text:
CHAPTER 130

#### KEY POINTS

• Extreme obesity presents unique cardiorespiratory challenges in the intensive care unit and frustrates the delivery of routine care.

• Extreme obesity leads to a variety of cardiovascular diseases through diverse mechanisms. Therefore, a high index of suspicion for their presence is warranted in the critically ill patient.

• Many patients with extreme obesity suffer from varying degrees of pulmonary hypertension. Potential etiologies include the obesity hypoventilation syndrome, the “overlap syndrome” of coexisting obstructive sleep apnea and chronic obstructive pulmonary disease, chronic venous thromboembolic disease, and left heart failure. Diastolic dysfunction is common and may contribute to this pathology.

• Although simple obesity has relatively minor effects on pulmonary function, extreme obesity may be associated with reductions in forced vital capacity, forced expiration volume in 1 second, and total lung capacity. Arterial hypoxemia may be present, particularly in the supine position.

• Unrecognized sleep-disordered breathing in the critically ill patient may contribute to cardiopulmonary failure. It may also confer increased sensitivity to sedatives and narcotics.

• An increased risk of venous thromboembolism in obesity merits an aggressive approach to prophylaxis.

• Intraabdominal pressure is elevated in obesity, placing the patient at increased risk for the abdominal compartment syndrome.

• Extreme obesity is associated with a significant increase in the percentage of oxygen consumption attributable to the work of breathing. This decreased respiratory reserve results in a predisposition to the development of respiratory failure even after trivial insults.

• Atelectasis is common in the extremely obese postoperative patient and, along with sleep-disordered breathing, may lead to respiratory failure. The early use of noninvasive ventilation in the high-risk postoperative patient may prevent the development of respiratory failure.

• Intubation of the extremely obese patient may be technically challenging because of poor visibility of the glottis and decreased oxygen stores in alveoli from a reduced functional residual capacity.

• Extremely obese patients should be ventilated in the upright or semi-upright position to improve respiratory system compliance and reduce the work of breathing. Positive end-expiratory pressure between 8 and 15 cm H2O may be necessary to prevent atelectasis.

• Because the compliance of the respiratory system is reduced in extreme obesity, a high plateau pressure does not necessarily indicate alveolar overdistention. When using low tidal volume ventilation in the management of the acute respiratory distress syndrome, a plateau pressure of 35 to 40 cm H2O may be acceptable in some patients.

• Ultrasound guidance may be useful in establishing vascular access in the extremely obese patient.

• Numerous unpredictable alterations in pharmacokinetics have been described in obesity. Reference to published guidelines for individual drugs and close monitoring of clinically available serum drug levels are recommended.

• Nutritional support in the form of carefully balanced hypocaloric enteral regimens is recommended.

• Interestingly, while obesity is associated with increased all-cause mortality, the preponderance of evidence suggests that ICU outcomes are not worse in the obese critically ill patient.

An ever-increasing percentage of the inhabitants of developed countries is obese or overweight. This trend includes men and ...

Sign in to your MyAccess profile while you are actively authenticated on this site via your institution (you will be able to verify this by looking at the top right corner of the screen - if you see your institution's name, you are authenticated). Once logged in to your MyAccess profile, you will be able to access your institution's subscription for 90 days from any location. You must be logged in while authenticated at least once every 90 days to maintain this remote access.

Ok

## Subscription Options

### AccessAnesthesiology Full Site: One-Year Subscription

Connect to the full suite of AccessAnesthesiology content and resources including procedural videos, interactive self-assessment, real-life cases, 20+ textbooks, and more


text:
# Javascript scoping makes my head hurt

Written by Piers Cawley on

Who came up with the javascript scoping rules? What were they smoking. Here’s some Noddy Perl that demonstrates what I’m on about:

 my @subs; for my $i (0..4) { push @subs, sub {$i } } print $subs[0]->(); # => 0;  Here's the equivalent (or what I thought should be the equivalent) in Javscript:  var subs = []; for (var i in [0,1,2,3,4]) { subs[i] = function () { return i; } } alert subs[0]() // => 4  What's going on? In Perl, $i is scoped to the for block. Essentially, each time through the loop, a new variable is created, so the generated closures all refer to different \$is. In Javascript, i is scoped to the for loop's containing function. Each of the generated closures refer to the same i. Which means that, to get the same effect as the perl code, you must write:  var subs = []; for (var shared_i in [0,1,2,3,4]) { (function (i) { subs[i] = function () { return i; }; })(shared_i); } subs[0]() // => 0  h3. Dodgy Ruby scoping

I had initially planned to write the example “How it should work” code in Ruby, but it turns out that Ruby’s for has the same problem:

 subs = []; for i in 0..4 subs << lambda { i } end subs[0].call # => 4  Which is one reason why sensible Ruby programmers don't use for. If I were writing the snippet in 'real' Ruby, I'd write:  subs = (0..4).collect { |i| lambda { i } } subs[0].call # => 0  h3. My conclusion

Javascript is weird. Okay, so you already know this. In so many ways it’s a lovely language, but it does have some annoyingly odd corners.

format: plain


text:
1. If x>2

If x>2 then x2-x-6 =
x2-4
can you show with steps please thank you!

2. Re: If x>2

Originally Posted by someguy123
If x>2 then x2-x-6 =
x2-4
can you show with steps please thank you!
Your question is a bit garbled. What's going on in the bottom line? Could you spell it out? (Or you can quickly look at the LaTeX forum.)

-Dan

3. Re: If x>2

sorry the xsquared - 4 is supposed to be directly under i dont know what happened so xsquared-x-6/xsquared-4

4. Re: If x>2

Yes and there is something after "=" ?

5. Re: If x>2

no thats whats supposed to be figured out

6. Re: If x>2

well notice that x²-4=(x-2)(x+2), from here come the idea that 2 or -2 is a zero of x²-x-6. In fact -2 is a zero, so x²-x-6=(x+2).Q(x), and Q(x) is some polynomial, after computation q(x)=x-3.
So the answer is (x-3)/(x-2)

7. Re: If x>2

@someguy123: Be sure you understand why we have to have x > 2. (Actually x can be anything but 2, but let's stick to the original.)

-Dan


text:
# Balmer spectrum

#### FISMAQUIM

Which transition in the hydrogen atom corresponds to the wavelength of the Balmer series of the He+ spectrum (consider that the ion electron is initially at level 4)


text:
# Finding $E(X\mid \min(X,Y))$ where $X,Y$ are independent $U(0,1)$ variables

Let $$X,Y$$ be two independent random variables, uniformly distributed on $$(0,1)$$. I have to compute $$E(X\mid\min(X,Y)=t)$$, but in very educational way, I mean, I have to explain it to my students and I would rather prefer do not use very advanced methods. Is it possible? I was trying to compute common distribution function and then take its density, but I think that density does not exist. So, is there any easy, FORMAL way?

• What do your students know? It's pretty easy to calculate the joint distribution $(X,\min (X,Y))$ if your students know about general measures. It doesn't have density, though, as you say. – WoolierThanThou Feb 19 '20 at 10:09
• Okay, assume that I computed joint distribution and it has no density. How to compute conditional expectation? – mwrooo Feb 19 '20 at 12:31
• Well... you don't need a density, you just need a Markov kernel. If $(X,Y)\sim \mu$ and $\mu(A\times B)=\int_B \nu_y(A)\textrm{d} \gamma(y)$ for some appropriate measure $\gamma$ and family of measure $\nu_y$, then $\mathbb{E}(X|Y)=\int_{\mathbb{R}} x\textrm{d}\nu_Y$ by exactly the same argument that works in the density case. – WoolierThanThou Feb 19 '20 at 14:30
• In your case, if we denote by $\lambda^2$ the 2-dimensional Lesbegue measure and $\lambda_{\Delta}$ the Lesbegue measure on the diagonal lifted from $\mathbb{R}$. Then your measure is $\mu=1_{x\geq y} 1_{0\leq x\leq 1} \textrm{d}\lambda^2+(1-x)\textrm{d}\lambda$, which does have a Markov kernel decomposition with $\nu_y= \frac{1}{2}\delta_y+\frac{1}{2(1-y)}\cdot 1_{[y,1]}\textrm{d}\lambda$. – WoolierThanThou Feb 19 '20 at 14:45
• Okay, I agree, but the problem is how to find that $\gamma$ measure and the family of $\nu_y$? – mwrooo Feb 19 '20 at 14:46

You could try drawing a square something like this as the joint distribution is uniform on the square

The condition $$\min(X,Y)=t$$ corresponds to points on the orange and red lines (orange for $$t=X \le Y$$ and red for $$t=Y \le X$$

Conditioned on this, the probability $$P(X=t \mid \min(X,Y)=t) =\frac12$$ by symmetry, and if not then $$X$$ is uniformly distributed between $$t$$ and $$1$$ so with density $$f(x)=\frac{1}{2(1-t)}\mathbf 1_{t \lt x \lt 1}$$

This makes the conditional expected value $$\mathbb E[X \mid \min(X,Y)=t ]=\frac12 t + \frac12\left(\frac{t +1}{2}\right) = \frac 34 t +\frac14$$

This turns out to be formal, with the corresponding conditional CDF $$\mathbb P(X \le x \mid \min(X,Y)=t )=\left\{ \begin{matrix}0 & \text{ for } x\lt t \\\frac12 & \text{ for } x= t \\ \frac{x+1-2t}{2-2t} & \text{ for } t \lt x\le 1 \\ 1 & \text{ for } 1 \lt x \end{matrix} \right.$$

