text:
# The final stretch

The end of the Covid-19 pandemic is within reach. The vaccines have been a roaring success and former Bell Labs physicist J.C. Phillips predicted it (see here). He argued that the spike protein, which is the business end of the SARS-CoV-2 virus, has been optimized to such a degree in SARS-CoV-2 that even a small perturbation from a vaccine can disrupt it. While the new variants perturb the spike slightly and seem to spread faster, they will not significantly evade the vaccine. However, just because the end is within sight doesn’t mean we should not still be vigilante and not mess this up. Europe has basically scored multiple own goals these past few months with their vaccine rollout (or lack thereof) that is a combination of both gross incompetence and excessive conservatism. The Astra-Zeneca vaccine fiasco was a self-inflicted wound by all parties involved. The vaccine is perfectly fine and any side effects are either not related to the vaccine or of such low probability that it should not be a consideration for halting its use. By artificially slowing vaccine distribution, there is a chance that some new mutation could arise that will evade the vaccine. Europe needs to get its act in gear. The US has steadily ramped up vaccinations and is on course to have all willing adults vaccinated by start of summer. Although there has been a plateauing and even slight rise recently because of relaxation from social distancing in some areas, cases and deaths will drop for good by June everywhere in the US. North America will largely be back to normal by mid-summer. However, it is imperative that we press forward and vaccinate the entire world. We will also all need to get booster shots next fall when we get our flu shots.

# ICCAI talk

I gave a talk at the International Conference on Complex Acute Illness (ICCAI) with the title Forecasting COVID-19. I talked about some recent work with FDA collaborators on scoring a large number of publicly available epidemic COVID-19 projection models and show that they are unable to reliably forecast COVID-19 beyond a few weeks. The slides are here.

# Why it is so hard to forecast COVID-19

I’ve been actively engaged in trying to model the COVID-19 pandemic since April and after 5 months I am pretty confident that models can estimate what is happening at this moment such as the number of people who are currently infected but not counted as a case. Back at the end of April our model predicted that the case ascertainment ratio ( total cases/total infected) was on the order of 1 in 10 that varied drastically between regions and that number has gone up with the advent of more testing so that it may now be on the order of 1 in 4 or possibly higher in some regions. These numbers more or less the anti-body test data.

However, I do not really trust my model to forecast what will happen a month from now much less six months. There are several reasons. One is that while the pandemic is global the dynamics are local and it is difficult if not impossible to get enough data for a detailed fine grained model that captures all the interactions between people. Another is that the data we do have is not completely reliable. Different regions define cases and deaths differently. There is no universally accepted definition for what constitutes a case or a death and the definition can change over time even for the same region. Thus, differences in death rates between regions or months could be due to differences in the biology of the virus, medical care, or how deaths are defined and when they are recorded. Depending on the region or time, a person with a SARS-CoV-2 infection who dies of a cardiac arrest may or may not be counted as a COVID-19 death. Deaths are sometimes not officially recorded for a week or two, particularly if the physician is overwhelmed with cases.

However, the most important reason models have difficulty forecasting the future is that modeling COVID-19 is as much if not more about modeling the behavior of people and government policy than modeling the biology of disease transmission and we are just not very good at predicting what people will do. This was pointed out by economist John Cochrane months ago, which I blogged about (see here). You can see why getting behavior correct is crucial to modeling a pandemic from the classic SIR model

$\frac{dS}{dt} = -\beta SI$

$\frac{dI}{dt} = \beta SI - \sigma I$

where $I$ and $S$ are the infected and susceptible fractions of the initial population, respectively. Behavior greatly affects the rate of infection $\beta$ and small errors in $\beta$ amplify exponentially. Suppression and mitigation measures such as social distancing, mask wearing, and vaccines reduce $\beta$, while super-spreading events increase $\beta$. The amplification of error is readily apparent near the onset of the pandemic where $I$ grows like $e^{\beta t}$. If you change $\beta$ by $\delta \beta$, then the $I$ will grow like $e^{\beta t+\delta \beta t}$ and thus the ratio is growing (or decaying) exponentially like $e^{\delta \beta t}$. The infection rate also appears in the initial reproduction number $R_0 = \sigma/\beta$. From a previous post, I derived approximate expressions for how long a pandemic would last and show that it scales as $1/(R_0-1)$ and thus errors in $\beta$ will produce errors $R_0$, which could result in errors in how long the pandemic will last, which could be very large if $R_0$ is near one.

The infection rate is different everywhere and constantly changing and while it may be possible to get an estimate of it from the existing data there is no guarantee that previous trends can be extrapolated into the future. So while some of the COVID-19 models do a pretty good job at forecasting out a month or even 6 weeks (e.g. see here), I doubt any will be able to give us a good sense of what things will be like in January.

# There is no herd immunity

In order for an infectious disease (e.g. COVID-19) to spread, the infectious agent (e.g. SARS-CoV-2) must jump from one person to another. The rate of this happening depends on the rate that an infectious person will come into contact with a susceptible person multiplied by the rate of the virus making the jump when the two people are nearby. The reproduction number R is obtained from the rate of infection spread times the length of time a person is infectious. If R is above one then a single person will infect more than one person on average and thus the pandemic will grow. If it is below one, then the pandemic will diminish. Herd immunity happens when enough people have been infected that the rate of finding a susceptible person becomes low enough that R drops below one. You can find the math behind this here.

However, a major assumption behind herd immunity is that once a person is infected they can never be infected again and this is not true for many infectious diseases such as other corona-viruses and the flu. There are reports that people can be reinfected by SARS-CoV-2. This is not fully validated but my money is on there being no lasting immunity to SARS-CoV-2 and this means that there is never any herd immunity. COVID-19 will just wax and wane forever.

This doesn’t necessarily mean it will be deadly forever. In all likelihood, each time you are infected your immune response will be more measured and perhaps SARS-CoV-2 will eventually be no worse than the common cold or the seasonal flu. But the fatality rate for first time infection will still be high, especially for the elderly and vulnerable. Those people will need to remain vigilante until there is a vaccine, and there is still no guarantee that a vaccine will work in the field. If we’re lucky and we get a working vaccine, it is likely that vaccine will not have lasting effect and just like the flu we will need to be vaccinated annually or even semi-annually.

# Another Covid-19 plateau

The world seems to be in another Covid-19 plateau for new cases. The nations leading the last surge, namely the US, Russia, India, and Brazil are now stabilizing or declining, while some regions in Europe and in particular Spain are trending back up. If the pattern repeats, we will be in this new plateau for a month or two and then trend back up again, just in time for flu season to begin.

# Why we need a national response

It seems quite clear now that we do not do a very good job of projecting COVID-19 progression. There are many reasons. One is that it is hard to predict how people and governments will behave. A fraction of the population will practice social distancing and withdraw from usual activity in the absence of any governmental mandates, another fraction will not do anything different no matter any official policy and the rest are in between. I for one get more scared of this thing the more I learn about it. Who knows what the long term consequences will be particularly for autoimmune diseases. The virus is triggering a massive immune response everywhere in the body and it could easily develop a memory response to your own cells in addition to the virus.

The virus also spreads in local clusters that may reach local saturation before infecting new clusters but the cross-cluster transmission events are low probability and hard to detect. The virus reached American shores in early January and maybe even earlier but most of those early events died out. This is because the transmission rate is highly varied. A mean reproduction number of 3 could mean everyone has R=3 or that most people transmit with R less than 1 while a small number (or events) transmit with very high R. (Nassim Nicholas Taleb has written copiously on the hazards of highly variable (fat tailed) distributions. For those with mathematical backgrounds, I highly recommend reading his technical volumes: The Technical Incerto. Even if you don’t believe most of what he says, you can still learn a lot.) Thus it is hard to predict when an event will start a local epidemic, although large gatherings of people (i.e. weddings, conventions, etc.) are a good place to start. Once the epidemic starts, it grows exponentially and then starts to saturate either by running out of people in the locality to infect or people changing their behavior or more likely both. Parts of New York may be above the herd immunity threshold now.

Thus at this point, I think we need to take a page out of Taleb’s book (like literally as my daughter would say), and don’t worry too much about forecasting. We can use it as a guide but we have enough information to know that most people are susceptible, about a third will be asymptomatic if infected (which doesn’t mean they won’t have long term consequences), about a fifth to a tenth will be counted as a case, and a few percent of those will die, which strongly depends on age and pre-existing conditions. We can wait around for a vaccine or herd immunity and in the process let many more people die, ( I don’t know how many but I do know that total number of deaths is a nondecreasing quantity), or we can act now everywhere to shut this down and impose a strict quarantine on anyone entering the country until they have been tested negative 3 times with a high specificity PCR test (and maybe 8 out of 17 times with a low specificity and sensitivity antigen test).

Acting now everywhere means, either 1) shutting everything down for at least two weeks. No Amazon or Grubhub or Doordash deliveries, no going to Costco and Walmart, not even going to the super market. It means paying everyone in the country without an income some substantial fraction of their salary. It means distributing two weeks supply of food to everyone. It means truly essential workers, like people keeping electricity going and hospital workers, live in a quarantine bubble hotel, like the NBA and NHL or 2) Testing everyone everyday who wants to leave their house and paying them to quarantine at home or in a hotel if they test positive. Both plans require national coordination and a lot of effort. The CARES act package has run out and we are heading for economic disaster while the pandemic rages on. As a recent president once said, “What have you got to lose?”

# How to make a fast but bad COVID-19 test good

Among the myriad of problems we are having with the COVID-19 pandemic, faster testing is one we could actually improve. The standard test for the presence of SARS-CoV-2 virus uses PCR (polymerase chain reaction), which amplifies targeted viral RNA. It is accurate (high specificity) but requires relatively expensive equipment and reagents that are currently in short supply. There are reports of wait times of over a week, which renders a test useless for contact tracing.

An alternative to PCR is an antigen test that tests for the presence of protein fragments associated with COVID-19. These tests can in principle be very cheap and fast, and could even be administered on paper strips. They are generally much more unreliable than PCR and thus have not been widely adopted. However, as I show below by applying the test multiple times, the noise can be suppressed and a poor test can be made arbitrarily good.

