(VDF-TrFE) layer were attributed to the dipole moments. The solvents with lower dipole moments could not provide the thermal energy required for the configurational and conformational entropically favored states. Rapid solvent evaporation can form a flat surface because the entropy is not transferred to the P(VDF-TrFE) layer from the solvents. On the other hand, solvents with high dipole moments form rougher surfaces because slow solvent evaporation transfers thermal energy to the P(VDF-TrFE)32,50,51. M. Benz et al. discussed the effect of polar solvents on the roughness of films prepared from PVF2 polymers and predicted that solvents with higher dipole densities might form rougher surfaces52. The improved P(VDF-TrFE) layer achieved satisfactory ferroelectricity, which was approximately 7.0 μC cm−2 of polarisation, and demonstrated obvious piezoelectric responses against an applied low pressure of 0.025 MPa. The sensor exhibited sufficient sensitivity for monitoring the pulse rate and had superior pressure sensitivity. The piezoelectricity of P(VDF-TrFE) depended on its crystallinity. In the fabrication of the P(VDF-TrFE) layer, we reduced the temperature of samples by the more common slow cooling process after annealing at 135 °C for 1 h. Fine piezoelectricity and homogeneity cannot be realised if it is cooled down rapidly53,54,55.

The sensor was attached to a human wrist and neck to demonstrate the sensing and measurement of the pulse rate by real-time scanning. Additionally, the printed sensor was connected to a flexible substrate with a communication amplifier circuit for the wireless detection of the pulse wave/rate. The sensor circuit satisfactorily detected the pulse rate and displayed the data clearly on a tablet PC.

In conclusion, we fabricated a fully printed vital sensor made of P(VDF-TrFE), which can be applied for pulse wave/rate monitoring on the skin. The printed sensor exhibited high responsivity to the applied pressure and mechanical stability. Moreover, we succeeded in wireless monitoring of the pulse wave using the sensor. The information obtained from the fabrication of the fully printed vital sensor used in this study further illustrates the potential of these devices in fabricating novel electronic devices such as wearable applications.

## Methods

### Fabrication process of the fully printed vital sensor

The fully printed vital sensor was fabricated on a 50-μm thick (Q65HA, DuPont) poly(ethylene naphthalate) (PEN) film substrate and fixed to a glass carrier. A cross-linkable poly(4-vinyl-phenol) (PVP) (436224, Sigma-Aldrich) solution consisting of a mixture of PVP and melamine resin (418560, Sigma-Aldrich) with 1-methoxy-2-propyl acetate (Kanto Chemicals 01948-00) as the solvent was spin-coated onto the PEN film as the planarization layer. The lower electrode made of PEDOT:PSS (Clevios SV4 STAB, Heraeus) was formed on the planarization layer by screen printing (MT320T, Micro-tec, Japan), and annealed at 150 °C for 30 min; the thickness was approximately 500 nm. Further, a 2000-nm layer of poly(vinylidene fluoride-co-trifluoroethylene) [P(VDF-TrFE)] (62-010, Piezotech, France; VDF:TrFE molar ratio of 75:25) was formed by screen printing and annealed at 135 °C for 1 h21. P(VDF-TrFE) was dissolved in several polar solvents at concentrations of 10 wt%. The upper electrode of the sensor was formed with PEDOT:PSS using a screen printer and annealed at 135 °C for 30 min. Finally, a 200-nm layer of Cytop (CTX-809A, Asahi Glass, Japan) was spin-coated as the passivation layer and annealed at 100 °C for 10 min.

### Screen printing Process

The electrodes and the P(VDF-TrFE) layer were formed by screen printing. The moving speed of the squeeze was 50 mm min−1. Room temperature was maintained for stable and reliable printing. The clearance between the substrate and mask was 1.2 mm.

### Performance measurements

All electric characteristics were measured under atmospheric conditions using an oscilloscope (MDO3000, Tektronix) and a waveform generator (AFG3101C, Tektronix). The P-E loops were estimated by the Sawyer–Tower method. The interface and surface morphology of the P(VDF-TrFE) layer were observed by SEM (7600-FE, JEOL, Japan) and AFM (5500, Agilent). The piezoelectricity of the sensor was measured by a mechanical compression testing machine (STD-203NB, Imada Seisakusho, Japan). We conducted XRD (SmartLab, Rigaku, Japan) measurements on the P(VDF-TrFE) layer to assess the crystallinity and analyse the crystal structures.