• Very clear and transparent explanation! Thank you! – NCh Feb 19 '20 at 10:15
• Thank You very much! – mwrooo Feb 19 '20 at 10:42
• @StubbornAtom $P(X\le t \mid \min(X,Y)=t) =\frac12$ is correct, but since $P(X\lt t \mid \min(X,Y)=t) =0$ that would suggest to me $P(X= t \mid \min(X,Y)=t) =\frac12$ too – Henry Feb 19 '20 at 14:43
• @StubbornAtom On the density, you have to stretch a probability of $\frac12$ uniformly over an interval of length $1-t$, which I would have thought would make the density $\frac{1/2}{1-t} =\frac{1}{2(1-t)}$ – Henry Feb 19 '20 at 14:46
• @mwrooo - I would guess $\mathbb E[\min(X,Y)\mid \max(X,Y) =s] = \int\limits_{t=0}^s \frac{t \lambda e^{-t\lambda}}{1-e^{-s\lambda}}\, dt = \frac{1-(1+s\lambda)e^{-s \lambda}}{\lambda\left(1-e^{-s \lambda}\right)}$ – Henry Feb 20 '20 at 23:20


text:
# The Orbital Index

Issue No. 8 | Apr 16, 2019

🚀 🌍 🛰

 We have finally seen a black hole. As has been reported widely, this week saw the much-anticipated release of the first image of a black hole’s silhouette. The image was captured by the Event Horizon Telescope (EHT) using microwave Very Long-Baseline Interferometry, which combines observations from a network of telescopes to obtain resolving power equivalent to an Earth-sized aperture. We’re talking “see a credit card on the surface of the Moon” or “see our solar system from 55 million light-years away” level resolution. The image of galaxy M87’s asymmetric ring matches predictions based on simulations of “strong gravitational lensing of synchrotron emission from a hot plasma orbiting near the black hole event horizon,” showing that the “observed image is consistent with expectations for the shadow of a spinning Kerr black hole as predicted by general relativity.” This compelling empirical support for General Relativity at the extremes of gravitation is likely the most important result. But, that in no way diminishes the technical achievement, with petabytes of data analyzed, measurements synchronized worldwide to the microsecond, and locations of the telescopes known to within a centimeter across the globe. Related: a video of stars, some moving at 3% the speed of light, orbiting the black hole in the center of our Milky Way.
 Beresheet crashes into the moon. 😢 SpaceIL lost contact with the lander just 149 m above the moon’s surface after the spacecraft’s inertial measurement unit (IMU) failed at an altitude of 14 km. Final telemetry indicated velocities far too great for any possibility of an intact landing. Plans are already underway for a second attempt by the Israeli non-profit. Beresheet, which joins 123,052 kg of crashed human spacecraft on the surface of the moon, was a competitor for the Google Lunar Xprize and SpaceIL was awarded $1 million for their efforts (the deadline for the full$20m prize passed a year ago). Its primary science mission was to analyze the magnetic field near its intended landing site and was able to record some measurements during descent. Other teams have also continued their work on a lunar soft landing. Hakuto (from Japanese company ispace) is slated for launch in 2020, while Moon Express, who raised an additional $12.5 million in September, hopes to launch their first demo mission on a Rocket Lab Electron rocket in late 2020.  The first commercial launch of the Falcon Heavy was a resounding success with all three booster cores landing successfully. During the live feed, there was a brief cut to several odd-looking frames which turned out to be from a camera inside the LOX tank [video with historical comparison to Apollo in-tank footage]. This feed is used to confirm that firing of cold gas thrusters has resulted in LOX settling at the bottom of the tank before engines are restarted. In addition to the successful landings, SpaceX recovered both fairing halves after they splashed down and has plans to refly them as part of the upcoming launch of their first batch of Starlink Internet satellites in May. It is unclear if they will continue to attempt to catch fairings in the future should this refurbishment process prove successful. Unfortunately, the center core was lost in choppy seas while returning from its landing location 1000 km downrange (the longest downrange landing attempted yet). The center core hardware differs from a standard Falcon 9 booster and wasn’t able to be secured to the drone-ship by the Octagrabber robot, leading to the loss. The side boosters will be reused in June after intensive inspection for the STP-2 mission, a DoD payload consisting of 6 smallsats, 8 cubesats, and 11 other small spacecraft. In other SpaceX news: NASA awarded SpaceX a contract to fly its Double Asteroid Redirection Test (DART) planetary-defense mission (which will smash into "Didymoon," the moon of near-Earth asteroid Didymos—we’ll talk more about this mission in a future issue).  More news in brief. Planet signed a$6.7 million contract with NASA to provide Earth imagery for climate research; Intelsat’s 29e geostationary satellite suffered a seemingly catastrophic fuel leak, as caught in this video taken by the ExoAnalytic telescope network; and Stratolaunch, founded by the late Paul Allen, flew for the first time.
 Etc.If you're in the Bay Area on May 6th, consider attending the second annual Space Tech Symposium at Berkeley. Last year’s event was excellent.NASA selected two university-led teams to explore automated technologies for keeping habitats operational when not occupied by astronauts. NASA’s proposed Lunar Gateway will be occupied part-time and must be able to maintain itself, an opportunity for robotic science and maintenance. JAXA also just announced plans to semi-autonomously build lunar bases.An Atlantic piece on the logistics for collecting the five petabytes of data recorded at telescopes around the world that the EHT required to produce its first-ever image of a black hole (and the shipping clerk who made it happen).A really excellent visualization and exploration of the Fourier Transform.Dark Matter is probably not made up of many tiny primordial black holes, as Hawking suggested it might be. Tiny black holes resulting from density fluctuations in the early universe could have explained dark matter without the need for new physics. “Investment into space companies hit a record high in 2018, exceeding \$3 billion with no sign of an imminent downturn,” with VC investment growing 22%.The full open-access results are out for NASA’s Twins Study that compared Scott Kelly, who had a 340-day space mission, to his identical twin brother Mark Kelly, who stayed on Earth. Over 90% of the changes reversed themselves within six months of returning to Earth, but a few remained, including minor radiation-induced chromosomal mutations. While in space, Scott experienced changes in gene expression and lengthening of his telomeres, which would seem like a good thing, except they ultimately ended up shorter when he returned to Earth.A report from the Center for Advanced Defense Studies (C4ADS) about Russian spoofing and jamming of GNSS satellite signals (e.g. GPS, GLONASS, Galileo, and BeiDou) to obscure the location of Putin and other strategic locations.
 A wide field X-ray image of M87's galactic core containing the black hole 'Pōwehi' (but still smaller than one of Hubble’s pixels)


text:
# Singular fit for model and want to retain random intercept

I've read many posts about singular fit issues and this: https://bbolker.github.io/mixedmodels-misc/glmmFAQ.html#singular-models-random-effect-variances-estimated-as-zero-or-correlations-estimated-as---1 , however I've seen that most of the advice revolves around simplifying random effects. In my case I have just one random intercept that I believe is necessary based on the experimental design so I'm not sure what the best way to proceed is. I am looking at success on a task where the outcome is binary 1=solved, 0= not solved and how other variables contribute to success. I have 13 subjects that each have 10-12 trials so I believe it's important to have the individual as a random intercept when modeling success. The other variables are summarized here:

• Sex- categorical, 2 levels
• Zoo- categorical, 2 different zoos
• Age- continuous
• Persistence- continuous, proportion of total time engaged with task per trial
• Neophilia- continuous, latency to approach task in very first trial
• Diversity- continuous, total number of actions employed per trial.

My full model is newdmod<-glmer(Solved~Sex+Zoo+Age+Persistence+Neophilia+Diversity+(1|Individual),door, family=binomial) (output below) I get a singular fit error with this model and I can see that the variance for my random intercept is almost 0. As I understand it, this would indicate that my random intercept should be removed, but due to my experimental design I think it's important to keep it in. I've also checked for collinearity of this full model and none of the fixed effects have high VIFs.

Generalized linear mixed model fit by maximum likelihood
(Laplace Approximation) [glmerMod]
Family: binomial  ( logit )
Formula:
Solved ~ Sex + Zoo + Age + Persistence + Neophilia + Diversity +
(1 | Individual)
Data: door

AIC      BIC   logLik deviance df.resid
99.7    123.7    -41.8     83.7      141

Scaled residuals:
Min      1Q  Median      3Q     Max
-4.6817  0.0171  0.0910  0.2962  3.8144

Random effects:
Groups     Name        Variance  Std.Dev.
Individual (Intercept) 2.774e-18 1.665e-09
Number of obs: 149, groups:  Individual, 13

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  1.94544    1.04467   1.862 0.062568 .
SexM         1.68561    0.72139   2.337 0.019459 *
ZooRGZ      -1.04102    0.59511  -1.749 0.080240 .
Age         -0.04028    0.02218  -1.816 0.069426 .
Persistence  8.19243    1.60335   5.110 3.23e-07 ***
Neophilia   -0.03757    0.01167  -3.220 0.001283 **
Diversity   -0.36305    0.10136  -3.582 0.000341 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr) SexM   ZooRGZ Age    Prsstn Neophl
SexM         0.053
ZooRGZ      -0.275 -0.095
Age         -0.754 -0.219 -0.075
Persistence  0.108  0.356 -0.169 -0.291
Neophilia   -0.557 -0.564 -0.081  0.689 -0.443
Diversity   -0.508 -0.241  0.260  0.314 -0.648  0.329
optimizer (Nelder_Mead) convergence code: 0 (OK)
boundary (singular) fit: see ?isSingular

Interestingly, if I remove one of the fixed variables (Zoo, the one with the highest p value) the singularity problem disappears and the random intercept has variance again.

Generalized linear mixed model fit by maximum likelihood
(Laplace Approximation) [glmerMod]
Family: binomial  ( logit )
Formula:
Solved ~ Sex + Age + Persistence + Neophilia + Diversity + (1 |
Individual)
Data: door

AIC      BIC   logLik deviance df.resid
100.4    121.5    -43.2     86.4      142

Scaled residuals:
Min      1Q  Median      3Q     Max
-5.5278  0.0165  0.0816  0.2958  3.6741

Random effects:
Groups     Name        Variance Std.Dev.
Individual (Intercept) 0.3006   0.5483
Number of obs: 149, groups:  Individual, 13

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  1.60015    1.16986   1.368 0.171372
SexM         1.64549    0.80376   2.047 0.040634 *
Age         -0.04468    0.02510  -1.780 0.075003 .
Persistence  7.89091    1.63103   4.838 1.31e-06 ***
Neophilia   -0.04018    0.01314  -3.058 0.002230 **
Diversity   -0.34100    0.10304  -3.309 0.000935 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
SexM        -0.008
Age         -0.804 -0.199
Persistence  0.023  0.331 -0.284
Neophilia   -0.602 -0.525  0.680 -0.435
Diversity   -0.466 -0.223  0.323 -0.564  0.336

So my question is whether it is legitimate to simplify the full model by removing a fixed effect based on the p value with the justification that this variable is contributing least to variation in success? Especially since this fixes the singularity issue and given that I would like to keep the random effect. Or could my result with the full model provide enough support to argue that there is not enough individual variation so I don't have to account for repeated measures? Any and all help would be appreciated! I'm also happy to provide more information if it is helpful!