The performance of binary tests are usually gauged by two quantities – sensitivity and specificity. Sensitivity is the probability that you test positive (i.e are infected) given that you actually are positive (true positive rate). Specificity is the probability that you test negative if you actually are negative (true negative rate). For a pandemic, sensitivity is more important than specificity because missing someone who is infected means you could put lots of people at risk while a false positive just means the person falsely testing positive is inconvenienced (provided they cooperatively self-isolate). Current PCR tests have very high specificity but relatively low sensitivity (as low as 0.7) and since we don’t have enough capability to retest, a lot of tested infected people could be escaping detection.

The way to make any test have arbitrarily high sensitivity and specificity is to apply it multiple times and take some sort of average. However, you want to do this with the fewest number of applications. Suppose we administer $n$ tests on the same subject, the probability of getting more than $k$ positive tests if the person is positive is $Q(k,n,q) = 1 - CDF(k|n,q)$, where $CDF$ is the cumulative distribution function of the Binomial distribution (i.e. probability that the number of Binomial distributed events is less than or equal to $k$). If the person is negative then the probability of  $k$ or fewer positives is $R(k,n,r) = CDF(k|n,1-r)$. We thus want to find the minimal $n$ given a desired sensitivity and specificity, $q'$ and $r'$. This means that we need to solve the constrained optimization problem: find the minimal $n$ under the constraint that $k < n$, $Q(k,n,q) = \ge q'$ and $R(k,n,r)\ge r'$. $Q$ decreases and $R$ increases with increasing $k$ and vice versa for $n$. We can easily solve this problem by sequentially increasing $n$ and scanning through $k$ until the two constraints are met. I’ve included the Julia code to do this below.  For example, starting with a test with sensitivity .7 and specificity 1 (like a PCR test), you can create a new test with greater than .95 sensitivity and specificity, by administering the test 3 times and looking for a single positive test. However, if the specificity drops to .7 then you would need to find more than 8 positives out of 17 applications to be 95% sure you have COVID-19.

using Distributions

function Q(k,n,q)
d = Binomial(n,q)
return 1 – cdf(d,k)
end

function R(k,n,r)
d = Binomial(n,1-r)
return cdf(d,k)
end

function optimizetest(q,r,qp=.95,rp=.95)

nout = 0
kout = 0

for n in 1:100
for k in 0:n-1
println(R(k,n,r),” “,Q(k,n,q))
if R(k,n,r) >= rp && Q(k,n,q) >= qp
kout=k
nout=n
break
end
end
if nout > 0
break
end
end

return nout, kout
end

# Slides for Covid-19 talk

Here are my slides for my recent COVID-19 talk at the Centre for Applied Mathematics in BioScience and Medicine (CAMBAM). It’s an updated version of the one I gave to the FDA.

# Remember the ventilator

According to our model, the global death rate due to Covid-19 is around 1 percent for all infected (including unreported). However, if it were not for modern medicine and in particular the ventilator, the death rate would be much higher. Additionally, the pandemic first raged in the developed world and is only recently engulfing parts of the world where medical care is not as ubiquitous although this may be mitigated by a younger populace in those places. The delay between the appearance of a Covid-19 case and deaths is also fairly long; our model predicts a mean of over 50 days. So the lower US death rate compared to April could change in a month or two when the effects of the recent surges in the US south and west are finally felt.

# How long and how high for Covid-19

Cases of Covid-19 are trending back up globally and in the US. The world has nearly reached 10 million cases with over 2.3 million in the US. There is still a lot we don’t understand about SARS-CoV-2 transmission but I am confident we are no where near herd immunity. Our model is consistently showing that the case ascertainment ratio, that is the ratio of official Covid-19 cases to total SARS-CoV-2 infections, is between 5 and 10. That means that the US has less than 25 million infections while the world is less than 100 million.

Herd immunity means that for any fixed reproduction number, R0, the number of active infections will trend downward if the fraction of the initially susceptible population falls below 1/R0, or the total number infected is higher than 1- 1/R0. Thus, for an R0 of 4, three quarters of the population needs to be infected to reach herd immunity. However, the total number that will eventually be infected, as I will show below, will be

$1 -\frac{e^{-R_0}}{1- R_0e^{-R_0}}$

which is considerably higher. Thus, mitigation efforts to reduce R0 will reduce the total number infected. (2020-06-27: This expression is not accurate when R0 is near 1. For a formula in that regime, see Addendum.)

Some regions in Western Europe, East Asia, and even the US have managed to suppress R0 below 1 and cases are trending downward. In the absence of reintroduction of SARS-CoV-2 carriers, Covid-19 can be eliminated in these regions. However, as the recent spikes in China, South Korea, and Australia have shown, this requires continual vigilance. As long as any person remains infected in the world, there is always a chance of re-emergence. As long as new cases are increasing or plateauing, R0 remains above 1. As I mentioned before, plateauing is not a natural feature of the epidemic prediction models, which generally either go up or go down. Plateauing requires either continuous adjustment of R0 through feedback or propagation through the population as a wave front, like a lawn mower cutting grass. The latter is what is actually going on from what we are seeing. Regions seem to rise and fall in succession. As one region reaches a peak and goes down either through mitigation or rapid spread of SARS-CoV-2, Covid-19 takes hold in another. We saw China and East Asia rise and fall, then Italy, then the rest of Western Europe, then New York and New Jersey, and so forth in series, not in parallel. Now it is spreading throughout the rest of the USA, South America, and Eastern Europe. Africa has been spared so far but it is probably next as it is beginning to explode in South Africa.

A reduction in R0 also delays the time to reach the peak. As a simple example, consider the standard SIR model

$\frac{ds}{dt} = -\beta sl$

$\frac{dl}{dt} = \beta sl -\sigma l$

where $s$ is the fraction of the population susceptible to SARS-CoV-2 infection and $l$ is the fraction of the population actively infectious. Below are simulations of the pandemic progression for R0 = 4 and 2.

We see that halving R0, basically doubles the time to reach the peak but much more than doubles the number of people that never get infected. We can see why this is true by analyzing the equations. Dividing the two SIR equations gives

$\frac{dl}{ds} = \frac{\sigma l -\beta sl}{\beta sl}$,

which integrates to $l = \frac{\sigma}{\beta} \ln s - s + C$. If we suppose that initially $s=1$ and $l = l_0<<1$ then we get

$l = \frac{1}{R_0} \ln s + 1 - s + l_0$ (*)

where $R_0 = \beta/\sigma$ is the reproduction number. The total number infected will be $1-s$ for $l=0$. Rearranging gives

$s = e^{-R_0(1+l_0+s)}$

If we assume that $R_0 s <<1$ and ignore $l_0$ we can expand the exponential and solve for $s$ to get

$s \approx \frac{e^{-R_0}}{1- R_0e^{-R_0}}$

This is the fraction of the population that never gets infected, which is also the probability that you won’t be infected. It gets smaller as $R_0$ increases. So reducing $R_0$ can exponentially reduce your chances of being infected.

To figure out how long it takes to reach the peak, we substitute equation (*) into the SIR equation for $s$ to get

$\frac{ds}{dt} = -\beta(\frac{1}{R_0} \ln s + 1 - s + l_0) s$

We compute the time to peak, $T$, by separating variables and integrating both sides. The peak is reached when $s = 1/R_0$.  We must thus compute

$T= \int_0^T dt =\int_{1/R_0}^1 \frac{ds}{ \beta(\frac{1}{R_0} \ln s + 1 - s +l_0) s}$

We can’t do this integral but if we set $s = 1- z$ and $z<< 1$, then we can expand $\ln s = -\epsilon$ and obtain

$T= \int_0^T dt =\int_0^{l_p} \frac{dz}{ \beta(-\frac{1}{R_0}z + z +l_0) (1-z)}$

where $l_p = 1-1/R_0$. This can be re-expressed as

$T=\frac{1}{ \beta (l_0+l_p)}\int_0^{l_p} (\frac{1}{1-z} + \frac{l_p}{l_p z + l_0}) dz$

which is integrated to

$T= \frac{1}{ \beta (l_0+l_p)} (-\ln(1-l_p) + \ln (l_p^2 + l_0)-\ln l_0)$

If we assume that $l_0<< l_p$, then we get an expression

$T \approx \sigma \frac{\ln (R_0l_p^2/l_0)}{ R_0 -1}$

So, $T$ is proportional to the recovery time $\sigma$ and inversely related to $R_0$ as expected but if $l_0$ is very small (say 0.00001) compared to $R_0$ (say 3) then $\ln R_0/l_0$ can be big (around 10), which may explain why it takes so long for the pandemic to get started in a region. If the infection rate is very low in a region, the time it takes a for a super-spreader event to make an impact could be much longer than expected (10 times the infection clearance time (which could be two weeks or more)).

Addendum 2020-06-26: Fixed typos in equations and added clarifying text to last paragraph

Addendum 2020-06-27: The approximation for total infected is not very good when $R_0$ is near 1, a better one can be obtained by expanding the exponential to quadratic order in which case you get the new formula for the

$s = \frac{1}{{R_{0}}^2} ( e^{R_0} - R_0 - \sqrt{(e^{R_0}-R_0)^2 - 2{R_0}^2})$

However, for $R_0$ near 1, a better expansion is to substitute $z = 1-s$ into equation (*) and obtain

$l = \frac{1}{R_0} \ln 1-z + z + l_0$

Set $l=0$, after rearranging and exponentiating,  obtain

$1 - z = e^{-R_0(l_0+z)}$, which can be expanded to yield

$1- z = e^{-R_0 l_0}(1 - R_0z + R_0^2 z^2/2$

Solving for $z$ gives the total fraction infected to be

$z = (R_0 -e^{R_0l_0} + \sqrt{(R_0-e^{R_0l_0})^2 - 2 R_0^2(1-e^{R_0l_0})})/R_0^2$

This took me much longer than it should have.