### Fabrication of the wireless communication circuit

The Cu electrode of the wireless communication circuit was fabricated on a flexible polyimide film by photolithography. For the amplifier circuit, an operational amplifier (LT6004, Linear Technology), chip resistors (2 mm × 1.2 mm), and a chip capacitor (2 mm × 1.2 mm) were mounted on the polyimide substrate by soldering. Further, the printed vital sensor and a sheet-type cell battery of 25 mAh (FDK Corporation, Japan) were connected to the substrate of the circuit using conductive tape (ALCARE Co., Ltd., Japan). A wireless BLE module (EYSGJNZWY, Taiyo Yuden Co., Ltd.) was similarly mounted on the substrate for transmitting the measured pulse rate data to a PC tablet. This sensor circuit was attached to the skin near the wrist of a volunteer using a skin-compatible adhesive patch. For the measurement of human pulse rate, the ethics statement in this manuscript includes a sentence confirming that written informed consent was obtained from all subjects.

### Data availability

All data generated or analysed during this study are included in this published article (and its Supplementary Information files). In this manuscript, demonstrations involved a volunteer with approval from the institutional review board of the Yamagata University (no. 29-2). The specific consents from the authors and volunteer have been obtained to publish the information in an open-access online publication. The authors confirmed that all experiments were performed in accordance with relevant guidelines and regulations of the institutional review board.


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# modulus of iota

Straight Lines and Circles. Answer and Explanation: 1. Therefore, the modulus of i is | i | = √(0 + 1²) = √1 = 1. Properties of multiplication. A 10 g l −1 gel formed in 0.25 M KCl has an elastic modulus of 0.32 × 10 4 Pa, while for a κ-carrageenan gel in 0.25 M KCl it is 6.6 × 10 4 Pa. Therefore, $\iota^2 = -1$ When studying Modulus, I was . are all imaginary numbers. 3i, 4i, -i, $$\sqrt[]{-9}$$ etc. Add your answer and earn points. The symbol {eq}i {/eq} is read iota. It includes: - eldoLED® drivers for flicker-free dimming and tunable white - nLight® networked lighting controls and embedded sensors - IOTA® power pack for emergency back-up power The Modulus system was designed with features from the best of Acuity Brands’ control and driver systems. Integral Powers of IOTA (i). Modulus and Conjugate of a Complex Number; Argand Plane and Polar Representation; Complex Quadratic Equations; Similarly, all the numbers that have ‘i’ in them are the imaginary numbers. Powers. The modulus of a complex number by definition is given that z = x + iy, then |z| = √(x² + y²), where x and y are real numbers. dshkkooner1122 dshkkooner1122 ∣w∣=1 ∣ z−i Ex5.2, 3 Convert the given complex number in polar form: 1 – i Given = 1 – Let polar form be z = (cosθ+ sinθ ) From (1) and (2) 1 - = r (cos θ + sin θ) 1 – = r cos θ + r sin θ Comparing real part 1 = r cos θ Squaring both sides if Z is equal to X + iota Y and U is equal to 1 minus iota Z upon Z + iota if modulus of U is equal to 1 then show that Z is purely real 1 See answer harsh0101010101 is waiting for your help. Stack Exchange Network. Subtraction of complex numbers. Solved Examples. Conjugate of complex numbers. The number i, is the imaginary unit. The elastic modulus increases when the ionic concentration increases up to 0.25 M and, at higher concentrations, it decreases due to a salting out effect. Imaginary quantities. Iota, denoted as 'i' is equal to the principal root of -1. Geometrically, that makes since because you can think of i has a unit vector, so it has unit length of 1. Here, {eq}c {/eq} is the real part and {eq}b {/eq} is the complex part. De Moivres Theorem. Equality of complex numbers. Complex numbers. management of the lighting; and an IOTA ® power pack for backup power specified in emergency applications. The modulus, which can be interchangeably represented by \(\left ... Introduction to IOTA. Free Modulo calculator - find modulo of a division operation between two numbers step by step If z and w are two complex numbers such that |zw| = 1 and arg (z) - arg(w) = π/2, then show that zw = -i. Modulus is the distance or length of a vector. Addition and Subtraction. Geometrical Interpretation. Modulus and Argument. Multiplication of complex numbers. Division of complex numbers. Addition of complex numbers. Modulus also supports controls systems with open protocols. But smaller luminaires and Distance and Section Formula. Properties of addition of complex numbers. Modulus takes lighting design to the next level Larger luminaires offer more space to embed LED drivers, sensors, and other technologies. Examples on Rotation.