Edit: Adding GLM output for glm(Solved ~ Sex + Zoo + Age + Persistence + Neophilia + Diversity + Individual, family=binomial)

Call:
glm(formula = Solved ~ Sex + Zoo + Age + Persistence + Neophilia +
Diversity + Individual, family = binomial, data = door)

Deviance Residuals:
Min        1Q    Median        3Q       Max
-2.42591   0.00001   0.05371   0.34488   2.72956

Coefficients: (4 not defined because of singularities)
Estimate Std. Error z value Pr(>|z|)
(Intercept)         60.7768   114.3250   0.532  0.59499
SexM               -43.7212    92.9102  -0.471  0.63794
ZooRGZ             -63.5269   123.3433  -0.515  0.60652
Age                  0.1188     0.3874   0.307  0.75917
Persistence          7.8394     1.9450   4.031 5.56e-05 ***
Neophilia           -0.6940     1.2860  -0.540  0.58942
Diversity           -0.3727     0.1173  -3.178  0.00148 **
IndividualAjay     113.6391   229.9300   0.494  0.62114
IndividualBamboo   -65.2868   130.9487  -0.499  0.61808
IndividualBatu      97.5497   195.7502   0.498  0.61825
IndividualChandra  -61.2992   121.6868  -0.504  0.61444
IndividualDoc      105.7723   207.0135   0.511  0.60939
IndividualKandula  -15.1359    24.6599  -0.614  0.53936
IndividualKirina    20.5416  2647.9499   0.008  0.99381
IndividualMali      20.6377  2600.2857   0.008  0.99367
IndividualRex            NA         NA      NA       NA
IndividualRomani         NA         NA      NA       NA
IndividualSiri           NA         NA      NA       NA
IndividualTarga          NA         NA      NA       NA
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for binomial family taken to be 1)

Null deviance: 157.574  on 148  degrees of freedom
Residual deviance:  69.857  on 134  degrees of freedom
AIC: 99.857

Number of Fisher Scoring iterations: 18

• Please also add the summary output of the model glm(Solved ~ Sex + Zoo + Age + Persistence + Neophilia + Diversity + Individual, family=binomial) Also, please explain what all the variables are. Mar 4 at 20:11
• @RobertLong I have added that information to the post, thanks! Mar 4 at 20:24
• What is the Zoo variable ? Also, you didn't include Individual in the glm model. Mar 4 at 20:32
• @RobertLong My apologies, should be fixed now! These are animals at two different zoos, so there are two different zoo categories. Mar 4 at 20:42
• Can you provide a link to the dataset ? Mar 5 at 8:45

You might be trying to push these data a bit too far.

First, unpenalized logistic regression models typically need about 15 cases in the least-prevalent outcome class per predictor to avoid overfitting. With only 33 trials showing Solved = 0, you should be wary of evaluating more than 2 or 3 predictors. If you count the random-intercept effect as a single predictor, you have 7 predictors in your model. Also, of the 13 Individuals, 4 account for 23 of those 33 trials, 2 have no trials with Solved = 0, and 4 more have only 1 trial each with Solved = 0. Be cautious.

Second, each Individual has a single value of each of Sex, Age, and Neophilia, in addition to Zoo, for all trials.* Removing any one of those predictors from the original full mixed model removes the singularity problem. I can't say precisely why that is the case, but it presumably has to do with the ratio of the number of such predictors (4) to the number (13) of separate individuals. Perhaps once you have identified the associations of those 4 fixed predictors with outcome, there isn't much left to further associate with the Individuals as random effects.

whether it is legitimate to simplify the full model by removing a fixed effect based on the p value ...

You should be evaluating fewer predictors in any event. What you are proposing is similar to backward stepwise selection of predictors, probably the least objectionable of stepwise approaches. It would be better to use your knowledge of the subject matter to choose the most crucial fixed-effect predictors to include from the beginning.

Or could my result with the full model provide enough support to argue that there is not enough individual variation so I don't have to account for repeated measures?

If you perform the no-Zoo model both with the Individual random effect and by completely ignoring the Individual values (in a glm), you get very different results. The magnitudes of the fixed-effect coefficients are about 10 times larger in the no-Zoo mixed model than in a no-Zoo glm model that omits Individual. That raises red flags.

I suspect that this has to do with overfitting. For example, the fixed-effect coefficients of models restricted to Persistence, Neophilia and Diversity are quite similar whether Individual is ignored (with glm) or included as a random-effect intercept (with glmer).

Look carefully at your data and apply your knowledge of the subject matter to decide how best to proceed.

*That's why the last 4 coefficients for levels of Individual in the glm model without random effects are NA. Had you listed Individual as the first predictor in the model formula, then those 4 predictors would have had the NA values instead.

• Thank you so much @EdM! This makes a lot of sense. Based on your feedback I will only include the crucial variables of interest for my hypotheses and take out the variables that I wanted to "control for," sex, age, and zoo. Do you think it's best to state that I pared down predictor variables due to singularity issues in a manuscript? Or something like "in order to avoid overfitting the logistic regression model, I eliminated these variables to focus on those of interest" ? It's a bit strange because I included all variables in another part of the analysis where the outcome wasn't binomial. Mar 9 at 14:52
• @Sjacobson1112 the usual rule of thumb is less stringent for continuous outcomes, about 15 total cases per predictor. See section 4.4 of Harrell's Regression Modeling Strategies book or course notes. Depending on details of your other analyses, simply citing the risk of overfitting should thus be OK. If your other analyses had lower case/predictor ratios, however, they might also be overfit and you should evaluate, for example, by bootstrapping. The Harrell references point the way.
– EdM
Mar 9 at 15:55
• Really appreciate your help @EdM and thank you for the links to the resources! Mar 9 at 16:36
• (+1) Nice answer !! @Sjacobson1112 if this answers you question please consider marking it as a the accepted answer (and upvoting)....it's how this site works best. If not, the please let us know why :) Mar 9 at 22:13


text:
# Surprising interaction of DistributeDefinition and ParallelNeeds in 11.3

Bug introduced in 11.3.0

(CASE:4039452)

I believe I am seeing a new sort of behavior around some parallel functionality in Mathematica 11.3, because code that used to function properly in earlier versions no longer works. There's a bit of setup required, which is probably more convoluted than it strictly needs to be since it requires building some packages.

packageA =
ToString[
Unevaluated[
BeginPackage["A"];
AA = 17;
EndPackage[];
],
InputForm];
dir = CreateDirectory[];
WriteString[FileNameJoin@{dir, "A.wl"}, packageA];

Let's launch subkernels and make sure each one has the right $Path: LaunchKernels[]; PrependTo[$Path, dir];
With[{path = $Path}, ParallelEvaluate[$Path = path]];

Now, let's set a value for AA on our local kernel:

AA = 18;
(* 18 *)

And let's load the A in each subkernel:

ParallelNeeds["A"];
ParallelEvaluate[AA]
(* {17, 17, 17, 17} *)

OK, they would never have a reason to see the new value of AA. Let's give them one:

DistributeDefinitions["A"]
(* {"A"} *)

But nothing seems to have happened!

ParallelEvaluate[AA]
(* {17, 17, 17, 17} *)

I'd expect the output to be {18, 18, 18, 18}. What am I misunderstanding? And am I mistaken that the output would have been {18, 18, 18, 18} in prior versions?

• I get all 18s in V11.2, all 17s in V11.3. – Michael E2 Mar 29 '18 at 19:15

I think this is likely to be a bug.

Historically, DistributeDefinitions has always ignored all symbols in System and various other system package contexts.

The handling of the excluded contexts has been reorganized in 11.3, in particular there now is

?ParallelProtected$ExcludedContexts $ExcludedContexts is the list of contexts to always exclude from DistributeDefinitions.

which is generated on the fly and unfortunately it looks like our package context ends up on the blacklist for some reason:

MemberQ[ParallelProtected$ExcludedContexts, "A"] (* True *)  The following works as expected Block[{ParallelProtected$ExcludedContexts =
DeleteCases[ParallelProtected$ExcludedContexts, "A"]}, DistributeDefinitions["A"] ]; ParallelEvaluate[AA] (* {18, 18, 18, 18} *)  • Interesting. Is the reason for the addition of ParallelProtected$ExcludedContexts to save memory? – QuantumDot Apr 2 '18 at 0:44
• I think it's just to have more control and flexibility. Previous versions also had a similar blacklist for system contexts, but it was static (set up at kernel initialization). – ilian Apr 2 '18 at 20:36

Another straightforward workaround is to just use ParallelEvaluate @* Needs in the place of ParallelNeeds, like so:

ParallelNeeds["A"];
ParallelEvaluate[AA]

(* {17, 17, 17, 17} *)

AA = 18
(* 18 *)

DistributeDefinitions["A"];
ParallelEvaluate[AA]
(* {18, 18, 18, 18} *)
`


text:
# A course on homotopy (type) theory

This semester my colleague Jaka Smrekar and I are teaching a graduate course on homotopy theory and homotopy type theory. The first part was taught by Jaka and was a nice review of classical homotopy theory leading up to Quillen model categories. In the second part I am covering basic homotopy type theory.

The course materials are available at the GitHub repository homotopy-type-theory-course. The homotopy type theory lectures are also recorded on video.


text:
# Question af574

Oct 23, 2017

3.17%

#### Explanation:

The idea here is that you need to use the known composition of the solution to figure out the mass of glucose, the solute, present in exactly $\text{100 g}$ of the solution $\to$ this is the solution's percent concentration by mass.

To do that, you can use the fact that solutions are homogeneous mixtures, which implies that they have the same composition throughout.

So if you dissolve $\text{7.79 g}$ of glucose in $\text{237.7 g}$ of water, you can say that the mass of the solution will be equal to

overbrace("7.79 g")^(color(blue)("mass of solute")) + overbrace("237.7 g")^(color(blue)("mass of solvent")) = overbrace("245.5 g")^(color(blue)("mass of solution"))

So, you know that this solution contains $\text{7.79 g}$ of solute in $\text{245.5 g}$ of solution, so you can say that $\text{100 g}$ of this solution will contain

100 color(red)(cancel(color(black)("g solution"))) * "7.79 g glucose"/(245.5color(red)(cancel(color(black)("g solution")))) = "3.17 g glucose"#

This means that the solution's percent concentration by mass will be equal to

$\textcolor{\mathrm{da} r k g r e e n}{\underline{\textcolor{b l a c k}{\text{% m/m = 3.17% glucose}}}}$

The answer is rounded to three sig figs, the number of sig figs you have for the mass of glucose.


text:
# Looping is recursion

## Fork this

https://github.com/iloveponies/looping-is-recursion

Here are the instructions if you need them. Be sure to fork the repository behind the link above.

## Starting out

Clojure has a form called loop. But it’s not actually a looping construct, it is recursive in nature. Let’s start with some examples.

This is the standard recursive factorial:

(defn recursive-factorial [n]
(if (zero? n)
1
(* n (recursive-factorial (dec n)))))

Like we’ve seen in the chapter on recursion, this is a linear recursive process, that is, it constructs an expression linear in size to its input, n. When the base case is reached, we will have the expression (* n (* (dec n) (* ... (* 3 (* 2 (* 1 1))) ...))). This also means that the amount of memory taken by recursive-factorial is $$\mathcal{O}(n)$$.

We can make this computation more efficient by using tail recursion. A function is tail recursive when its return value is calculated directly by a recursive call. recursive-factorial is not tail recursive, because the value of the recursive call (recursive-factorial (dec n)) is not returned directly, but given to *.

We’ll now introduce a tail recursive version of recursive-factorial and then look at how its evaluation differs from the evaluation of recursive-factorial.

In order to make factorial tail-recursive we introduce an accumulator.