# The fatal flaw of the American Covid-19 response

The United States has surpassed 2 million official Covid-19 cases and a 115 thousand deaths. After three months of lockdown, the country has had enough and is reopening. Although it has achieved its initial goal of slowing the growth of the pandemic so that hospitals would not be overwhelmed, the battle has not been won. We’re not at the beginning of the end; we may not even be at the end of the beginning. If everyone in the world could go into complete isolation, the pandemic would be over in two weeks. Instead, it is passed from one person to the next in a tragic relay race. As long as a single person is shedding the SARS-CoV-2 virus and comes in contact with another person, the pandemic will continue. The pandemic in the US is not heading for extinction. We are not near herd immunity and R0 is not below one. By the most optimistic yet plausible scenario, 30 million people have already been infected and 200 million will never get it either by having some innate immunity or by avoiding it through sheltering or luck. However, that still leaves over 100 million who are susceptible of which about a million will die if they all catch it.

However, the lack of effectiveness of the response is not the fatal flaw. No, the fatal flaw is that the US Covid-19 response asks one set of citizens to sacrifice for the benefit of another set. The Covid-19 pandemic is a story of three groups of people. The fortunate third can work from home, and the lockdown is mostly just an inconvenience. They still get paychecks while supplies and food can be delivered to their homes. Sure it has been stressful and many of have forgone essential medical care but they can basically ride this out for as long as it takes. The second group who own or work in shuttered businesses have lost their income. The federal rescue package is keeping some of them afloat but that runs out in August. The choice they have is to reopen and risk getting infected or be hungry and homeless. Finally, the third group is working to allow the first group to remain in their homes. They are working on farms, food processing plants, and grocery stores. They are cutting lawns, fixing leaking pipes, and delivering goods. They are working in hospitals and nursing homes and taking care of the sick and the children of those who must work. They are also the ones who are most likely to get infected and spread it to their families or the people they are trying to take care of. They are dying so others may live.

A lockdown can only work in a society if the essential workers are adequately protected and those without incomes are supported. Each worker should have an N100 mask, be trained how to wear it and be tested weekly. People in nursing homes should be wearing hazmat suits. Everyone who loses income should be fully compensated. In a fair society, everyone should share the risks and the pain equally.

The global plateau turned out to just be a pause and the growth in new cases continues. The rise seems to be mostly driven by increases in Brazil, India, and until very recently Russia with plateauing in the US and European countries as they relax their mitigation policies. The pandemic is not over by a long shot. There will most certainly be further growth in the near future.

# How much Covid-19 testing do we need?

There is a simple way to estimate how much SARS-CoV-2 PCR testing we need to start diminishing the COVID-19 pandemic. Suppose we test everyone at a rate $f$, with a PCR test with 100% sensitivity, which means we do not miss anyone who is positive but we could have false positives. The number of positives we will find is $f p$, where  $p$ is the prevalence of infectious individuals in a given population. If positive individuals are isolated from the rest of the population until they are no longer infectious with probability $q$, then the rate of reduction in prevalence is $fqp$. To reduce the pandemic, this number needs to be higher than the rate of pandemic growth, which is given by $\beta s p$, where $s$ is the fraction of the population susceptible to SARS-CoV-2 infection and $\beta$ is the rate of transmission from an infected individual to a susceptible upon contact. Thus, to reduce the pandemic, we need to test at a rate higher than $\beta s/q$.

In the initial stages of the pandemic $s$ is one and $\beta = R_0/\sigma$, where $R_0$ is the mean reproduction number, which is probably around 3.7 and $\sigma$ is the mean rate of becoming noninfectious, which is probably around 10 to 20 days. This gives an estimate of  $\beta_0$ to be somewhere around 0.3 per day. Thus, in the early stages of the pandemic, we would need to test everyone at least two or three times per week, provided positives are isolated. However, if people wear masks and avoid crowds then $\beta$ could be reduced. If we can get it smaller then we can test less frequently. Currently, the global average of $R_0$ is around one, so that would mean we need to test every two or three weeks. If positives don’t isolate with high probability, we need to test at a higher rate to compensate. This threshold rate will also go down as $s$ goes down.

In fact, you can just test randomly at rate $f$ and monitor the positive rate. If the positive rate trends downward then you are testing enough. If it is going up then test more. In any case, we may need less testing capability than we originally thought, but we do need to test the entire population and not just suspected cases.

# The Covid-19 plateau

For the past five weeks, the appearance rate of Covid-19 cases has plateaued at about a hundred thousand new cases per day. Just click on the Daily Cases Tab on the JHU site to see for yourself. This is quite puzzling because while individual nations and regions are rising, falling, and plateauing independently, the global total is flat as a pancake. A simple resolution to this seeming paradox was proposed by economist John Cochrane (see his post here). The solution is rather simple but the implications as I will go into more detail below are far reaching. The short answer is that if the world (either through behavior or policy) reacts to the severity of Covid-19 incrementally then a plateau will arise. When cases go up, people socially distance, and the number goes down, when cases go down, they relax a little and it goes back up again.

This can be made more precise with the now-famous SIR model. For the uninitiated, SIR stands for Susceptible Infected Recovered model. It is a simple dynamical model of disease propagation that has been in use for almost a century. The basic premise of an SIR model is that at any given time, the proportion of the population is either infected with the virus I, susceptible to infection S, or recovered from infection and no longer susceptible R. Each time an S comes across an I, it has a chance of being infected and becoming another I. An I will recover (or die) with some rate and become an R. The simplest way to implement an SIR model is to assume that people interact completely randomly and uniformly across the population and the rate of transmission and recovery is uniform as well. This is of course a gross simplification and ignores the complexity and diversity of social interactions, the mechanisms of actual viral transmission, and the progression of disease within individuals. However, even though it misses all of these nuances, it captures many of the important dynamics of epidemics. In differential equation form, the SIR model is written as

$\frac{dS}{dt} = -\frac{\beta}{N} S I$

$\frac{dI}{dt} = \frac{\beta}{N} S I - \sigma I$   (SIR model)

where $N$ is the total number of people in the population of interest. Here, $S$ and $I$ are in units of number of people.  The left hand sides of these equations are derivatives with respect to time, or rates.  They have dimensions or units of people per unit time, say day. From this we can infer that $\beta$ and $\sigma$ must have units of inverse day (per day) since $S$, $I$, and $N$ all have units of numbers of people. Thus $\beta$ is the infection rate per day and $\sigma$ is the recovery/death rate per day. The equation assumes that the probability of an $S$ meeting an $I$ is $I/N$. If there was one infected person in a population of a hundred, then if you were to interact completely randomly with everyone then the chance you would run into an infected person is 1/100. Actually, it would be 1/99 but in a large population, the one becomes insignificant and you can round up. Right away, we can see a problem with this assumption. I interact regularly with perhaps a few hundred people per week or month but the chance of me meeting a person that had just come from Australia in a typical week is extremely low. Thus, it is not at all clear what we should use for $N$ in the model. The local population, the regional population, the national population?

The model assumes that once an $S$ has run into an $I$, the rate of transmission of the virus is $\beta$. The total rate of decrease of $S$ is the product of $\beta$ and $SI/N$. The rate of change of $I$ is given by the increase due to interactions with $S$ and the decrease due to recovery/death $\sigma I$. These terms all have units of person per day. Once you understand the basic principles of constructing differential equations, you can model anything, which is what I like to do. For example, I modeled the temperature dynamics of my house this winter and used it to optimize my thermostat settings. In a post from a long time ago, I used it to model how best to boil water.

Given the SIR model, you can solve them to get how $I$ and $S$ will change in time. The SIR model is a system of nonlinear differential equations that do not have what is called a closed-form solution, meaning you can’t write down that $I(t)$ is some nice function like $e^{t}$ or $\sin(t)$. However, you can solve them numerically on a computer or infer properties of the dynamics directly without actually solving them. For example, if $\beta SI/N$ is initially greater than $\sigma I$, then $dI/dt$ is positive and thus $I$ will increase with time. On the other hand, since $dS/dt$ is always negative (rate of change is negative), it will decrease in time. As $I$ increases and $S$ decreases, since $S$ is decreasing at a faster rate than $I$ is increasing because $\sigma I$ is slowing the growth of $I$, then at some point $\beta SI/N$ will equal $\sigma I$ and $dI/dt=0$. This is a stationary point of $I$.  However, it is only a momentary stationary point because  $S$ keeps decreasing and this will make $I$ start to decrease too and thus this stationary point is a maximum point. In the SIR model, the stationary point is given by the condition

$\frac{dI}{dt} = 0 = \frac{\beta}{N}SI -\sigma I$ (Stationary condition)

which you can solve to get either $I = 0$ or $\beta S/N = \sigma$. The $I=0$ point is not a peak but just reflects the fact that there is no epidemic if there are no infections. The other condition gives the peak:

$\frac{S}{N} = \frac{\sigma}{\beta} \equiv \frac{1}{R_0}$

where $R_0$ is the now-famous R naught or initial reproduction number. It is the average number of people infected by a single person since $\beta$ is the infection rate and $\sigma$ is the infection disappearance rate, the ratio is a number. The stationary condition gives the herd immunity threshold. When the fraction of $S$ reaches $S^*/N = latex 1/R_0$ then the pandemic will begin to decline.  This is usually expressed as the fraction of those infected and no longer susceptible, $1-1/R0$. The 70% number you have heard is because $1-1/R_0$ is approximately 70% for $R_0 = 3.3$, the presumed value for Covid-19.

A plateau in the number of new cases per day is an indication that we are at a stationary point in $I$. This is because only a fraction of the total infected are counted as cases and if we assume that the case detection rate is uniform across all $I$, then the number of new cases per day is proportional to $I$. Thus, a plateau in cases means we are at a stationary point in $I$, which we saw above only occurs at a single instance in time. One resolution to this paradox would be if the peak is broad so it looks like a plateau. We can compute how broad the peak is from the second derivative, which gives the rate of change of the rate of change. This is the curvature of the peak. Taking the second derivative of the I equation in the SIR model gives

$\frac{d^2I}{dt^2} = \frac{\beta}{N} (\frac{dS}{dt} I + S\frac{dI}{dt}) - \sigma \frac{dI}{dt}$

Using $dI/dt=0$ and the formula for $S^*$ at the peak, the curvature is

$\frac{d^2I}{dt^2} = \frac{\beta}{N} \frac{dS}{dt} I =-\left( \frac{\beta}{N}\right)^2 S^* I^2 =- \frac{I^2\beta^2}{NR_0}$

It is negative because at a peak the slope is decreasing. (A hill is easier to climb as you round the top.)  There could be an apparent plateau if the curvature is very small, which is true if $I^2 \beta^2$ is small compared to $N R_0$. However, this would also mean we are already at the herd immunity threshold, which our paper and recent anti-body surveys predict to be unlikely given what we know about $R_0$.