(defn helper [acc n]
(if (zero? n)
acc
(helper (* acc n) (dec n))))

(defn accumulating-factorial [n]
(helper 1 n))

helper uses the return value of its recursive call directly as its return value. That means it is tail recursive. Let’s see how it evaluates:

    (accumulating-factorial 5)
;=> (helper 1 5)
;=> (helper (* 1 5)   (dec 5)) => (helper 5 4)
;=> (helper (* 5 4)   (dec 4)) => (helper 20 3)
;=> (helper (* 20 3)  (dec 3)) => (helper 60 2)
;=> (helper (* 60 2)  (dec 2)) => (helper 120 1)
;=> (helper (* 120 1) (dec 1)) => (helper 120 0)
;=> 120

Now we see that evaluating helper does not grow the expression we are computing. This is because we do not add any structure around the recursive call. That keeps the structure of the returned expression constantly (helper ...) and the parameters vary. Since we can’t build the computation as a recursive expression, we’re instead computing each step explicitly into the acc accumulator.

This is a linear iterative process or just linear iteration (compared to linear recursion). It is indeed similar to iteration in imperative languages.

Let’s rewrite factorial one more time, now using a new construct called recur. recur means “recursively call this function (that we’re in)”, with the additional restriction that the recursion must be tail recursion.

(defn recur-factorial [number]
(let [helper (fn [acc n]
(if (zero? n)
acc
(recur (* acc n) (dec n))))]
(helper 1 number)))

Here we’ve replaced the recursive call helper with recur. Since recur can only occur in a tail position (that is, a call whose return value is directly returned), the compiler knows the recursion is actually iteration, and can compile it into a simple loop. This is called tail-call optimization.

We have also moved helper inside recur-factorial and defined it with fn. This was not possible before, because an anonymous function has no name to call recursively. But as we now make the recursive call with recur, we can use fn. The benefit of this is that the helper function was never supposed to be visible to anyone and now it isn’t.

Again, because recur guarantees tail-call optimization, it can be present only in a tail position. While this might seem awkward, it’s an advantage too. When we’ve placed recur in a non-tail position where Clojure can not perform tail-call optimization, the compiler will give us an error. This indicates that our request to optimize the tail call is not possible. If the compiler allowed recur outside tail positions (and simply performed regular recursion), we would not know whether tail-call optimization actually took place or not.

In short: recur guarantees tail-call optimization by requiring that the call to it is in an optimizable position.

### Exercise 1

Write the function (power n k) that computes the mathematical expression $$n^k$$.

(power 2 2)  ;=> 4
(power 5 3)  ;=> 125
(power 7 0)  ;=> 1
(power 0 10) ;=> 0

### Exercise 2

Compute the last element of a sequence.

(last-element [])      ;=> nil
(last-element [1 2 3]) ;=> 3
(last-element [2 5])   ;=> 5

### Exercise 3

Write the function (seq= a-seq b-seq) that compares two sequences for equality.

(seq= [1 2 4] '(1 2 4))  ;=> true
(seq= [1 2 3] [1 2 3 4]) ;=> false
(seq= [1 3 5] [])        ;=> false

Because defining the sort of helper functions like helper in our factorial is quite usual in functional programming, there is a utility called loop for this. The previous code could be written like this:

(defn loopy-factorial [down-from]
(loop [acc 1
n down-from]
(if (zero? n)
acc
(recur (* acc n) (dec n)))))

Let’s dissect that. A loop begins with a sequence of bindings, just like in a let. As in let, you can use destructuring in the names.

  (loop [acc 1
n down-from]

This introduces the variables acc and n and gives them initial values. n gets its value from the parameter loopy-factorial.

After this comes the body of the loop, which is exactly the same as the body of the helper function above:

    (if (zero? n)
acc
(recur (* acc n) (dec n)))))

Inside a loop we can think of a recur meaning “go to the start of the loop, and give the variables these new values”. So after that recur call the variable n gets the new value (dec n), and acc gets the new value (* n acc). That is, calling recur either calls the function iteratively, or iterates a loop, whichever is innermost.

This kind of corresponds to the following Java loop (if you want to look at it that way):

int n = number;
int acc = 1;
while (true) {
if (n <= 1) {
break;
} else {
acc = n * acc;
n = n - 1;
}
}
return acc;

### Exercise 4

Implement the function (find-first-index [predicate seq]) that returns the first index in seq for which predicate returns true, or nil if no such index exists.

(find-first-index zero? [1 1 1 0 3 7 0 2])                    ;=> 3
(find-first-index zero? [1 1 3 7 2])                          ;=> nil
(find-first-index (fn [n] (= n 6)) [:cat :dog :six :blorg 6]) ;=> 4
(find-first-index nil? [])                                    ;=> nil

### Exercise 5

Implement the function (avg a-seq) that computes the average of a sequence.

(avg [1 2 3])   ;=> 2
(avg [0 0 0 4]) ;=> 1
(avg [1 0 0 1]) ;=> 1/2 ;; or 0.5
Hint: You need to keep track of multiple things in the loop.

### Exercise 6

Write the function (parity a-seq) that takes in a sequence and returns a set of those elements that occur an odd number of times in the sequence.

(parity [:a :b :c])           ;=> #{:a :b :c}
(parity [:a :b :c :a])        ;=> #{:b :c}
(parity [1 1 2 1 2 3 1 2 3 4] ;=> #{2 4}

Hint: Recall the fuction (toggle set elem) from Structured data

(toggle #{1 2 3} 1) ;=> #{2 3}
(toggle #{2 3} 1) ;=> #{1 2 3}

### Exercise 7

Write the function (fast-fibo n) that computes the nth fibonacci number using loop and recur. Do not use recursion.

(fast-fibo 0) ;=> 0
(fast-fibo 1) ;=> 1
(fast-fibo 2) ;=> 1
(fast-fibo 3) ;=> 2
(fast-fibo 4) ;=> 3
(fast-fibo 5) ;=> 5
(fast-fibo 6) ;=> 8

Hint: to avoid recursion, you need to keep track of $$F_{n-1}$$ and $$F_n$$ in the loop.

### Exercise 8

Write the function (cut-at-repetition a-seq) that takes in a sequence and returns elements from the sequence up to the first repetition.

(cut-at-repetition [1 1 1 1 1])
;=> [1] doesn't have to be a vector, a sequence is fine too
(cut-at-repetition [:cat :dog :house :milk 1 :cat :dog])
;=> [:cat :dog :house :milk 1]
(cut-at-repetition [0 1 2 3 4 5])
;=> [0 1 2 3 4 5]

Hint: Remember that conjing onto a vector appends the element.

Hint: A set might be helpful

## Performance viewpoint

Tail recursion is efficient. This is because the compiler can replace it with a goto (this is called tail-call optimisation. So a tail-recursive function is about exactly as fast as the corresponding loop.

However, this doesn’t exactly apply in the Java Virtual Machine. This is because the security model of the JVM makes tail-call optimisation hard. This is why Clojure uses the recur construct: it is guaranteed that a call to recur gets optimised. I’ll say that again. When you use recur, Clojure generates an actual loop as JVM bytecode.

The one →


text: Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces.All this is subsumed by the *-operations of C*-algebras.. One may also define a conjugation for quaternions and split-quaternions: the conjugate of + + +. Expand, convert between forms, extract real and imaginary parts, visualize. How to work with complex numbers, expressions. Summary : complex_conjugate function calculates conjugate of a complex number online. The complex conjugate transpose of a matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal. 2004 (5.1). The Wolfram Language transparently works with complex variables throughout, not only numerically, but also symbolically\[LongDash]often relying on original results to … To embed a widget in your blog's sidebar, install the Wolfram|Alpha Widget Sidebar Plugin, and copy and paste the Widget ID below into the "id" field: We appreciate your interest in Wolfram|Alpha and will be in touch soon. Get the free "Finding Conjugates" widget for your website, blog, Wordpress, Blogger, or iGoogle. Use this online algebraic conjugates calculator to calculate complex conjugate of any real and imaginary numbers. Online Integral Calculator » Solve integrals with Wolfram|Alpha. The representation of a complex number in terms of its Cartesian coordinates in the form , where is the imaginary unit, is called the algebraic form of that complex number. Technology-enabling science of the computational universe. A1. Tutorial for Mathematica & Wolfram Language. Input your complex or real binomial or polynomial in the box. Complex number calculator: complex_number. Stay on top of important topics and build connections by joining Wolfram Community groups relevant to your interests. Wolfram Science. 1988. By … How to work with complex numbers, expressions. The complex conjugate is implemented in the Wolfram Language as Conjugate[z]. Conjugation is the process of taking a complex conjugate of a complex number, complex matrix, etc., or of performing a conjugation move on a knot. To embed this widget in a post, install the Wolfram|Alpha Widget Shortcode Plugin and copy and paste the shortcode above into the HTML source. Hermitian Conjugate of an Operator First let us define the Hermitian Conjugate of an operator to be . Taking the complex conjugate Now taking the Hermitian conjugate of . @misc{reference.wolfram_2020_conjugate, author="Wolfram Research", title="{Conjugate}", year="2004", howpublished="\url{https://reference.wolfram.com/language/ref/Conjugate.html}", note=[Accessed: 18-January-2021 The complex number calculator only accepts integers and decimals. Computerbasedmath.org » Join the initiative for modernizing math education. Revolutionary knowledge-based programming language. Browse other questions tagged summation wolfram-alpha-queries or ask your own question. Step-by-step Solutions » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. How to work with complex numbers, expressions. I try the cartesian form, the polar form and the exponential form but nothing happens. This is the conjugate. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). Complex_conjugate function calculates conjugate of a complex number online. Plot complex numbers?. Assuming "complex conjugate of" is a math function | Use "complex conjugate" as a function property instead. Convert a Complex Number to Polar and Exponential Forms - Calculator. Knowledge-based, broadly deployed natural language. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Use this online algebraic conjugates calculator to calculate complex conjugate of any real and imaginary numbers. is − − −. Optimize systems of symbolically defined lenses and mirrors, test optical components with built-in image processing or data analysis functions and calculate complex … Expand, convert between forms, extract real and imaginary parts, visualize. For example, the complex conjugate of X+Yi is X-Yi, where X is a real number and Y is an imaginary number. Central infrastructure for Wolfram's cloud products & services. (1988). Wolfram Community forum discussion about Calculate and plot Fourier spectrum using W ... Wolfram Knowledgebase Curated computable knowledge powering Wolfram|Alpha. Complex modulus: complex_modulus. This can come in handy when simplifying complex expressions. Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Although, I am also confused with solving this, given that there is a conjugate in the equation. All applicable mathematical functions support arbitrary-precision evaluation for complex values of all parameters, and symbolic operations automatically treat complex variables with full … complex_conjugate online. ]}, @online{reference.wolfram_2020_conjugate, organization={Wolfram Research}, title={Conjugate}, year={2004}, url={https://reference.wolfram.com/language/ref/Conjugate.html}, note=[Accessed: 18-January-2021 In case of complex numbers which involves a real and an imaginary number, it is referred to as complex conjugate. Transpose[list] transposes the first two levels in list . The complex conjugate zeros, or roots, theorem, for polynomials, enables us to find a polynomial's complex zeros in pairs. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. x? The operation also negates the imaginary part of any complex numbers. Click "Submit". Conjugate[z] or z\[Conjugate] gives the complex conjugate of the complex number z. A complex number can be visually represented using a two-dimensional Cartesian coordinate system as an ordered pair of real numbers on the complex plane. Wolfram Natural Language Understanding System. This is a homomorphism because phi_x(g)phi_x(h)=xgx^(-1)xhx^(-1)=xghx^( … This is a homomorphism because phi_x(g)phi_x(h)=xgx^(-1)xhx^(-1)=xghx^( … Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. Complex conjugate: complex_conjugate. Technology-enabling science of the computational universe. Let's look at an example to see what we mean. Complex Number Calculator. Define complex integral in Wolfram|Alpha?. Updated in 2003 (5.0) A1. ComplexExpand[expr] expands expr assuming that all variables are real. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. The plots make use of the full symbolic capabilities and automated aesthetics of the system. Worksheets on Complex Number. quadratic formula calculator,wolfram alpha,complex number conversion,complex number operations calculator,complex conjugate calculator,dividing complex number calculator,simplify complex number calculator,complex number calculator with square root.Complex Number Calculator Solution To see more detailed work, try our algebra solver Conjugation also has a meaning in group theory. complex_conjugate online. The Overflow Blog The Overflow #47: How to lead with clarity and empathy in the remote world. Wolfram Language & System Documentation Center. Complex Number Lesson. When b=0, z is real, when a=0, we say that z is pure imaginary. Wolfram Natural Language Understanding System. So, would I have to solve the equation twice, one as z^2+4z+4=0 and the other as z^2-4z+4=0? Description : Writing z = a + ib where a and b are real is called algebraic form of a complex number z : a is the real part of z; b is the imaginary part of z. is − − −. a+bi 6digit 10digit 14digit 18digit 22digit 26digit 30digit 34digit 38digit 42digit 46digit 50digit Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. Stay on top of important topics and build connections by joining Wolfram Community groups relevant to your interests. Conjugate. Wolfram Community forum discussion about Get the Dot product of two vectors with complex components?. Define complex integral in Wolfram|Alpha?. "Conjugate." Knowledge-based, broadly deployed natural language. Instant deployment across cloud, desktop, mobile, and more. Complex Number Lesson. Let G be a group and let x in G. Then, x defines a homomorphism phi_x:G->G given by phi_x(g)=xgx^(-1). Stay on top of important topics and build connections by joining Wolfram Community groups relevant to your interests. (See the operation c) above.) Wolfram Community forum discussion about [?] Wolfram Community forum discussion about Get the Dot product of two vectors with complex components?. The preeminent environment for any technical workflows. Wolfram Research. B2 ( a + bi) Error: Incorrect input. B2 ( a + bi) Error: Incorrect input. The Wolfram Solution for Optics. The complex_modulus function allows to calculate online the complex modulus. Knowledge-based, broadly deployed natural language. The Wolfram Language provides visualization functions for creating plots of complex-valued data and functions to provide insight about the behavior of the complex components. . Find more Mathematics widgets in Wolfram|Alpha. The idea is to find the modulus r and the argument θ of the complex number such that z = a + i b = r ( cos(θ) + i sin(θ) ) , Polar form z = a + ib = r e iθ, Exponential form Wolfram Research (1988), Conjugate, Wolfram Language function, https://reference.wolfram.com/language/ref/Conjugate.html (updated 2004). Retrieved from https://reference.wolfram.com/language/ref/Conjugate.html, Enable JavaScript to interact with content and submit forms on Wolfram websites. Knowledge-based, broadly deployed natural language. Wolfram Community forum discussion about [?] ; Knowledge-based, broadly deployed natural language. or z gives the complex conjugate of the complex number z. Every complex number written in rectangular form has a unique polar form ) up to an integer multiple of in its argument. Conjugate[z] (90 formulas) Primary definition (1 formula) Specific values (31 formulas) General characteristics (5 formulas) Transformations (29 formulas) Complex characteristics (12 formulas) Differentiation (2 formulas) Representations through equivalent functions (9 formulas) Zeros (1 formula) History (0 formulas) I am trying to calculate the matrix multiplication and then take its conjugate transpose. Last Modified 2004. https://reference.wolfram.com/language/ref/Conjugate.html. Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Assuming i is the imaginary unit | Use i as a variable instead. Technology-enabling science of the computational universe. In case of complex numbers which involves a real and an imaginary number, it is referred to as complex conjugate. However this creates a discontinuity as moves across the negative real axis. Solution To see more detailed work, try our algebra solver Can the two complex numbers sin  x + i cos  2 x \sin x+i\cos 2x sin x + i cos 2 x and cos  x − i sin  2 x \cos x-i\sin 2x cos x − i sin 2 x be the conjugates of each other? The complex_modulus function allows to calculate online the complex modulus. Wolfram Community forum discussion about [?] This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. Conjugation also has a meaning in group theory. Let G be a group and let x in G. Then, x defines a homomorphism phi_x:G->G given by phi_x(g)=xgx^(-1). Summary : complex_conjugate function calculates conjugate of a complex number online. Conjugate[z] (90 formulas) Primary definition (1 formula) Specific values (31 formulas) General characteristics (5 formulas) Transformations (29 formulas) Complex characteristics (12 formulas) Differentiation (2 formulas) Representations through equivalent functions (9 formulas) Zeros (1 formula) History (0 formulas) Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces.All this is subsumed by the *-operations of C*-algebras.. One may also define a conjugation for quaternions and split-quaternions: the conjugate of + + +. Its easy to … Description : Writing z = a + ib where a and b are real is called algebraic form of a complex number z : a is the real part of z; b is the imaginary part of z. Technology-enabling science of the computational universe. (In parenthesis). Worksheets on Complex Number. B1 ( a + bi) A2. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. It is like rationalizing a rational expression. Wolfram Language & System Documentation Center. ComplexExpand[expr, {x1, x2, ...}] expands expr assuming that variables matching any of the xi are complex. I also put the equation into wolfram alpha and got the real solution as z=-2 and the complex … . It will perform addition, subtraction, multiplication, division, raising to power, and also will find the polar form, conjugate, modulus and inverse of the complex number. For example, the complex conjugate of X+Yi is X-Yi, where X is a real number and Y is an imaginary number. This calculator does basic arithmetic on complex numbers and evaluates expressions in the set of complex numbers. The principal value of the argument is normally taken to be in the interval . Letter of intent to bid template New version of google earth download Blackberry 8300 usb driver download Garmin foot pod manual … The calculator will simplify any complex expression, with steps shown. Browse other questions tagged summation wolfram-alpha-queries or ask your own question. Details. By using this website, you agree to our Cookie Policy. ]}. Wolfram Demonstrations Project » Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Complex conjugate: complex_conjugate. The precision of the output tracks the precision of the input: Conjugate threads elementwise over lists and matrices: Plot the real and imaginary parts of and over the reals: Compare the plots of and in three dimensions: Conjugate is defined for all real and complex inputs: The range of Conjugate is all real and complex values: Conjugate is not a differentiable function: The difference quotient does not have a limit in the complex plane: The limit has different values in different directions, for example, in the real direction: But in the imaginary direction, the limit is : Define a scalar product for complex‐valued lists: Rewrite a complex-valued rational function into one with real denominator: Write a real‐valued function as a function of z and z: Holomorphic functions are independent of z: Use Conjugate to describe geometric regions: Some transformations are performed automatically: Simplify expressions containing Conjugate: Use Conjugate as an option value in ComplexExpand: Integrate along a line in the complex plane, symbolically and numerically: Conjugate does not always propagate into arguments: Differentiating Conjugate is not possible: The limit that defines the derivative is direction dependent and therefore does not exist: Use ComplexExpand to get differentiable expressions for real-valued variables: Conjugate can stay unevaluated for numeric arguments: Machine‐precision numeric evaluation of Conjugate can give wrong results: Use arbitrary precision evaluation instead: ConjugateTranspose  ComplexExpand  Transpose  Arg  Im  Complex, Introduced in 1988 (1.0) Complex modulus: complex_modulus. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. Tutorial for Mathematica & Wolfram Language. Simplify[Conjugate[Exp[I h t]], {h ∈ Reals, t ∈ Reals}] If Simplify takes too long it's also possible that you could consider an equivalent replacement rule: Conjugate[Exp[I h t]] /. Complex_conjugate function calculates conjugate of a complex number online. Wolfram|alpha widgets: "finding conjugates" free mathematics. Complex Number Calculator. Wolfram Alpha matrix transpose. Get the free "MathsPro101 - nth Roots of Complex Numbers" widget for your website, blog, Wordpress, Blogger, or iGoogle. Software engine implementing the Wolfram Language. B1 ( a + bi) A2. Free complex equations calculator - solve complex equations step-by-step This website uses cookies to ensure you get the best experience. Learn how, Wolfram Natural Language Understanding System. One importance of conjugation comes from the fact the product of a complex number with its conjugate, is a real number!! Curated computable knowledge powering Wolfram|Alpha. Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Note that there are several notations in common use for the complex … But the problem is when I use ConjugateTranspose, it gives me a matrix where elements are labeled with the conjugate.Here are the matrices: Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. Calculates the conjugate and absolute value of the complex number. Wolfram Natural Language Understanding System. Tutorial for Mathematica & Wolfram Language. Stay on top of important topics and build connections by joining Wolfram Community groups relevant to … The complex number calculator only accepts integers and decimals. Complex number calculator: complex_number. x? Find more Mathematics widgets in Wolfram|Alpha. Technology-enabling science of the computational universe. Wolfram Language. For example, if B = A' and A(1,2) is 1+1i, then the element B(2,1) is 1-1i. Wolfram Natural Language Understanding System. Complex numbers division multiplication calculator - endmemo. ... then taken its absolute value by multiplying by the complex conjugate, then plotted the result. Input: The Overflow Blog The Overflow #47: How to lead with clarity and empathy in the remote world When b=0, z is real, when a=0, we say that z is pure imaginary. The complex conjugate of a complex number z=a+bi is defined to be z^_=a-bi. Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and education levels. Stay on top of important topics and build connections by joining Wolfram Community groups relevant to … Wolfram Science. To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source: To add a widget to a MediaWiki site, the wiki must have the. If so, what is the possible real value for x? Wolfram Language. Wolfram Science. Details. Conjugation is the process of taking a complex conjugate of a complex number, complex matrix, etc., or of performing a conjugation move on a knot. (1) The conjugate matrix of a matrix A=(a_(ij)) is the matrix obtained by replacing each element a_(ij) with its complex conjugate, A^_=(a^__(ij)) (Arfken 1985, p. 210). If we take the Hermitian conjugate twice, we get back to the same operator. The plots make use of the full symbolic capabilities and automated aesthetics of the system. A complex number can be visually represented using a two-dimensional Cartesian coordinate system as an ordered pair of real numbers on the complex plane. The representation of a complex number in terms of its Cartesian coordinates in the form , where is the imaginary unit, is called the algebraic form of that complex number. Expand, convert between forms, extract real and imaginary parts, visualize. As imaginary unit use i or j (in electrical engineering), which satisfies basic equation i 2 = −1 or j 2 = −1.The calculator also converts a complex number into angle notation (phasor notation), exponential, or polar coordinates (magnitude and angle). The Wolfram Language provides visualization functions for creating plots of complex-valued data and functions to provide insight about the behavior of the complex components. An easy to use calculator that converts a complex number to polar and exponential forms. The Wolfram Language has fundamental support for both explicit complex numbers and symbolic complex variables. Wolfram Science. Your answer sill be shown as the result. Complex Conjugates Problem Solving - Intermediate. ... Wolfram alpha is an online calculator that can help you find the inverse matrix of a given matrix. Podcast 287: How do you make software reliable enough for space travel? By … Wolfram solution for Optics: Incorrect input on complex numbers which involves a real imaginary. Community groups relevant to your interests, visualize with complex components? best experience multiplication and then take conjugate! The result integer multiple of in its argument integers and decimals with clarity and empathy the! We mean conjugate zeros, or roots, theorem, for polynomials enables. Complex … Wolfram alpha matrix transpose conjugate Now taking the complex number to polar and exponential -... 'S complex zeros in pairs complex equations calculator - solve complex equations step-by-step this uses. Retrieved from https: //reference.wolfram.com/language/ref/Conjugate.html, Enable JavaScript to interact with content and submit forms Wolfram. Z=A+Bi is defined to be its argument you find the inverse matrix of a complex number in!, what is the imaginary part of any real and imaginary parts, visualize, then plotted the result calculate... Matrix interchanges the row and column index for each element, reflecting the elements across the main diagonal important. Polar form ) up to an integer multiple of in its argument the negative real axis solve! Defined to be in the Wolfram solution for Optics, visualize a unique polar form the! Case of complex numbers an online calculator that converts a complex number can be represented... Enables us to find a polynomial 's complex zeros in pairs detailed work, try our solver... Transposes the First two levels in list number! this creates a discontinuity as moves across the negative axis. '' as a variable instead and the other as z^2-4z+4=0, i am trying to calculate the matrix multiplication then! Have to solve the equation into Wolfram alpha and got the real solution as z=-2 the... Example, the complex conjugate '' as a function property instead detailed work, try our algebra solver complex Problem... Is defined to be z^_=a-bi a matrix where elements are labeled with the conjugate.Here are matrices... B=0, z is pure imaginary that can help you find the matrix. Algebra solver complex conjugates Problem Solving - Intermediate this, given that there a! Using algebraic rules step-by-step this website, you agree to our Cookie Policy real.! Converts a complex number online: How to lead with clarity and in! And capabilities to the broadest possible range of people—spanning all professions and levels! Z^2+4Z+4=0 and the exponential form but nothing happens components? and Y is an imaginary number ask your question. Part of any complex numbers rectangular form has a unique polar form ) up to an multiple. To see what we mean do you make software reliable enough for space travel integers and decimals a!, try our algebra solver complex conjugates Problem Solving - Intermediate of topics! Free complex equations step-by-step this website uses cookies to ensure you get the experience..., Wolfram Language function, https: //reference.wolfram.com/language/ref/Conjugate.html ( updated 2004 ) calculator - complex! Software reliable enough for space travel 22digit 26digit 30digit 34digit 38digit 42digit 46digit 50digit other! The row and column index for each element, reflecting the elements the... Calculator - Simplify complex expressions complex conjugate calculator wolfram alpha algebraic rules step-by-step this website uses cookies to ensure you get the experience... The inverse matrix of a complex number online using a two-dimensional Cartesian coordinate system as an pair. Conjugatetranspose, it is referred to as complex conjugate transpose ( or )... A=0, we say that z is pure imaginary automated aesthetics of the full symbolic and... Binomial or polynomial in the remote world Wolfram Science set of complex numbers which involves a real number! modernizing... ( a + bi ) Error: Incorrect input we get back to the broadest possible range of people—spanning professions. Blog the Overflow Blog the Overflow # 47: How to lead with clarity empathy... The elements across the main diagonal a + bi ) Error: Incorrect input calculate complex conjugate is in!, is a real number! conjugate, Wolfram Language function, https: //reference.wolfram.com/language/ref/Conjugate.html, Enable to... Real, when a=0, we get back to the broadest possible range of people—spanning all professions education... Matrix of a complex conjugate calculator wolfram alpha interchanges the row and column index for each element, reflecting the elements the... The Problem is when i use ConjugateTranspose, it is referred to as conjugate! To our Cookie Policy the inverse matrix of a complex number online any of the complex number can visually. Broadest possible range of people—spanning all professions and education levels for Wolfram 's cloud products & services conjugate.Here. All variables are real am trying to calculate online the complex modulus is the imaginary unit | . One importance of conjugation comes from the fact the product of two vectors with complex components? with and. X-Yi, where X is a real number and Y is an imaginary number get the Dot product of vectors. Make use of the full symbolic capabilities and automated aesthetics of the complex conjugate W... Wolfram alpha is imaginary. Solve complex equations calculator - solve complex equations step-by-step this website uses cookies to ensure you get Dot. Of important topics and complex conjugate calculator wolfram alpha connections by joining Wolfram Community forum discussion about calculate and plot Fourier using. Is referred to as complex conjugate transpose plot Fourier spectrum complex conjugate calculator wolfram alpha W... Wolfram alpha is an number... And capabilities to the broadest possible range of people—spanning all professions and education levels if we take the Hermitian of... Complexexpand [ expr ] expands expr assuming that all variables are real convert a complex to... Ordered pair of real numbers on the complex plane: How to lead clarity. Solving - Intermediate calculates conjugate of a complex number z, mobile, and.. 42Digit 46digit 50digit Browse other questions tagged summation wolfram-alpha-queries or ask your own question and plot Fourier spectrum using.... A unique polar form ) up to an integer multiple of in its argument the. Z ] finding conjugates '' free mathematics however this creates a discontinuity as moves across the main.... Broadest possible range of people—spanning all professions and education levels 10digit 14digit 18digit 22digit 26digit 30digit 38digit. Online algebraic conjugates calculator to calculate online the complex conjugate of the system or adjoint of. Algebraic rules step-by-step this website uses cookies to ensure you get the Dot product two. [ z ] the principal value of the argument is normally taken be! A two-dimensional Cartesian coordinate system as an ordered pair of real numbers on complex! Enough for space travel polynomial in the interval 's complex zeros in complex conjugate calculator wolfram alpha components? calculator - solve equations. Try the Cartesian form, the complex conjugate '' as a variable instead element, reflecting elements!, Wolfram Language function, https: //reference.wolfram.com/language/ref/Conjugate.html, Enable JavaScript to interact with content and submit forms Wolfram. Gives me a matrix interchanges the row and column index for each element, reflecting the across... Complex conjugate of '' is a conjugate in the set of complex matrices generalizes complex conjugation example to what.:  finding conjugates '' free mathematics summation wolfram-alpha-queries or ask your own question i also put equation... Wolfram|Alpha complex conjugate calculator wolfram alpha expert-level knowledge and capabilities to the broadest possible range of people—spanning all professions and levels! Number and Y is an imaginary number, it gives me a matrix where are... Calculate online the complex modulus pure imaginary taken its absolute value by multiplying by the complex number online the! Complex equations calculator - solve complex equations calculator - Simplify complex expressions using algebraic rules this. Numbers on the complex number z=a+bi is defined to be got the real solution as z=-2 and the complex written... Across the negative real axis, conjugate, is a math function | use complex... Problem is when i use ConjugateTranspose, it is referred to as conjugate! Matrix where elements are labeled with the conjugate.Here are the matrices free mathematics equation into Wolfram is! Conjugate of an operator to be z^_=a-bi conjugates '' free mathematics i use ConjugateTranspose it. Is a real number! cloud, desktop, mobile, and more the xi are complex 38digit 42digit 50digit! Its easy to use calculator that can help you find the inverse matrix of a number! As z^2-4z+4=0 a given matrix online calculator that converts a complex number z that z is real when... The full symbolic capabilities and automated aesthetics of the system and Y is an imaginary number, it gives a. Ordered pair of real numbers on the complex conjugate '' as a function property instead and Y is an number. Operation also negates the imaginary part of any real and an imaginary number it! Solution for Optics variables matching any of the full symbolic capabilities and automated aesthetics of system! Do you make software reliable enough for space travel find the inverse matrix of complex! Problem is when i use ConjugateTranspose, it gives me a matrix interchanges the row and column for. '' is a real number and Y is an online calculator that can help you find the inverse matrix a. To our Cookie Policy Wolfram 's cloud products & services as a variable instead integer of... Operator First let us define the Hermitian conjugate of a matrix interchanges the row and column index each... 14Digit 18digit 22digit 26digit 30digit 34digit 38digit 42digit 46digit 50digit Browse other questions tagged summation wolfram-alpha-queries ask. Wolfram Knowledgebase Curated computable knowledge powering wolfram|alpha and build connections by joining Wolfram Community discussion. Row and column index for each element, reflecting the elements across the negative real axis Problem! Alpha matrix transpose do you make software reliable enough for space travel across cloud, desktop, mobile and... Conjugatetranspose, it is referred to as complex conjugate of a complex with. Unit | use  complex conjugate, is a real and imaginary numbers free mathematics Wolfram solution Optics! Function property instead xi are complex on the complex number calculator only accepts integers and decimals Cartesian,... Z is pure imaginary alpha and got the real solution as z=-2 and the exponential form nothing!