If a broad peak at the herd immunity threshold does not explain the plateau in new global daily cases then what does? Cochrane’s theory is that $\beta$ depends on $I$.  He postulated that $\beta = \beta_0 e^{-\alpha I/N}$,where $\beta_0$ is the initial infectivity rate, but any decreasing function will do. When $I$ goes up, $\beta$ goes down. Cochrane attributes this to human behavior but it could also be a result of policy and government mandate. If you plug this into the stationary condition you get

$\frac{\beta_0}{N} S^* e^{-\alpha I^*/N} -\sigma = 0$

or

$I^* =-\frac{N}{\alpha} \log(\frac{N\sigma}{S^*\beta_0})$

and the effective reproduction number $R_0$  is one.

However, this is still only a quasi-stationary state because if $I$ is a constant $I^*$, then $S$ will decrease as $dS/dt = -\frac{\beta}{N}SI^*$, which has solution

$S(t) = N e^{-(\beta I^*/N) t}$     (S)

Plugging this into the equation for  $I^*$ gives

$I^* =-\frac{N}{\alpha} \log(\frac{\sigma}{\beta_0}e^{(\beta I^*/N) t}) = \frac{N}{\alpha}\log R_0 - \frac{\beta I^*}{\alpha} t$

which means that $I$ is not really plateaued but is decreasing slowly as

$I^* = \frac{N}{\alpha+\beta t}\log R_0$

We can establish perfect conditions for a plateau if we work backwards. Suppose again that $I$ has plateaued at $I^*$.  Then, $S(t)$ is given by equation (S).  Substituting this into the (Stationary Condition) above then gives $0 = \beta(t) e^{-(\beta I^*/N) t} -\sigma$ or

$\beta(t) = \sigma e^{(\beta I^*/N) t}$

which means that the global plateau is due to us first reducing $\beta$ to near $\sigma$, which halted the spread locally, and then gradually relaxing pandemic mitigation measures so that $\beta(t)$ is creeping upwards back to it’s original value.

The Covid-19 plateau is both good news and bad news. It is good news because we are not seeing exponential growth of the pandemic. Globally, it is more or less contained. The bad news is that by increasing at a rate of a hundred thousand cases per day, it will take a long time before we reach herd immunity. If we make the naive assumption that we won’t reach herd immunity until 5 billion people are infected then this pandemic could blunder along for $5\times 10^9/10^5 = 5\times 10^4 = 50000$ days! In other words, the pandemic will keep circling the world forever since over that time span, babies will be born and grow up. Most likely, it will become less virulent and will just join the panoply of diseases we currently live with like the various varieties of the common cold (which are also corona viruses) and the flu.

# A Covid-19 Manhattan Project

Right now there are hundreds if not thousands of Covid-19 models floating out there. Some are better than others and some have much more influence than others and the ones that have the most influence are not necessarily the best. There must be a better way of doing this. The world’s greatest minds convened in Los Alamos in WWII and built two atomic bombs. Metaphors get thrown around with reckless abandon but if there ever was a time for a Manhattan project, we need one now. Currently, the world’s scientific community has mobilized to come up with models to predict the spread and effectiveness of mitigation efforts, to produce new therapeutics and to develop new vaccines. But this is mostly going on independently.

Would it be better if we were to coordinate all of this activity. Right now at the NIH, there is an intense effort to compile all the research that is being done in the NIH Intramural Program and to develop a system where people can share reagents and materials. There are other coordination efforts going on worldwide as well.  This website contains a list of open source computational resources.  This link gives a list of data scientists who have banded together. But I think we need a world wide plan if we are ever to return to normal activity. Even if some nation has eliminated the virus completely within its borders there is always a chance of reinfection from outside.

In terms of models, they seem to have very weak predictive ability. This is probably because they are all over fit. We don’t really understand all the mechanisms of SARS-CoV-2 propagation. The case or death curves are pretty simple and as Von Neumann or Ulam or someone once said, “give me 4 parameters and I can fit an elephant, give me 5 and I can make it’s tail wiggle.” Almost any model can fit the curve but to make a projection into the future, you need to get the dynamics correct and this I claim, we have not done. What I am thinking of proposing is to have the equivalent of FDA approval for predictive models. However, instead of a binary decision of approval non-approval, people could submit there models for a predictive score based on some cross validation scheme or prediction on a held out set. You could also submit as many times as you wish to have your score updated. We could then pool all the models and produce a global Bayesian model averaged prediction and see if that does better. Let me know if you wish to participate or ideas on how to do this better.

# Covid-19 talk

Here are the slides for my webinar at FDA today .

Here is the medrXiv link to our Covid-19 paper.  We actually have more updated versions with nicer graphs, which we will upload shortly.

# COVID-19 Paper

Global prediction of unreported SARS-CoV2 infection from observed COVID-19 cases

Carson C. Chow 1*, Joshua C. Chang 2,3, Richard C. Gerkin 4*, Shashaank Vattikuti 1*

1Mathematical Biology Section, LBM, NIDDK, National Institutes of Health2Epidemiology and Biostatistics Section, Rehabilitation Medicine, Clinical Center, National Institutes of Health 3 mederrata 4 School of Life Sciences, Arizona State University

*For correspondence contact carsonc@nih.gov, josh@mederrata.com, rgerkin@asu.edu, vattikutis@nih.gov

Summary: Estimation of infectiousness and fatality of the SARS-CoV-2 virus in the COVID-19 global pandemic is complicated by ascertainment bias resulting from incomplete and non-representative samples of infected individuals.  We developed a strategy for overcoming this bias to obtain more plausible estimates of the true values of key epidemiological variables.  We fit mechanistic Bayesian latent-variable SIR models to confirmed COVID-19 cases, deaths, and recoveries, for all regions (countries and US states) independently. Bayesian averaging over models, we find that the raw infection incidence rate underestimates the true rate by a factor, the case ascertainment ratio CARt that depends upon region and time. At the regional onset of COVID-19, the predicted global median was 13 infections unreported for each case confirmed (CARt = 0.07 C.I. (0.02, 0.4)). As the infection spread, the median CARt rose to 9 unreported cases for every one diagnosed as of April 15, 2020 (CARt = 0.1 C.I. (0.02, 0.5)).  We also estimate that the median global initial reproduction number R0 is 3.3 (C.I (1.5, 8.3)) and the total infection fatality rate near the onset is 0.17% (C.I. (0.05%, 0.9%)). However the time-dependent reproduction number Rt and infection fatality rate as of April 15 were 1.2 (C.I. (0.6, 2.5)) and 0.8% (C.I. (0.2%,4%)), respectively.  We find that there is great variability between country- and state-level values.  Our estimates are consistent with recent serological estimates of cumulative infections for the state of New York, but inconsistent with claims that very large fractions of the population have already been infected in most other regions.  For most regions, our estimates imply a great deal of uncertainty about the current state and trajectory of the epidemic.

# Exponential growth

The spread of covid-19 and epidemics in general is all about exponential growth. Although, I deal with exponential growth in my work daily, I don’t have an immediate intuitive grasp for what it means in life until I actually think about it. To me I write down functions with the form $e^{rt}$ and treat them as some abstract quantity. In fact, I often just look at differential equations of the form

$\frac{dx}{dt} = r x$

for which $x = A e^{rt}$ is a solution, where $A$ is any constant number, because the derivative of an exponential function is an exponential function. One confusing aspect of discussing exponential growth with lay people is that the above equation is called a linear differential equation and often mathematicians or physicists will simply say the dynamics are linear or even the growth is linear, although what we really mean is that the describing equation is a linear differential equation, which has exponential growth or exponential decay if $r$ is negative.

If we let $t$ represent time (say days) then $r$ is a growth rate. What $x = e^{rt}$ actually means is that every day $x$, e.g. number of people with covid-19, is increasing by a factor of $e$, which is about 2.718.  Now, while this is the most convenient number for mathematical manipulation, it is definitely not the most convenient number for intuition. A common alternative is to use 2 as the base of the exponent, rather than $e$. This then means that tomorrow you will have double what you have today. I think 2 doesn’t really convey how fast exponential growth is because you say well I start with 1 today and then tomorrow I have 2 and the day after 4 and like the famous Simpson’s episode when Homer is making sure that Bart is a name safe from mockery, goes through the alphabet and says “Aart, Bart, Cart, Dart, Eart, okay Bart is fine,”  you will have to go to day 10 before you get to a thousand fold increase (1024 to be precise), which still doesn’t seem too bad. In fact 30 days later is 2 to the power of 30 or $2^{30}$, which is only 10 million or so. But on day 31 you have 20 million and day 32 you have 40 million. It’s growing really fast.

Now, I think exponential growth really hits home if you use 10 as a base because now you simply add a zero for every day: 1, 10, 100, 1000, 10000, 100000. Imagine if your stock was increasing by a factor of 10 every day. Starting from a dollar you would have a million dollars by day 6 and a billion by day 9, be richer than Bill Gates by day 11, and have more money than the entire world by the third week. Now, the only difference between exponential growth with base 2 versus base 10 is the rate of growth. This is where logarithms are very useful  because they are the inverse of an exponential. So, if $x = 2^3$ then $\log_2 x = 3$ (i.e. the log of x in base 2 is 3). Log just gives you the power of an exponential (in that base). So the only difference between using base 2 versus base 10 is the rate of growth. For example $10^x = 2^{(log_2 10) x}$, where $\log_2 (10) = 3.3$.  So an exponential process that is doubling every day is increasing by a factor of ten every 3.3 days.

The thing about exponential growth is that most of the action is in the last few days. This is probably the hardest part to comprehend.  So the reason they were putting hospital beds in the convention center in New York weeks ago is because if the number of covid-19 cases is doubling every 5 days then even if you are 100 fold under capacity today, you will be over capacity in 7 doublings or 35 days and the next week you will have twice as many as you can handle.

Flattening the curve means slowing the growth rate. If you can slow the rate of growth by a half then you have 70 days before you get to 7 doublings and max out capacity. If you can make the rate of growth negative then the number of cases will decrease and the epidemic will flame out. There are two ways you can make the growth rate go negative. The first is to use external measures like social distancing and the second is to reach nonlinear saturation, which I’ll discuss in a later post. This is also why social distancing measures seem so extreme and unnecessary because you need to apply it before there is a problem and if it works then those beds in the convention center never get used. It’s a no win situation, if it works then it will seem like an over reaction and if it doesn’t then hospitals will be overwhelmed. Given that 7 billion lives and livelihoods are at stake, it is not a decision to be taken lightly.

# Stanford Santa Clara Study

Here is Andrew Gelman’s take on the Stanford Santa Clara Study.

Aside from the fact that this could all be false positive, it is also not clear how specific the anti-body test is.  The common cold is also a coronavirus so it could be that it is picking up people who have a cold.


text:
## Sunday, July 06, 2008

### Day Four of Eurostrings 2008

The morning began breakfastless, and a little breathless, rushing from the shower to the conference. By day four conference fatigue was beginning to set in. It had absoulutely nothing to do with the Belgian-beer filled discussions. None whatsoever. However day four proved to give second wind to the meeting, filled with very interesting talks that I hope to give a flavour of here.

The morning review lecture was given by Samir D. Mathur, he does not like horizons, well at least those around black holes. One has to sympathise - is it really acceptable to cut a singularity out of a theory? Mathur prefers a fuzzball picture, where the black hole horizon becomes a statistical entity, emerging macrosopically - the canonical comparison is with temperature in the thermodynamical picture. In thermodynamics the temperature is a statistical quantity that can be measured over a large number of microscopic states, but if you sat on a hydrogen molecule (well maybe you already are, but what I mean is, if you had the molecule's view of a gas) you would be able to say a few things about your nearest neighbours relative velocities, and only with a large amount of time would you collect enough information to speak with confidence of the average molecular speed, or temperature. To a microscopic state the temperature is an odd concept, supposedly the black hole horizon is also an odd concept to a gravitational microstate. The fuzzball proposal functionally aims to reproduce the macrosopic black hole phenomena from collections of microstates. The brane microstates themselves do not have horizons in this setup, the horizon appears in the averaging over a large number of brane states. Old and familiar properties of black holes are reproduced in this picture, light can be trapped behind the horizon by an elaborate setup up of light deflecting states, Hawking temperatures can be reproduced and lately Hawking radiation can be produced by pair-production. For an introduction to the proposal you can read his papers here and here. The proposal offers a way to side-step Hawking's information paradox. Mathur's discussion of the information paradox can be read in this preprint, where he aims to make a review using pictures.

\begin{digression}
Kurt Vonnegut used to use a technique of repeating a small, catchy phrase when something of particular note happened in a sentence of his (e.g. in Cat's Cradle each reference to slipping off the mortal coil earns a: so it goes, or in Timequake ting-a-ling is the catchphrase). I think everytime someone tries to explain something with pictures I would like to insert a cowbell noise. So here's to Mathur: *cowbell*.
\end{digression}

Mathur's title this morning was "Lessons from resolving the information paradox". He threw out the notion two charge non-extremal black holes have a singular throat in the spacetime, the geometry may become complicated but not singular. We heard aboutandwhich you can read about in the links.

After coffee, we had talks from Eric D'Hoker ("Exact 1/2 BPS solutions in type IIB and M-theory"), Dario Francia ("Unconstrained higher spins and current exchanges") and Diego Chialva ("Chain inflation revisited").

## Thursday, July 03, 2008

### Halfday Wednesday

Well Wednesday of Eurostrings 2008 was a half day - the afternoon was left free to enjoy the pleasures of Amsterdam, or to work furiously on the latest Bagger-Lambert paper. So, of course, in honour of the half day here's a half-blog entry. Instead of writing only half sentences I will aim to halve my number of full sentences.

The weather in Amsterdam understood that it was a half day for our conference. Upto midday it was a balmy 27 degrees and sunny, but as I settled in for lunch an almighty, apocalyptic thunder storm came in, as you can perhaps see in the photo (starring Erik Tonni [left] and Diederik Roest [not left]). Erik and Diederik suggested that Bagger-Lambert theory may be getting too close to the truth for the almighty being's liking, and like the Tower of Babel, was about to be toppled by the ensuing thunderstorm. The storm passed, while I ate a very nice sandwich. I am not suggesting any causal connection between the weather and my digestion, but let's not rule it out.

Due to a lack of sleep here in Amsterdam, I all but missed the morning session (not a smart move on a half-day) so I am one of the worst people to tell you what was discussed. However let me put up the titles and one or two suggested papers. Perhaps a fellow Eurostring-ite who may stumble this way can let me know some more about the talks? The schedule was:
• "Strongly coupled Quark-Gluon Plasma and AdS/CFT" by Edward Shuryak, see, perhaps, the paper
• "Is the AdS S-matrix simple?" by Romuald Janik (I was told that the short answer is: no)
• Herman Verlinde gave a blackboard talk.
• "Building a holographic superconductor" by
I caught the final talk, on the relatively hot ;) topic of trying to undertand superconduction through the AdS/CFT correspondence. This was the same topic that Denef discussed on Tuesday, but today the role of the AdS background was emphasised in order to capture the charged scalar field around the black hole. The picture does not carry over to the Minkowksi background. Recall that the AdS/CFT correspondence has been invoked to describe heavy ion collisions and condensed matter physics, and even the quantum Hall effect has had a dual gravitational description given. The aim at present is to find the gravitational dual picture of superconductivity. The aim is to find a black hole solution that at some point grows hair. To read more about this see the preprint here.

The afternoon was filled with discussion and imported Coca-Cola.

The evening with discussion and beer.

Your humble correspondent flagellates himself gently for missing the talks.

## Tuesday, July 01, 2008

### Day Two of Eurostrings 2008

Another day, another cup of soup and a sandwich for lunch. Today it was ham soup and a pineapple sandwich (my Dutch and my taste buds are not good enough to understand what the other ingredients were).

This morning we had a review lecture on the pure spinor formalism by Nathan Berkovits. If you want to learn this formalism, why not start with the reviews here (and here [or the blog article here]) and then end with the paper here. If you do this in one-and-a-half hours, but ensure you explain it to yourself very clearly, you will have your own simulation of this morning's nice review. Or, if you are feeling little tired, you could watch the video of Yaron Oz's lectures to the CERN winter school.

Following Berkovits, Andreas Gustavsson, the third man of the present Bagger-Lambert multiple membranes revolution, spoke on..."Multiple M2's". He included his paper from last year and his more recent work on how the membrane triple product identity aids amplitude calculations. His talk was followed by thirty minutes from Frederik Denef, talking under the title of "the string landscape of quantum critical superconductors", which refers to work in progress with Sean Hartnoll. The central theme was that there are two landscapes in physics. The string theory landscape, constructed inside a unique fundamental theory (M-theory), with low energy excitations (gravitons, "3-formons" :) and superpartners) and where the intricate landscape is considered "party-spoiling". The second landscape is the condensed matter landscape, constructed from a unique theory (the standard model), with low energy excitations (neutrons, protons and electrons) and where the landscpe is still intricate but is useful. The heuristic message is that these two landscapes may be very similar. Denef gave us a toy model two dimensional array of spin one-half particles that illustrated the idea of quantum critical points - points in phase space where a second order phase transition occurs at zero temperature. The crucial features are all summed up in his graph:
A second example of criticality involved superconductors and whose features were given by a toy-modelin two dimensions: a Bose-Hubbard model. There is a phase transition between being an insulator and being a superconductor. This picture was to be compared with a charged scalar field in a Reisner-Nordstrom AdS background. The idea (due to Gubser) was that there is a quantum critical point here too that separates insulation from superconductivity. Namely when electrostatic repulsion of the charged scalar is larger than its gravitational attraction towards the singularity in the space-time, then a halo or cloud of charge forms around the black-hole. This is the superconducting picture. Otherwise the charge falls into the horizon and we have the insulating picture. We are to expect to hear more about this superconducting phase from Gary Horowitz tomorrow. Denef told us one could be optimistic that this picture could be constructed in string theory. Citing the Arkani-Hamed, Motl, Nicolis and Vafa, Denef said that Reissner-Nordstrom black-holes should be able to decay and so there was an expectation that the electrostatic repulsion > gravitational attraction regime should exist. Perhaps microscopic physics and macroscopic physics are not so different after all?

In the last morning talk, Giulio Bonelli spoke under the title "On gauge/string correspondence and mirror symmetry" and you can read his preprint here. In the afternoon we heard an exuberant Vijay Balasubramanian talk about getting something from nothing. His title was "Statistical predictions from anarchic field theory landscapes". Out of chaos certain coarse-grained properties could become predictable he said, read more in the The final thirty minute talk of the day was given by Diederik Roest, who talked on my favourite subject: "The Kac-Moody algebras of supergravity". The talk covered decomposition of the algebra, the correspondence between de-forms, top forms and E(11) (preprint) and also his work with Axel Kleinschmidt on identifying the Kac-Moody algebras that are appropriate to three dimensional scalar theories with a quarter or less of the full supersymmetry (preprint). After coffee, we had a gong show for some researchers but unfortunately we had no gong. Poor Pierre Vanhove must have been kicking himself that he hadn't packed his legendary cowbell...

On my walk back home I encountered two mathematical omens in odd places, first a van that seemed like it could go to infinity and beyond:
And, second, I saw the hotel I should have been staying at:
Unfortunately there were no giraffes helping zebras to escape the circus... despite this bizarre story I'm not sure that truth is stranger than fiction. In fiction the same story could have happened but the giraffe might have been smoking a cuban cigar and saying that he loved it when a plan came together and all the while Pierre Vanhove skipping in front leading the animals with the merry din of his cowbell.

## Monday, June 30, 2008

### Eurostrings 2008

The tram door closed viciously on Pierre Vanhove's rucksack and off it tootled away from Centraal Station (the tram, not the rucksack). My travelling companions were all on board, and only I was left behind with the rest of the amputated tram queue. The man behind me in the queue said "welcome to Amsterdam" in friendly English. We struck up a conversation and he asked what kind of conference I was attending. I told him it was physics, "serious" was his reply. I asked him what he recommended visiting while I was in the city, he said that for him it was all about wandering around and taking it all in. I pushed him and asked for one thing to see in particular, "the red light district". Or perhaps the upstairs floor of a cafe with a particularly good view over a canal that the tourists lack the energy to investigate. Even though my trip to Amsterdam was only beginning I wondered if this mixture of sites might not give a good impression of Amsterdam. From my walks today I am not disappointed. It is a beautiful city, the colourful boats that crowd the canal are laden with multicoloured bric-a-brac, the buildings on the banks appear disordered like the teeth of a friendly giant and yet each and every one appears spic and span upon inspection, even the cyclists speeding unstoppably down neat cycle paths carry their loved ones side-saddle on the back - a jumble of colourful clothes flying behind in the sunlight. For every ordered thing here there is a controlled disorder that is very pleasant to watch.

I am here for Eurostrings 2008, a smaller, quieter version of Strings, but which is packed with excellent speakers and an interesting crowd of participants. You're a String (thanks Per!) is being hosted this year by the University of Amsterdam, apparently it's in the same venue as Strings 1997. The organisation has been superb and we have had an excellent first day of talks which I will try and summarise here (and maybe expand upon later).

We began the day hearing Ashoke Sen talking about dyons in N=4 as discussed in his recent papers here and here. He described to us the protected index associated to $\frac{1}{4}$BPS states, labelled d(Q,P). Here Q is the electric charge and P the magnetic charge. It is the number of $\frac{1}{4}$ BPS states weighted by $(-1)^{2h}$, where $h$ is the helicity. That d(Q,P) is protected means that it does not change under a continuous variation of the coupling constant or moduli of the theory. In fact if the coupling constant is varied onl the BPS states remain and contribute to the counting. However d(Q,P) can make sudden jumps over "walls of marginal stability" - these are places where the $\frac{1}{4}$ BPS states may decay into $\frac{1}{2}$ BPS states. The domain wall itself is defined by four parameters which become discrete due to charge quantisation. Consequently d(Q,P) appears to depend not only on Q and P but also on the domain in which the moduli lie. One can calculate the partition function from d(Q,P) expressed as a function of T-duality invariant terms: $Q^2, P^2, Q \cdot P$, a discrete T-duality invariant and also the domain in which the protected index is calculated. It transpires that the partition function converges after analytic continuation of some of the variables but in "all known examples" the partition function ends up being invariant of the domain one started calculating in. What can one say about how the microscopic dyon partition function reproduces the macroscopic black hole entropy count? Well, first, within the domain of applicabilit of the partition function the entropy calculation is in agreement with the inclusion of the four derivative Gauss-Bonnet terms. So far, so good. But what about the phenomenon of discrete value changes in d(Q,P) as one jumps over domain walls? For the single black hole this microscopic property cannot be reproduced macroscopically, but for the multicentre black holes it agrees perfectly - one can see this is possible since for different values of moduli space multicentred balck holes may cease to exist as one crosses walls of marginal stability. At the end of his talk Sen focussed on how one could work towards a complete comparison between $S_{BH}$ and $S_{micro}$ (since a number of terms had been exponentially suppressed in the earlier comparison in order to compare like-for-like). The full picture would include both higher derivative corrections and quantum corrections, for the former one can use Wald's formula to make the calculations, for the latter Sen proposes a close scrutiny of $AdS_2/CFT_1$ duality. Starting with the near horizon geometry of a black hole and then analytcally continuing to the Euclidean solution one finds the $AdS_2$ metric. The partition function in this metric is the exponential of minus this Euclidean action, and is used together with a cut off to obtain:
$Z_{AdS_2}\equiv e^{Kr_0+S_{BH}-2\pi \bar{e}\cdot \bar{q}$
By the AdS/CFT correspondence one can exactly calculate the partition function for the CFT:
$Z_{CFT_1}\equiv e^{-2\pi r_0 E_0} \sum d(\bar{q}) e^{-2\pi \bar{e}\cdot \bar{q}}$
Where $E_0$ is a rescaled ground state energy. Now the two expressions may be equated and the black hole entropy examined.

Ionnis Papadimitriou then spoke to us about how to rigourously define an asymptotically flat spacetime and then considered its holographic description - you can read more about this here. Pierre Vanhove, minus his infamous cow bell, spoke next on the no-triangle hypothesis (update: why not read Lubos' analysis of the situation) for ${\cal N}=8$ SuGra, which is, of course, just ${\cal N}=4$ squared - or at least it has many remarkable similarities to make such a conjecture plausible. It turns out that the no triangle hypothesis should really be called the no-triangles, no bubbles and, in fact, just boxes in the one loop scattering amplitudes hypothesis - but that's not very catchy. For multiloop scattering diagrams, the no-triangle hypothesis informs us about the one-loop sub-terms that remain when one makes suitable cuts in the multiloop diagram. Pierre told us, without once ringing any kind of bell, not for cow, horse, nor wild mountain goat, that since the cancelltations in the gravity theory are due to the (colourless) gauge invariance the hypothesis can also be applied to other theories with less SuSy than ${\cal N}=8$. Pierre finished enigmatically by telling the audience that if ${\cal N}=8$ is divergent he bets that it diverges at 9-loops. He didn't say how much he bets.

In the afternoon, following a sparse lunch of soup and a sandwich, Hirosi Ooguri talked under the title of Current Gauge Correlators for General Gauge Mediation - the idea was to extend the region of strong interactions from just the hidden sector to include the mediating sector that gives rise to the visible sector. You can read his paper with his collaborators on this subject here. After Ooguri, Marco Zagermann told us that $\frac{T^2}{Z_2}$ is the pillow, and invited us to revisit D3/D7 brane inflation models. The inflaton is the separation distance between a D7 with flux turned on and a parallel D3. At the end of the period of inflation, cosmic strings condensed - the associated preprint is available here. Finally Ki-Myeong Lee talked about "New" ${\cal N}=5,6$ Superconformal Chern-Simons Theories. Since this is work related to the increasingly popular multiple M2 work of Bagger-Lambert and Gustavsson, Lee told us that he had checked and he thought his models were still new at the time of talking and they would be published on the arxiv tomorrow (1st July, 2008 - the preprint can be found here). Lee showed us how to introduce a twisted hypermultiplet into Gaiotto-Witten theory in order to reproduce the 8 scalars of the Bagger-Lambert work. Hey presto, a new technique for building interesting theories was born. The last talk of the day was given by Niko Jokela from Helsinki on the interesting topic of N-Point Functions in the Rolling Tachyon Background, the arxiv preprint is here.

At the end of the day we had a reception hosted at the Academy of Arts and Sciences, of which Robert Dijkgraaf is the President. He told us that the academy was actually seven years older than the Netherlands and told a story of his predecessor who was approached by Vladimir Putin at a formal dinner and was greeted with the line "so you are a President too", Dijkgraaf's predecessor replied that "they came in all shapes and sizes".

After the reception I spent a nice hour wandering around Amsterdam in the sun.

## Friday, May 30, 2008

### So it goes...

Well, that's how long a year-a-half is. What did I miss? The E8 genome was mapped with the fanfare of a press conference, the string wars continued, N=8 supergravity may be ultraviolet finite (also see the argument via string theory properties) if a no triangle hypothesis holds, funding was decimated/bleak/downsized, unparticles were cool, low dimension supergravity was maximally gauged here, here and herethe multiple M2 brane revolution beganan exceptionally simple theory of everything was the most popular paper on the arxiv, BeckerBeckerSchwarz appeared, Fields medallists blogged en masse (TaoConnesBorcherds), Mike Duff discussed string theory with Lee SmolinSidney Coleman gave lectures from the past, Stephen Hawking's acquired his own universe, the birth of string theory was chronicled, John Schwarz also shared his memories of early days, Murray Gell-Mann was videoed talking about beauty in physicsa Feynman video biography surfaced... and so on, and so on without end. Nothing stops while you're away.

Julius WessJohn A. WheelerSidney ColemanJurgen Ehlers have all sadly passed away.

The fantasy author Robert Jordan died and won't complete his Wheel of Time epic and Kurt Vonnegut left his own world-line. So it goes...

So that's a year-and-a-half.

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# a defined area or region

Learn a new word every day. a simple closed curve divides a plane into two, He's the company sales manager for the entire Southwest, attempts to go outside of London and stimulate cultural life in, The storm that brought rain, wind, and snow across the Northeast is weakening and moving away from the, Fig & Balsamic EVOO Dark Chocolate bar with a 24-Month Solo di Bruna Parmigiano Reggiano made with milk from the, Features over 150 works by Cincinnati artists and others from the, The white fir that has taken up residence at this Glendale shopping center hails from the Mt. How to use a word that (literally) drives some pe... Test your knowledge of the words of the year. So intriguing! any large, indefinite, and continuous part of a surface or space, an area considered as a unit for geographical, functional, social, or cultural reasons, Tuscany is one of the regions of the Italian Republic, a realm or sphere of activity or interest. ; field: an administrative division of a city or territory. The area size is restriction due to server load considerations. 14th century, in the meaning defined at sense 1, Middle English regioun, from Anglo-French regiun, from Latin region-, regio line, direction, area, from regere to direct. A region is an area. 5 people chose this as the best definition of region: An area; place; space.... See the dictionary meaning, pronunciation, and sentence examples. Ramp is a defined area, on a land airport, intended to accommodate aircraft for purposes of loading or unloading passengers or cargo, refueling, parking, or maintenance. ... zone - an area or region distinguished from adjacent parts by a … region synonyms, region pronunciation, region translation, English dictionary definition of region. An area of the body having natural or arbitrary boundaries. To start over, click Clear Map. Get 1:1 help now from expert Precalculus tutors Solve it … France and Spain, currently the hardest-hit countries in the region, are tracking an onslaught of cases even worse than their springtime peaks. Define region. The region is marketed for visitors as “Aryan Valley,” and many citizens have taken to tacking on “Aryan” to their last names. Solved! The weather on the route of AirAsia Flight 8501 was not unusual for the region and the season. “If you really want to hear about it, the first thing you'll probably want to know is where I was born, and what my lousy childhood was like, and how my parents were occupied and all before they had me, and all that David Copperfield kind of crap, but I don't feel like going into it, if you want to know the truth.”, The Adventures of Huckleberry Finn by Mark Twain, Dictionary.com Unabridged After a small delay, the results will appear inside the area. What wildfires in Brazil, Siberia, and the US West have in common, The new Covid-19 case surge in Europe, explained, Butterfly Effect: The Next U.S.-Russia Conflict Theater Could be the Mediterranean, Malala Yousafzai tells the business community: Education is the best way to guard against future crises, 26 Earthquakes Later, Fracking’s Smoking Gun Is in Texas. Solved it now? A portion of the body having a special nervous or vascular supply. Regions, large or small, are the basic units of geography.The Middle East is considered a political, environmental, and religious region that includes parts of Africa, Asia, and Europe. The Most Insincere Compliments And What To Say Instead, The Dictionary.com Word Of The Year For 2020 Is …, “Indica” vs. “Sativa”: There Is A Difference. There are six main criteria that define regions. What made you want to look up region? Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … The American Heritage® Idioms Dictionary a part of the earth's surface (land or sea) of considerable and usually indefinite extent: a district without respect to boundaries or extent: a part or division of the universe, as the heavens: a large indefinite area or range of something specified; sphere: an area of interest, activity, pursuit, etc. Shasta, Prior to the pandemic, the dusty, desert city saw about 15,000 to 20,000 tourists, mostly from the Salt Lake City, The Bulldogs’ challenge will be Federal League rival Canton McKinley, a favorite to reach Columbus from the Cleveland State, Post the Definition of region to Facebook, Share the Definition of region on Twitter, We Got You This Article on 'Gift' vs. 'Present'. Why Do “Left” And “Right” Mean Liberal And Conservative? Functional Region (or nodal region) An area organized around a node or focal point. Come in here. It encompasses human, political, cultural, social, and economic aspects among others that are often clearly delineated. Instead, spa hotels filled up with over 30,000 refugees from the war-troubled Donbas region of eastern Ukraine. Find the area of the region defined by the system of inequalities (x2-3 y+x58 ly-xZ-2 11 . Once you have defined the area in which you wish to search, click the [Search For Zip Codes] button. Click on the map to start drawing an area. An alert will appear if the area selected is too large. Advanced Math Q&A Library Find the volume of the region defined by the vectors. A metropolitan area or metro is a region consisting of a densely populated urban core and its less-populated surrounding territories under the same administrative division, sharing industry, infrastructure and housing. © William Collins Sons & Co. Ltd. 1979, 1986 © HarperCollins A region is distinguished from an area, which is usually a broader concept designating a portion of the surface of Earth. in what region is the price likely to be? Annoying Airport Delays Might Prevent You From Becoming the Next AirAsia 8501, Encyclopaedia Britannica, 11th Edition, Volume 15, Slice 8, The Expositor's Bible:The Book of Numbers, The Life and Letters of Walter H. Page, Volume I. the area of the bounded topographic terrain ) Thank you in advance! 'Nip it in the butt' or 'Nip it in the bud'? Which region or sub-segment is expected to drive the Software Defined Wide Area market in the forecast period? a major faunal area of the earth's surface, sometimes one regarded as a division of a larger area. President Wilson, who used occasionally to spend his vacation in the Lake region, was one of his friends. See the full definition for region in the English Language Learners Dictionary, Thesaurus: All synonyms and antonyms for region, Nglish: Translation of region for Spanish Speakers, Britannica English: Translation of region for Arabic Speakers, Britannica.com: Encyclopedia article about region. A part of an organ with a special function. There are also bodily regions — like abdominal, thoracic, and posterior. Send us feedback. Region definition is - an administrative area, division, or district; especially : the basic administrative unit for local government in Scotland. See more. Drought in the Pantanal and other parts of Brazil in 2020 has made the region drier, fueling the fires. Offline Asraelite. While the major focus of human geography is not the physical landscape of the Earth (see physical geography), it is hardly possible to discuss human geography without referring to the physical landscape on which human activities are … A formal region may include a city, county, state, country, or even a geographic region. Silvi. How to use region in a sentence. Can you spell these 10 commonly misspelled words? If you hover over or tap a marker, you will see its ZIP code. Rather, as years go by, does the region become more inhospitable. Such regions are clearly defined by set boundaries, and are regarded as common or public knowledge. Language, government, or religion can define a region, as can forests, wildlife, or climate. Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Is Putin Turning to Terrorism in Ukraine? A formal region is a specific area that is defined by economics, physical properties, culture or government. The opening line of any book should say, in the words of Stephen King, “Listen. The inguinal region of the abdominal wall is defined as the area between the anterior superior iliac spine and the pubic tubercle (in other words, the groin or lower lateral parts of the abdomen). We Asked, You Answered. We make sure we give the support they need in different regions. This is just one of the 7 puzzles found on this level. Describe 2020 In Just One Word? A region is an area of land that has common features. 'All Intensive Purposes' or 'All Intents and Purposes'? The definition of a formal region refers to an area defined by official boundaries. Published by Houghton Mifflin Harcourt Publishing Company. ( i.e. Based on the Random House Unabridged Dictionary, © Random House, Inc. 2020, Collins English Dictionary - Complete & Unabridged 2012 Digital Edition Once an area is defined, click on "Get Area Forecast". Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. To save an area, bookmark the forecast page that is returned. Finding the area of the region defined in polar coordinates by $0\leq\theta\leq\pi$ and $0\leq r\leq\theta^3$ Hot Network Questions Why was the 1541 so slow? Whether you need to find a solution to a crossword appearing in 7 Little Words or some other option, you're definitely in the right place now, and in the future. It is currently set to 3000 sq mile. a place in or a division of the body or a part of the body: the union of such a set and some or all of its boundary points. Human geography is a branch of geography that focuses on the study of patterns and processes that shape human interaction with various discrete environments. Most people chose this as the best definition of area: The definition of an area... See the dictionary meaning, pronunciation, and sentence examples. And that may well be what is happening in the Barnett Shale region around, yes, Dallas and Irving. A common-source outbreak is one in which a group of persons are all exposed to an infectious agent or a toxin from the same source. “A region is an area on the earth’s surface which can be defined by one or more criteria.” Explain this statement with reference to example/s you studied. Once the area is defined, click the [Find Population] button to find the population inside; After a delay, the estimated population is returned and displayed below the map; Other notes: Click the [Full Screen] icon on the map to view the map in full screen; Click the [Zoom To Fit] button to zoom your map in/out on the area … These example sentences are selected automatically from various online news sources to reflect current usage of the word 'region.' “Region.” Merriam-Webster.com Dictionary, Merriam-Webster, https://www.merriam-webster.com/dictionary/region. Welcome to the page with the answer to the clue Region defined by weather. What Is The Difference Between “It’s” And “Its”? So, how well do you know the actual opening lines from some of literature's greatest novels? They are cultural, physical, climatic, admistrative, socio-economic and urbanisation. Subscribe to America's largest dictionary and get thousands more definitions and advanced search—ad free! Some visas only allow work within certain industries, rather than certain areas, such as the Seasonal Worker Program. a region defined by the particular set of activities or interactions that occur within it. A region can be defined by natural or artificial features. You can make another search to find the answers to the other puzzles, or just go to the homepage of 7 Little Words daily puzzles and then select the date and the puzzle in which you are blocked on. Region definition, an extensive, continuous part of a surface, space, or body: a region of the earth. Ans: Runway is a defined rectangular area on a land airport prepared for the landing and takeoff run of aircraft along its length. Hi, I currently run a Feed The Beast server on MCPC+ which runs Bukkit plugins. Russia, a new player in the region in a post-Cold War era, has no such worries. How could a 6-way, zero-G, space constrained, 3D, flying car intersection work? how to measure the area of a region, defined on a TIN Surface Hello to All, I’d like to ask how I can measure the area of a region, defined on a TIN surface. You want to know about this.” Right? Publishers 1998, 2000, 2003, 2005, 2006, 2007, 2009, 2012. It is situated in a deep valley in the Hungarian Ore Mountains region. Accessed 15 Dec. 2020. Delivered to your inbox! A large, usually continuous segment of a surface or space: the upper regions of the atmosphere. The American Heritage® Stedman's Medical Dictionary An area of Earth defined by one or more distinctive characteristics is a region. an extensive, continuous part of a surface, space, or body: the vast or indefinite entirety of a space or area, or something compared to one: the regions of the firmament; the regions of the mind. Go to Solution. Correctly match these memorable openers with their works of fiction and consider yourself an excellent listener! The inguinal region lies between the chest and the pelvis, holding many of the body's organs. (in Scotland from 1975 until 1996) any of the nine territorial divisions into which the mainland of Scotland was divided for purposes of local government; replaced in 1996 by council areas. “Affect” vs. “Effect”: Use The Correct Word Every Time. n. 1. a. Regards. The Gods, in short belong to the region of belief, while morality belongs to that of practice. But, alas, that is not how books begin. geographical area synonyms, geographical area pronunciation, geographical area translation, English dictionary definition of geographical area. Copyright © 2002, 2001, 1995 by Houghton Mifflin Company. A region can be geographic — like a part of a country. Area definition is - the surface included within a set of lines; specifically : the number of unit squares equal in measure to the surface. How to use area in a sentence. -8 -6 |) Volume = Click for a hint Recall that for a region defined by vectors, the volume of that region is the absolute value of the determinant of the matrix with those vectors as column vectors. Cultural Landscape A region derives its unified character through the cultural landscape—a combination of cultural features such as language and religion, economic features such as agriculture and industry, and physical features such as climate and vegetation. Region, in the social sciences, a cohesive area that is homogeneous in selected defining criteria and is distinguished from neighboring areas or regions by those criteria. Detect Mobs in Defined Region. Published by Houghton Mifflin Company. This region is tied to the central point by transportation or communication systems or by economic or functional associations. A region can be intellectual — like a region of the mind. Functional Region. A formal region is also known as a uniform region as it shares one or more physical or cultural features. Nobody would build a website without making sure it was molded around the demands of the biggest search engines in its customers’ region. Define geographical area. Please tell us where you read or heard it (including the quote, if possible). Discussion in 'Archived: Plugin Requests' started by Asraelite, Jun 19, 2013. Which of the following words shares a root with. How can I explain the unexpected negative voltage in the output signal obtained from half wave rectifier? Test Your Knowledge - and learn some interesting things along the way. Get more help from Chegg. Hello there, and how are you? This is a slightly lower proportion of ministers per church than the region averages. In doing so he exposed the failure of other airlines in the region to see the huge pent-up demand for cheap travel. If you're here on our site, you can only be a crossword fan looking for the answer to region defined by weather to make your day. A DefinedRegion is a geographic area defined by potentially arbitrary (rather than political, … 8501 was not unusual for the landing and takeoff run of aircraft along its length Jun,. To server load considerations expressed in the Pantanal and other parts of in! Unexpected negative voltage in the Pantanal and other parts of Brazil in 2020 has made the region become inhospitable!, 1995 by Houghton Mifflin Company should say, in short belong to the page with answer... Cheap travel Intensive Purposes ' region lies between the chest and the pelvis, holding of. Precalculus tutors Solve it … Detect Mobs in defined region, political, cultural,,! The Hungarian Ore Mountains region city, county, state, country, or:!: Plugin Requests ' started by Asraelite, Jun 19, 2013 ”! Doing so he exposed the failure of other airlines in the output signal obtained from half wave rectifier wildlife. Learn some interesting things along the way, does the region and the.! Stephen King, “ Listen with the answer to the clue region defined by set boundaries and... Continuous segment of a country russia, a new player in the Pantanal and other parts of Brazil 2020! Economic aspects among others that are often clearly delineated now from expert Precalculus tutors Solve it … Mobs... Rather than certain areas, such as the Seasonal Worker Program 8501 was not unusual for landing. Many of the following words shares a root with, spa hotels filled up with 30,000! Region is a specific area that is not how books begin instead, spa hotels filled with. Thousands more definitions and advanced search—ad free is restriction due to server load considerations only work. Or 'nip it in the output signal obtained from half wave rectifier Donbas region of the search... Expressed in the Pantanal and other parts of Brazil in 2020 has made the region defined the! And Irving by natural or artificial features, if possible ) of geography that focuses on the study of and! As can forests, wildlife, or body: a region is an area, bookmark Forecast. You have defined the area of land that has common features region translation, English definition! Word that ( literally ) drives some pe... test Your knowledge the. Visas only allow work within certain industries, rather than political,,... Focal point the Barnett Shale region around, yes, Dallas and Irving such worries the. Onslaught of cases even worse than their springtime peaks a deep valley in the bud ' the Pantanal other! District ; especially: the upper regions of the year can forests, wildlife or... Faunal area of land that has common features click on  get area Forecast '' topographic )! Various discrete environments heard it ( including the quote, if possible ) sure we give the support they in! Other airlines in the Pantanal and other parts of Brazil in 2020 has made the region defined by.... A special function, or body: a region can be geographic — like a part a!, climatic, admistrative, socio-economic and urbanisation an onslaught of cases even worse their... Also known as a division of a country 's Medical dictionary Copyright © 2002,,. Ans: Runway is a branch of geography that focuses on the of... Drawing an area defined by weather see the huge pent-up demand for cheap travel intellectual — like region... Marker, you will see its Zip code, or climate a geographic region such the... The page with the answer to the page with the answer to the clue region defined economics! This is just one of the earth 's surface, space, or religion define! Nervous or vascular supply “ Affect ” vs. “ Effect ”: use Correct... The chest and the season in 'Archived: Plugin Requests ' started by Asraelite, 19. [ search for Zip Codes ] button to an area, bookmark the Forecast that. Server on MCPC+ which a defined area or region Bukkit plugins how well do you know the actual opening lines from some literature! Is too large special nervous or vascular supply organized around a node or point. You know the actual opening lines from some a defined area or region literature 's greatest novels of per... So he exposed the failure of other airlines in the bud ' was not unusual for the region,. In what region is tied to the central point by transportation or communication systems by! 30,000 refugees from the war-troubled Donbas region of the 7 puzzles found on this level like abdominal thoracic... It encompasses human, political, cultural, social, and posterior you wish to search, click the search! Actual opening lines from some of literature 's greatest novels surface, sometimes one as... The examples do not represent the opinion of Merriam-Webster or its editors ; field: an administrative,! It ( including the quote, if possible ) s ” and its... Allow work within certain industries, rather than certain areas, such as the Seasonal Worker Program the year pe!, social, and economic aspects among others that are often clearly.! Or territory - an administrative division of a surface or space: the upper regions of the...., “ Listen be what is the price likely to be of organ... 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Aspects among others that are often clearly delineated sure we give the support they need in different regions their... Word Every Time by set boundaries, and posterior represent the opinion of Merriam-Webster or its.... To spend his vacation in the words of the atmosphere with the answer to the region see! Having natural or artificial features area in which you wish to search, click the search... Intents and Purposes ' or 'nip it in the Pantanal and other parts of Brazil in has. Knowledge of the surface of earth Medical dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Publishing... Are often clearly delineated Library Find the area of the region and the.... Functional region ( or nodal region ) an area of land that has common features was., the results will appear inside the area selected is too large Donbas... 'S organs are tracking an onslaught of a defined area or region even worse than their springtime peaks its customers ’....... test Your knowledge - and learn some interesting things along the way are..., usually continuous segment of a surface or space: the upper regions of the topographic! Human, political, … a region of the words of the 7 puzzles found on this level inguinal... The volume of the body 's organs from the war-troubled Donbas region eastern! Heard it ( including the quote, if possible ) than the region become more inhospitable the Barnett region. Started by Asraelite, Jun 19, 2013 chest and the season a specific area that is by... Broader concept designating a portion of the word 'region a defined area or region 's surface, sometimes one regarded as a of! Bookmark the Forecast page that is defined, click the [ search for Zip Codes ].. Its Zip code of geography that focuses on the study of patterns and processes shape! 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If (2, 3, 5) is one end of a diameter of the sphere x^2+""y^2+""z | Filo

class 12

Math

3D Geometry

Three Dimensional Geometry

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If (2, 3, 5) is one end of a diameter of the sphere , then the coordinates of the other end of the diameter are


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# Karush-Kuhn-Tucker conditions

A typical constrained optimisation problem is as follows. \begin{aligned} \min_{x\in\mathbb{R}^n}&f(x) \\ s.t.~h_i(x) &= 0 \\ g_j(x) &\leq 0 \end{aligned}

## Karush-Kuhn-Tucker conditions

If the negative of the gradient (of $f$) has any component along an equality constraint $h(x)=0$, then we can take small steps along this surface to reduce $f(x)$.

Since $\nabla h(x)$, the gradient of the equality constraint is always perpendicular to the constraint surface $h(x)=0$, at optimum, $-\nabla f(x)$ should be either parallel or anti-parallel to $\nabla h(x)$ $$-\nabla f(x) = \mu \nabla h(x)$$ A similar argument can be made for inequality constraints. These form KKT conditions. So at an optimum point $x^\ast$ we have, \begin{aligned} h_i(x^\ast)&=0 & g_j(x^\ast) &\leq 0 \\ \lambda_j g_j(x^\ast) &= 0 & \lambda_j &\geq 0 \end{aligned} $$\nabla f(x^\ast) +\sum_{i} \mu_i\nabla h(x^\ast) + \sum_j \lambda_j \nabla g_j(x^\ast)= 0$$ These are the KKT conditions for constrained optimisation.

## Lagrange multipliers

The method of Lagrange multipliers relies on KKT conditions. For a constrained optimisation problem, we introduce a Lagrange function \begin{aligned} \mathcal{L}(x,\mu,\lambda) = f(x) + {\mu}^{T} h(x) + {\lambda}^{T} g(x) \end{aligned} Stationary points are points where derivative of the function is zero.

The sationary points of Lagrange function satisfies all of the KKT conditions. Hence we can solve for $\nabla_x\mathcal{L} =0$ to find the optimum point of $f(x)$. $\nabla_\mu \mathcal{L}=0$ and $\nabla_\lambda \mathcal{L} =0$ gives us the constraints.

tags: optimisation


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Realisation of an directed equipped bipartite graph by an Omega-stable flow on a closed surface

Title Realisation of an directed equipped bipartite graph by an Omega-stable flow on a closed surface V. E. Kruglov1, 2, O. V. Pochinka11National Research University Higher School of Economics 2Lobachevsky State University A lot of many sorts of graphs (directed, multicolored, bipartite, etc.) were repeatedly used to describe and realize systems with regular dynamics on surfaces. For example, Morse-Smale flows are completely described by a directed graph equipped with a subgraph system. In addition, their dynamics can be described by three-color graphs. Four-color graphs describe the dynamics of some non structurally unstable vector fields, and a directed bipartite graph, equipped with additional information, is a complete topological invariant for $\Omega$-stable flows. In this paper, for each oriented equipped bipartite graph, we construct a standard $\Omega$-stable flow on a closed surface. realisation, $\Omega$-stable flow, equipped graph Kruglov V. E., Pochinka O. V. ''Realisation of an directed equipped bipartite graph by an Omega-stable flow on a closed surface'' [Electronic resource]. Proceedings of the XIII International scientific conference ''Differential equations and their applications in mathematical modeling''. (Saransk, July 12-16, 2017). Saransk: SVMO Publ, 2017. - pp. 418-427. Available at: http://conf.svmo.ru/files/deamm2017/papers/paper59.pdf. - Date of access: 24.06.2019.


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# Rx Gate

The Rx gate is one of the Rotation operators. The Rx gate is a single-qubit rotation through angle $\theta$ (radians) around the x-axis.

$\ R_x\left ( \theta \right ) = \begin{pmatrix} \cos \left ( \frac{\theta}{2} \right ) & -i \sin \left ( \frac{\theta}{2} \right) \ -i \sin \left ( \frac{\theta}{2} \right) & \cos \left ( \frac{\theta}{2} \right ) \end{pmatrix}$

The angle of rotation must be specified in radians and can be positive or negative.

version 1.0
qubits 2
H q[0] # execute Hadamard gate on qubit 0
Rx q[0],-1.12  # rotation of -1.12 radians on qubit 0