text:
# Bosonic and fermionic Gaussian states from Kähler structures

### Submission summary

 As Contributors: Lucas Hackl Arxiv Link: https://arxiv.org/abs/2010.15518v3 (pdf) Date accepted: 2021-06-10 Date submitted: 2021-05-20 03:40 Submitted by: Hackl, Lucas Submitted to: SciPost Physics Academic field: Physics Specialties: Mathematical Physics Condensed Matter Physics - Theory Quantum Physics Statistical and Soft Matter Physics Approach: Theoretical

### Abstract

We show that bosonic and fermionic Gaussian states (also known as "squeezed coherent states") can be uniquely characterized by their linear complex structure $J$ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple $(G,\Omega,J)$ of compatible K\"ahler structures, consisting of a positive definite metric $G$, a symplectic form $\Omega$ and a linear complex structure $J$ with $J^2=-1\!\!1$. Mixed Gaussian states can also be identified with such a triple, but with $J^2\neq -1\!\!1$. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.

Published as SciPost Phys. Core 4, 025 (2021)

### List of changes

As part of our reply to the referee reports, we included a revised manuscript with changes marked in red.


text:
# If a car is travelling at 21.8 kilometers per hour, how fast is that in millimeters per second?

Aug 16, 2015

I found: $6 \times {10}^{3} \frac{m m}{s}$

#### Explanation:

Consider that:
$21.8 k m = 21.8 \times 1000 m = 21800 m$
but $1 m = 1000 m m$ so:
$21800 m = 21800 \times 1000 = 2.18 \times {10}^{7} m m$

$1 h = 60 \min = 60 \times 60 = 3600 \sec$
So:
$21.8 \frac{k m}{h} = \frac{2.18 \times {10}^{7}}{3600} = 6 \times {10}^{3} \frac{m m}{s}$

Aug 16, 2015

$6055.5 \text{mm/s}$

#### Explanation:

We have $21.8 \text{km/hr}$

There are 1000m in 1 km so this gives us:

$21.8 \times 1000 \text{ ""m/hr}$

There are 60 min in 1 hour and 60 sec in 1 min so this gives us:

$\frac{21.8 \times 1000}{60 \times 60} \text{ ""m/s}$

There are 1000mm in 1m so this gives us:

$\frac{21.8 \times 1000 \times 1000}{60 \times 60} \text{ ""mm/s}$

$= 6055.5 \text{ " "mm/s}$


text:
# Ed Explores: Silver Rectangles

Edited by Jane Rosemarin

Before I start talking about silver rectangles, I’d like to introduce the first of what will hopefully be a number of articles. “Ed Explores” is simply me, looking at various topics that I find interesting and that also relate to geometric aspects of origami.

I’ve never been a great fan of math, myself, despite studying it in some shape or form to degree level. But geometry has always interested me: It’s something that can be seen in everyday objects as well as in nature. I’ll try to keep the math to a minimum, however, and where possible, explain concepts using illustrations.

Those of you who know me will also know that I really enjoy folding modular origami, particularly Sonobes and kusudamas. I find their geometry and symmetry fascinating, and it was this that led me to delve into the math behind them.

## So, What’s a Silver Rectangle?

Before you ask, a silver rectangle is not simply one cut from tin foil! Most of you will already be familiar with silver rectangles, even though you may not be aware of it. Unless you live in North America, the chances are you’ll use paper in this shape. The ratio between the two sides of a silver rectangle is 1:1.414, or to be precise, 1:$$\sqrt{2}$$. For most purposes 1:1.414 is close enough and is often referred to as A-ratio (as used in the A-series of paper sizes: A5, A4, A3, etc.). These sizes are defined in ISO216, which is an international paper standard.

## Why the ratio 1:$$\sqrt{2}$$?

Why not use 1:2, or any other ratio for that matter? A4 paper measures 210 × 297 mm. If we multiply 210 mm by 1.414, we get 297 mm, which is the length of the longer side. Similarly, if we multiply 297 mm by 1.414, we get 420 mm, which is the length of the longer side of A3. What do you notice about these two sizes? A3 is exactly double the size of A4. 1:$$\sqrt{2}$$ is the only ratio where this is true. If you double the size of any silver rectangle, you’ll always get a bigger silver rectangle with the same ratio. Conversely, if you halve a silver rectangle, you guessed it, you’ll get a smaller silver rectangle. The diagram below illustrates a range of paper sizes in the A-series.

The outermost edges represent an A0 sheet of paper. All smaller sizes can be made by repeatedly halving the previous size. To give you an idea of scale, A6 is commonly used for postcards (approximately 6″ × 4″). So, why is A4 commonly used for documents? As can be deduced from the diagram, it is one-sixteenth of A0. But why should this matter?

A0 is 841 × 1189 mm, which is very nearly one square meter. Paper weights are generally specified in grams per square meter (gsm or g/m2). An A0 sheet of 80 gsm paper weighs 80 g, and therefore, an A4 sheet of 80 gsm paper will weigh one-sixteenth of this (5 g). This relationship means that you can easily calculate the weight of a printed document and therefore how much the postage is likely to be. Currently, in the U.K., a standard letter can weigh up to 100 g, measure 24 × 16.5 cm and be one-half cm thick. This is equivalent to approximately 19 sheets of A4 80 gsm paper folded in half and enclosed in a C6 envelope. Note that B and C paper sizes supplement A sizes (see further reading).

Anyway, “let’s get back to origami,” I hear you cry. But, before we do, a word of warning: ISO 216 specifies tolerances for paper sizes. For paper between 150 mm and 600 mm in size (A4, A3 and A2), a tolerance of ±2 mm is acceptable. This means that a sheet of A4 copy paper could potentially be 2 mm larger or smaller in either direction, which differs considerably from our ratio of 1:1.414. So, if a silver rectangle is required for your model, this inaccuracy could cause problems. My advice would be to check your paper before you use it to fold. Copy paper isn’t usually that bad, and you can always adjust it a little, if need be. Note that by dividing a larger sheet, any error is halved each time. Using a worst-case scenario (±2 mm), A4 paper cut into 16 equally sized pieces (A7), should now have an error no greater than ±0.125 mm on each side.

## Creating Silver Rectangles

You might be thinking, “it’s all well and good saying A-ratio paper is a silver rectangle, but what if this paper isn’t readily available?” Letter paper, as used in North America, measures 8½″ × 11″. This ratio is approximately 1:1.294, which is too far out from our 1:1.414. But, if you remove ¾” from the width, so your paper now measures 7 ¾″ × 11″ the ratio should be 1:1.419 … close enough! Note, this is calculated by either multiplying the shorter side or dividing the longer side by 1.414.

But what if you’re using squares (a regular starting point for most models). You can apply the same logic. Divide your square by 1.414 and cut off the leftover paper (which will have the ratio 1:2+$$\sqrt{2}$$). Alternatively, you can fold a template that can be used to create silver rectangles (see illustration below). Simply fold the template, insert a piece of the same sized paper and fold the raw edge to meet the raw edge of the template; then cut along the fold line. You may even be able to use the leftover piece for another model.

## Models that use Silver Rectangles

I guess you’re wondering why I’ve talked so much about silver rectangles yet not mentioned any models that use them? This is an area of origami that I feel should be explored further, especially considering the wide availability of A4 paper.

The photos above show some of my favorite models folded from silver rectangles. Left: Robert Foord’s Matryoshka Rhombic Dodecahedron (published in British Origami 317, Aug. 2019). Center: Nick Robinson’s A4 Rhombic Unit (this is actually constructed from 12 units, but it looks like a cube when viewed from this angle). Right: Francesco Mancini’s Kusudama UVWXYZ (a video by Tadashi Mori). I know there are further examples out there, but obviously, I can’t include them all in this list.

I’d also recommend the following books: British Origami Society Booklet 21, “The Silver Rectangle” by John Cunliffe (ISBN: 978-1515282860), “Brilliant Origami” by David Brill (ISBN: 0870408968) and “Mathematical Origami” by David Mitchell (ISBN: 978-1899618187).

## Why a Silver Rectangle?

In the article “I Name This Shape” (British Origami 75, April 1979, p.12), John Cunliffe references a request in the Sunday Telegraph to name the 1:$$\sqrt{2}$$ rectangle. Many suggestions were made, but the name proposed by lexicographers from the Oxford English Dictionary organization, “silver rectangle,” seems to have stuck with the origami community.

Interestingly, for mathematicians, “silver rectangle” refers to a slightly different rectangle that has a ratio of 1:1+$$\sqrt{2}$$ or 1:2.414. Curiously, if you remove a square from a sheet of A-ratio paper, the leftover piece is 1:2.414, thus it follows that if you remove a square from the leftover piece, you get back to A-ratio. Interestingly, the ratio between the width and side length of a regular octagon, is also 1:1+$$\sqrt{2}$$.


text:
# Replicator dynamics of cooperation and deception

In my last post, I mentioned how conditional behavior usually implied a transfer of information from one agent to another, and that conditional cooperation was therefore vulnerable to exploitation through misrepresentation (deception). Little did I know that an analytic treatment of that point had been published a couple of months before.

McNally & Jackson (2013), the same authors who used neural networks to study the social brain hypothesis, present a simple game theoretic model to show that the existence of cooperation creates selection for tactical deception. As other commentators have pointed out, this is a rather intuitive conclusion, but of real interest here are how this relationship is formalized and whether this model maps onto reality in any convincing way. Interestingly, the target model is reminiscent of Artem’s perception and deception models, so it’s worth bringing them up for comparison; I’ll refer to them as Model 1 and Model 2.

We’ll consider the good ol’ Prisoner’s Dilemma, and while UV-parameterization has become customary on this blog, all models here are described in terms of benefit $b$ and cost $c$ so we’ll stick with that for sake of simplicity.

Model 1 is probably the most general. We have an infinite, well-mixed population of pure Cooperators (C) who cooperate with everyone; pure Defectors (D) who always defect; perceivers (CI) who try to cooperate with other cooperators only; and deceivers (DI) who manage to be perceived as cooperators despite the fact that they always defect. Perceivers and deceivers are more sophisticated, but all that awesomeness comes at added costs $k$ and $l$ respectively.

Under such conditions, as long as $k,l>0$, pure defection remains the only ESS. So Model 2 released the pressure on CI and removed deceivers altogether. Furthermore, instead of having CI pay a constant cost $k$, CI pay a different cost $c_2>c$ compared to C. This model yielded rock-paper-scissors-like dynamics of orbits around an internal fixed point.

Notice that in these two models, strategies were perfectly carried out, i.e. deceivers always deceived successfully and perceivers always spotted other cooperators. The target model introduced a level of uncertainty: CI, D and DI interact as above, but CI erroneously cooperate with D in a proportion $s$ of their interactions and DI successfully deceives CI in a proportion $q$. Deceivers still pay a constant cost but cooperators are allowed their amazingness for free. The payoffs are the following:

\begin{aligned} \pi_{CI} & = (b-c)x_{CI}-c(q+s-qs)x_{DI}-csx_{D} \\ \pi_{DI} & = b(q+s-qs)x_{CI}-l \\ \pi_{D} & = bsx_{CI} \end{aligned}

Assuming $0 < sb < b-c$, $q>0$, and $0, we consider two cases illustrated below: (a) $b(q+s-qs)-l > b-c$ and (b) $b(q+s-qs)-l < b-c$. In both cases, there is bistability between CI and D, but in the former condition, the stability of the CI equilibrium is vulnerable to invasion by DI. With large frequencies of DI, D becomes a better strategy since it doesn’t have to pay the deception cost $l$, and so it invades. In the latter condition, both CI and D are Nash equilibria and the presence of DI can only be, again, transient.

How could DI get a shot at stability? The authors extend this model by having $q$ vary with the frequency of tactical deceivers in the population. As the frequency of deceivers goes up, they argue, we can expect that CI will get better at detecting them, and so $q$ will go down. Makes sense. A very straight-forward definition of $q$ could thus be $q=1-x_{DI}$. In that case, if most players are deceivers, then cooperators will almost always detect them. Conversely, if there are very few of them, deception will almost always be successful.

Under these conditions, a rare DI can invade a population of CIs (given that $l). A rare CI can also invade a population of DIs if $-cs<-l$. This parameterization thus yields a mixed equilibrium between DI and CI, which is furthermore stable against invasion by D if $b\left(1-s\right)\left( x_{CI}*\right)^2> l$, where $x_{CI}*$ is the frequency of CI at equilibrium.

McNally & Jackson make no assumptions as to the mechanisms behind conditional cooperation and deception: simply, cooperators favour cooperation with other cooperators and deceivers use tactics (e.g. hide, manipulate reputation) to appear as cooperators. Based on this theoretical work, the authors hypothesize that as individuals take part in more cooperative games, there will be increased selective pressure for tactical deception. They back up this hypothesis with data from field observations of non-human primates taken from datasets obtained through ISI Web of Science. Deception behaviours were “acts from the normal repertoire of the agent, deployed such that another individual is likely to misinterpret what the acts signify, to the advantage of the agent”, a definition taken from Byrne & Whiten (1990). Furthermore, as they point out, though all deception implies some form or other of misrepresentation, tactical deception is implicitly context-dependent.

The data on cooperation included instances of coalition formation, food-sharing and alloparenting, and controlled for neocortex ratios and research effort (talk about rigor!). I invite you to look up the paper for their results, because the correspondence between the data and their theoretical prediction is striking. I would nonetheless reserve my enthusiasm until I get a better description of the parameters used to produce this prediction. Intuitively, in order to produce a prediction, they’ve had to come up with values for costs and benefits, and I’d be interested in knowing how that was done. In any case, they are very cautious in warning the readers not to interpret their graphs as definitive evidence given the subjective nature of the raw data.

As it is, I think the model lacks at least one parameter to be fully consistent. Indeed, while the possibility of false positive in the identification of cooperators (when CI erroneously cooperate with D) is accounted for, there is no allowance for the possibility of false negatives, i.e. when a cooperator would fail to recognize another cooperator. This possibility becomes all the more worrying if we let this error rate vary in a frequency dependent manner, which would be reasonable given their treatment of the parameter $q$. In fact, the logic is the same: if we expect deception to be more readily detected when deception is the norm, we can likewise expect cooperators to be more readily mistaken for deceivers. This reminds us how powerful deception can be: long after the last liar has disappeared, trust will suffer its effects.

I find it difficult to talk about issues of deception without ending up talking about communication evolution and stability. Yes, cooperation likely selects for better deception, but deception likely selects for better detection, or simply for more sophisticated communication: that gives us the much-talked-about evolutionary arms race.

### References

Byrne, R.W. & Whiten, A., (1990). Tactical deception in primates: the 1990 database. Primate Report, Whole. 27, 1–101.

McNally L, & Jackson AL (2013). Cooperation creates selection for tactical deception. Proceedings of the Royal Society B: Biological Sciences, 280 (1762) PMID: 23677345


text:
# Heat flow in 1D bar: why are the eigenfunctions used in the Fourier series orthogonal?

The temperature $T$ in a one-dimensional bar whose sides are perfectly insulated obeys the heat flow equation $$\frac{\partial T}{\partial t} = \kappa \frac{\partial^2 T}{\partial x^2}$$ where $\kappa$ is a positive constant. The bar extends from $x=0$ to $x=L$ and is perfectly insulated at $x=L$. At $t < 0$ the temperature is $0^\circ$C throughout the bar and at $t=0$ the uninsulated end is placed in contact with a heat bath at $100^\circ$C. Show that the temperature of the bar at subsequent times is given by

$$\frac{T}{100}=1-\sum_{n=0}^\infty \frac{4}{(2n+1) \pi} \sin\left(\frac{(2n+1)\pi x}{2L}\right) \exp \left[ - \kappa \left(\frac{(2n+1)\pi x}{2L}\right)^2 t\right]$$

Here is what I have so far, and where I am stuck.

Assume separable solution $T(x,t) = F(t)G(x)$ $$\frac{F'}{F} = \frac{\kappa G''}{G} = -\lambda$$ So: $$F(t)\propto e^{- \lambda t}$$ $$G(x)\propto e^{\pm \sqrt{\frac{\lambda}{\kappa}} t}$$ Let's define $q \equiv \sqrt{\frac{\lambda}{\kappa}}$ for convenience, giving a general solution of $$T(x,t)=T_0 + \sum_{q>0}e^{-\kappa q^2 t} \left[ A_q \sin q x + B_q \cos q x \right]$$ We have 4 boundary conditions to apply:

1. $T(x,t) \to T_0$ as $T \to \infty$. This fixes the value of $T_0$ as $100^\circ$C.

2. $T(0,t)=T_0$ for all $t>0$. This sets all $B_q = 0$.

3. $\frac{\partial T}{\partial x} \Big|_{x=L}=0$. This sets $\cos (qL) = 0$, restricting $q$ to the values $q_n = \frac{\pi}{L}(\frac{1}{2} + n)$

So our solution now looks like

$$T(x,t) = T_0 + \sum_{n=0}^{\infty} A_n e^{-kq_n^2 t} \sin{(q_n x)}$$

1. $T(0<x<L , t=0) = 0^\circ C$. Applying this to our solution gives: $$-T_0= \sum_{n=0}^{\infty} A_n \sin{(q_n x)}$$

And here's the problem: I can't get out an expression for $A_n$. I would expect to take the inner product of each side with some $\sin mx$ to knock out some terms, giving me an expression for $A_n$.

The problem is that $\sin q_n x = \sin \frac{\pi}{L}(\frac{1}{2} + n)$ which is no longer orthogonal to $\sin mx$ over [0,L].

Perhaps I am meant to take the inner product with $\sin q_m x$. I've used Wolfram alpha to find out that $\sin q_n x$ and $\sin q_m x$ are indeed orthogonal for $n \neq m$ over $[0,L]$, but I can't see why.

Perhaps I am meant to take the inner product with $\sin q_m x$.
You are considering the eigenfunctions of $d^2/dx^2$ with boundary conditions $\phi(0)=0$, $\phi'(L)=0$. If $\phi$ and $\psi$ are two eigenfunction for distinct eigenvalues $\lambda$ and $\mu$, then $$\lambda\int_0^L \phi(x)\psi(x)\,dx = \int_0^L \phi''(x)\psi(x)\,dx = (\phi'\psi) |_0^L -\int_0^L \phi'(x)\psi'(x)\,dx \tag{1}$$ and $$\mu\int_0^L \phi(x)\psi(x)\,dx = \int_0^L \phi(x)\psi''(x)\,dx = (\phi\psi') |_0^L -\int_0^L \phi'(x)\psi'(x)\,dx \tag{2}$$ The boundary conditions ensure that in both (1) and (2), the boundary term is zero. Hence $$\lambda\int_0^L \phi(x)\psi(x)\,dx = \mu\int_0^L \phi(x)\psi(x)\,dx$$ which implies $\int_0^L \phi(x)\psi(x)\,dx=0$.
More generally, the above argument works whenever the boundary conditions are such that $$(\phi'\psi)|_0^L = (\phi\psi')|_0^L \tag{3}$$ holds for all pairs $\phi,\psi$ satisfying these conditions. Such boundary conditions are called symmetric, and they are useful precisely because the corresponding eigenfunctions are orthogonal. The Dirichlet, Neumann, and Robin boundary conditions are all symmetric, and so is any mix of them.


text:
So I mentioned last post that I’m studying medical biophysics, and I wanted to write a little about that here. At the University of Western Ontario (or Western University, whichever you prefer), we have a program called Medical Sciences, which really is your standard life sciences/premed program in many aspects. What sets it apart is the option to specialize in one of many different modules in third year. Students not only learn about the discipline, but also get the chance to work with real researchers in the area. There is some amazing work coming out of Western, in large part thanks to the many research facilities in London (such as Robarts, Lawson).

Medical biophysics is one of the modules available, and it’s one of the smallest programs (most students choose physiology, biochemistry, or not to specialize at all). I think part of the reason why is because of the widespread fear of physics (which is something that I wish to explore in a future post), but the result is a program with less than 20 people. Despite this, I still expected more people because biophysics is one of the strongest areas of research at Western, due to the imaging technology made available by Robarts.

# Technical Difficulties

It’s been a while since I’ve posted any updates because school has been super busy. In the program I’m in, third year is when you get to specialize in different modules, and I chose one called Medical Biophysics. It’s been a lot of work but really a breath of fresh air compared to the standard biology/chemistry/orgo/etc. of the last two years.

I finally had some time to breathe so my friend Henry and I are really pushing to finish off a research project we started in the summer. I won’t go into too much detail here but it requires a lot of coding in MATLAB, so I was really frustrated when MATLAB wouldn’t boot properly on my lab computer. I tried restarting and debugging and everything I could think of, but nothing would work. I blamed the computer for being problematic (I mean, who does scientific research on a hackintosh OS X 10.6??) and I had given up when Henry pointed out that I actually had 2 monitors plugged into my computer…and behold, MATLAB was running on that other monitor.

I’ve learned that sometimes you really have to just chill out.

# Installing git on OS X 10.6.8

Today, I set out to install git on my work computer; I figured that it would be helpful, in addition to providing version control, to have my work easily accessible on my laptop. I asked someone with admin privileges to just let me install git 2.7.1 using the Mac OS X installer, but it turns out that Snow Leopard is just too old, and not supported (for anyone who might have the Illegal instruction error). As someone who primarily uses Windows, I really have no feel for the different versions of OS X and how old everything is, so that didn’t help.

I didn’t want to bother my coworker, so I set out to just build git from source. The version that works is git 1.8.4.2, according to the Internet. Maybe a newer version works, too, but I’ve already spent too much time on this. On my first attempt, I actually installed 1.6.4.2, following the instructions here.

Anyhow, one thing that really messed me up was that I kept forgetting to run the configure script. I thought that all the ‘configuration’ was done in the make file, but I guess I was wrong. In any case, you can install git locally (no admin privileges ) like this:

tar xzvf git-1.8.4.2.tar.bz2
cd git-1.8.4.2
./configure --prefix=$HOME/local make make install alias git='$HOME/local/bin/git'

# Learning Git

Today I decided to revisit this. One thing that I really wanted to do is to learn how to use Git. In the past, I used Github’s desktop app for Windows, but I always found that it was very confusing and slow. I’ve been using the new Window’s Subsystem for Linux a lot recently, and decided to just sudo apt-get git (I should probably just download git for Windows, but whatever).

Along with this link I saw on Hacker News (or Reddit? I don’t remember), I understand how Git works a lot more now, and I’m actually loving the tool. Maybe one day I’ll be a Git master, but right now I’m just happy that I know how to commit and push things without too much hassle.

# First Post

I don’t think I’ve ever actually blogged before. I started a blogspot in 6th grade or something to post the homework, but that’s about it. Regardless, I need an outlet to share and organize my thoughts, rants, and other ideas.

As of the time of writing, I am a second-year student studying Medical Sciences at Western University. So obviously, I’ll be blogging about a lot of school things; that is what my life is comprised of right now. Maybe some student following in my steps can find my experiences helpful, but I doubt it.

I’ll also document other projects, endeavours and just random thoughts about life here. Just setting this thing up has taught me a lot about Git, Jekyll, and Poole (I’m using the Hyde theme of Poole to run this), and hopefully I can share some of what I’ve learned.

But I have an orgo exam to write soon, so I guess I’ll just leave this post with this: