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# B How to measure Gravitational waves

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1. Nov 1, 2017

### Number 42

Why are clocks not used to measure gravity waves?

Spacetime wriggle and time must also vary. It is possible to measure time very accurately and with a high resolution. Så why not use clocks to detect gravity waves?

2. Nov 1, 2017

Staff Emeritus
What exactly do you intend to time?
How good a clock do you need?
How do you compare two clocks far away from each other?

3. Nov 1, 2017

### Ibix

Not necessarily. Gravitational waves (not gravity waves - that's a kind of surface wave on water) can affect the spatial part of the metric only. I don't think they can ever affect the time part of the metric only.

4. Nov 1, 2017

### Staff: Mentor

In appropriately chosen coordinates, yes. Those coordinates (the transverse-traceless gauge) are the ones standardly used to analyze gravitational wave detectors like LIGO, but it's worth keeping in mind that it's still a choice of coordinates.

That's correct; to state it more precisely, there is no possible choice of coordinates for which a GW only affects $g_{00}$ and not any other metric coefficient.

5. Nov 1, 2017

### vanhees71

One should emphasize however that the deformation of LIGO's arms due to the gravitational wave is of coarse gauge independent since it's a real physical effect of gravity. The choice of coordinates is completely arbitrary. Any choice will lead to the same outcome of the measurable physics, including the deformation of LIGO's arms and the change of the interference pattern of the laser light finally measured.

6. Nov 1, 2017

### Ibix

Understood. But whatever you choose to mean by "time" shouldn't there be a way to pick a gravitational wave so that it doesn't have any component in the time-like direction?

7. Nov 1, 2017

### vanhees71

You need to define some physical quantity to get a sensible answer to the question about the physical meaning of a particular solution of Einstein's equations (here gravitational waves in the usual approximation of the linearized theory). That's analogous to the solutions of Maxwell's equations in terms of the potentials. The potentials have no physical meaning a priori, but you have to calculate physically measurable quantities (like the electromagnetic field components) or evaluate how particles move in the so defined electromagnetic field (which of course also only depends on the field components but not the potentials). For sure, any "gauge dependent" quantity in gauge theories is not directly physically interpretable but you have to use gauge independent measureable quantities to get physics sense from it.

8. Nov 1, 2017

### Staff: Mentor

For weak (i.e., linear) gravitational waves, yes, they are purely transverse, so you can always choose coordinates (the transverse traceless ones) that express all of the spacetime curvature due to the waves as purely spatial. But of course that choice will be different for different waves (i.e., waves going in different directions).

9. Nov 1, 2017

### Ibix

Right. So doesn't that answer why you wouldn't use clocks to detect gravitational waves? There are a whole family of waves you just can't detect without sensors in relative motion.

10. Nov 1, 2017

### Staff: Mentor

I would say it shows why you can't use a set of clocks all at rest relative to each other to detect gravitational waves. (The OP's question probably had the part I put in italics as an unstated assumption, but it's still an assumption.)

11. Nov 2, 2017

### Number 42

I would think that the clocks should be spaced so that some are far enough away to be in different part of the wave. Say 1/4 of a wavelength.
Clocks close together will ofcourse not show any difference.
Transmitting the clock signal would not be a problem , but are clockes accurate enough?

12. Nov 2, 2017

### timmdeeg

Lets assume LIGO's mirrors carry synchronized clocks. Aren't they no more synchronous in case a GW passes by?

13. Nov 2, 2017

Staff Emeritus
I repeat:

What exactly do you intend to time?
How good a clock do you need?
How do you compare two clocks far away from each other?

14. Nov 2, 2017

### Staff: Mentor

This is not something you can just assume. You have to describe what "synchronized" means and how it will be accomplished. (For example, Google "Einstein clock synchronization".)

15. Nov 2, 2017

### pervect

Staff Emeritus
Yes. On a practical note, one can define clock synchronization is to choose some coordinates. With defined coordinates, clocks can be defined as synchronized if they have the same time coordinate. But in general, the results will depend on one's choice of coordinates.

Clock synchronization is presented as a human choice, and to a large extent it is. However, if one wants decent compatibility with Newtonian phsyics, so one can use most Newtonian formulae, one needs to use something like Fermi normal coordinates and make sure that the region one is analyzing is "small enough". I've seen a lot of posters who deal with the clock synchnoziation issue by ignoring them :(, this almost always leads to confusion in the end.

Conventional wisdom is more along the lines of "don't use coordinate dependent methods at all". This is OK, if one can get the necessary physical intuition that way, but if one is seeking some intuitive insight, using the coordinate methods and the right set of coordinates can be helpful. Contrawise, using the wrong set of coordinates can lead to more confusion.

The biggest problem with using Fermi normal coordinates in this context is converting the GW solution to these coordinates - while the GW has a nice closed form solution in the TT coordinates, attempts to convert them to Fermi Normal coordinates quicly results in a real mess. I do recall vaguely seeing some paper on the topic that talked about this issue and (I believe) they managed to deal with the mess, somehow. BUt I don't know how.

The aproach of "Fermi coordinates are too hard so I'll just use TT coordinates and interpret them in a Newtonian manner is just wrong :(.

16. Nov 4, 2017

### timmdeeg

I've read that but am thinking about a more-straight forward approach to see how the proper time is affected by the so-called ripples in spacetime. The proper time of a free clock in flat spacetime is recorded during the passage of a gravitational wave. The clock frequency shall be much higher than the frequency of the gravitational wave. After the wave has passed by the recorded ticks are compared with the ticks of the clock. Are the recorded ticks slower and faster according to the phase of the gravitational wave?

17. Nov 4, 2017

### Staff: Mentor

What "recorded ticks"? Other than the clock itself, what else is "ticking"?

18. Nov 4, 2017

### timmdeeg

The ticks which have been recorded by the clock during the passage of the GW will then be compared with the ticks of the same clock in flat spacetime however without a GW passing by.

19. Nov 4, 2017

### Staff: Mentor

How would you make such a comparison? Think carefully.

20. Nov 4, 2017

### timmdeeg

I would check if the frequency of the recording is periodically lower and higher than the frequency of the the clock. But how to do this technically, I would have to ask my technician.


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Filters cases based on the precedence relations between two sets of activities.

filter_precedence(eventlog, antecedents, consequents, precedence_type,
filter_method, reverse)

# S3 method for eventlog
filter_precedence(eventlog, antecedents, consequents,
precedence_type = c("directly_follows", "eventually_follows"),
filter_method = c("all", "one_of", "none"), reverse = FALSE)

# S3 method for grouped_eventlog
filter_precedence(eventlog, antecedents, consequents,
precedence_type = c("directly_follows", "eventually_follows"),
filter_method = c("all", "one_of", "none"), reverse = FALSE)

ifilter_precedence(eventlog)

## Arguments

eventlog The dataset to be used. Should be a (grouped) eventlog object. The set of antecendent and consequent activities. Both are character vectors containing at leaste one activity identifier. All pairs of antecedents and consequents are turned into seperate precedence rules. When directly_follows, the consequent activity should happen immediately after the antecedent activities. When eventually_follows, other events are allowed to happen in between. When all, only cases where all the relations are valid are preserved. When one_of, all the cases where at least one of the conditions hold are preserved. When none, none of the relations are allowed. Logical, indicating whether the selection should be reversed.

## Value

When given an eventlog, it will return a filtered eventlog. When given a grouped eventlog, the filter will be applied in a stratified way (i.e. each separately for each group). The returned eventlog will be grouped on the same variables as the original event log.

## Details

In order to extract a subset of an event log which conforms with a set of precedence rules, one can use the filter_precedence method. There are two types of precendence relations which can be tested: activities that should directly follow each other, or activities that should eventually follow each other. The type can be set with the precedence type argument. Further, the filter requires a vector of one or more antecedents (containing activity labels), and one or more consequents. Finally, also a filter method argument can be set. This argument is relevant when there is more than one antecedent or consequent. In such a case, you can specify that all possible precedence combinations must be present (all), at least one of them (one of), or none (none).

## Methods (by class)

• eventlog: Filter event log

• grouped_eventlog: Filter grouped event log

vignette("filters", "edeaR")

format: plain


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# Seeds of Helianthus, Fibonacci numbers in Nature

Every spring morning, the flower of the Helianthus plant turns facing eastward towards the sun. Throughout the day it slowly moves west, tracking the sun until nighttime, earning this plant the name sunflower. As amazing as this is, the sunflower also has a fascinating connection with the golden ratio.

The seeds at the head of the sunflower form into spirals, the number of spirals going clockwise and the number of spirals going counterclockwise form two adjacent Fibonacci numbers. (Numbers that are part of the Fibonacci sequence 1, 2 3, 5,8 13,…) The ratio of any two adjacent Fibonacci numbers approximate the golden ratio. For example $21/13 = 1.61538$ and  $43/21 = 1.61905$ which are very close to the golden ratio or $\phi = \frac{1+\sqrt5}{2} = 1.61803$.

This is believed to occur because seeds are formed at the center of the head of the sunflower and migrate outward as new seeds are formed. By approximating the golden spiral pattern the sunflower would maximize the number of seeds for its surface area.

Sunflowers are not the only plants that exhibit the golden ratio. Pine cones, also have clockwise and counterclockwise patterns that equal two adjacent Fibonacci numbers. The same pattern also exists pine apples. In some plants, their leaves and branches form a spiral pattern which also exhibits the same Fibonacci patterns.

Flowers also generally have a Fibonacci number as the number of petals such as 5 in the case of the cherry blossom, daisies and some sunflowers have 34 petals.

Sources used


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# Question b13d6

Sep 14, 2015

$\left(a\right) . 4.9 \times {10}^{- 3} \text{mol/l}$

$\left(b\right) . 0.66 \text{g/l}$

#### Explanation:

$C a S {O}_{4 \left(s\right)} r i g h t \le f t h a r p \infty n s C {a}_{\left(a q\right)}^{2 +} + S {O}_{4 \left(a q\right)}^{2 -}$

${K}_{s p} = \left[C {a}_{\left(a q\right)}^{2 +}\right] \left[S {O}_{4 \left(a q\right)}^{2 -}\right] = 2.4 \times {10}^{- 5} m o {l}^{2.} {L}^{- 6}$

Since $\left[C {a}_{\left(a q\right)}^{2 +}\right] = \left[S {O}_{4 \left(a q\right)}^{2 -}\right]$ and the solubility $s$ is equal to $\left[C {a}_{\left(a q\right)}^{2 +}\right]$ we can write:

${K}_{s p} = {s}^{2}$

So:

$s = \sqrt{{K}_{s p}} = \sqrt{2.4 \times {10}^{- 5}} = 4.9 \times {10}^{- 3} \text{mol/L}$

${M}_{r} = 136.1$

So solubility in $\text{g/L"=4.9xx10^(-3)xx136.1=0.66"g/L}$

The temperature was not given. It must always be specified for these type of questions involving $K$.

Sep 14, 2015

Molar solubility: $4.9 \cdot {10}^{- 3} \text{M}$

#### Explanation:

Calcium sulfate, ${\text{CaSO}}_{4}$, will not dissociate completely in aqueous solution to form calcium cations, ${\text{Ca}}^{2 +}$, and sulfate anions, ${\text{SO}}_{4}^{2 -}$.

What actually goes on is that the solid calcium sulfate will be in equilibrium with the amounts of ions that do dissolve. The position of this equilibrium, i.e. how much of the solid will actually dissolve to form ions in aqueous solution, is determined by the magnitude of the solubility product constant, ${K}_{s p}$.

The smaller the value of the ${K}_{s p}$, the less soluble will an ionic compound be.

To find the molar solubility of the calcium sulfate, use an ICE table

${\text{CaSO"_text(4(s]) " "rightleftharpoons" " "Ca"_text((aq])^(2+) " "+" " "SO}}_{\textrm{4 \left(a q\right]}}^{2 -}$

color(purple)("I")" " " " " " - " " " " " " " " " " "0" " " " " " " " " "0
color(purple)("C")" " " " " " - " " " " " " " "(+s)" " " " "(+s)
color(purple)("E")" " " " " " - " " " " " " " " " "s" " " " " " " "s

By definition, ${K}_{s p}$ is equal to

${K}_{s p} = \left[{\text{Ca"^(2+)] * ["SO}}_{4}^{2 -}\right]$

${K}_{s p} = s \cdot s = {s}^{2}$

This means that the molar solubility of calcium sulfate is

$s = \sqrt{{K}_{s p}} = \sqrt{2.4 \cdot {10}^{- 5}} = \textcolor{g r e e n}{4.9 \cdot {10}^{- 3} \text{M}}$

To get the mass solubility of calcium sulfate, use the compound's molar mass

4.9 * 10^(-3)color(red)(cancel(color(black)("moles")))/"L" * "136.141 g"/(1color(red)(cancel(color(black)("mole")))) = color(green)("0.67 g/L")#


text:
If the sides of a triangle are sina, cosa,1+sinα cosα,0<α <π2 the largest angle is

If the sides of a triangle are ,,$0$ the largest angle is

1. A

$60°$

2. B

$90°$

3. C

$120°$

4. D

$150°$

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Solution:

The largest side is

$\mathrm{cos}C=\frac{{a}^{2}+{b}^{2}-{c}^{2}}{2ab}$

.

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## Kinisot

A Python program to compute kinetic isotope effects from two Gaussian output files, one of which is a ground state and the other a transition state. The level of theory and basis set are detected from in the output files and the program will attempt to assign the appropriate scaling factor based on data from the Truhlar group. The program diagonalizes the mass-weighted Hessian matricies to obtain harmonic vibrational frequencies and Bigeleisen-Mayer Reduced Isotopic Partition Function Ratios.

Kinisot is freely available (CC-BY license) from GitHub; alternatively, it can be installed from the command line using pip install kinisot

kinetic isotope effect gaussian python

#### Mechanistic insight into palladium-catalyzed cycloisomerization − A combined experimental and theoretical study

##### Mekareeya, A.; Walker, P. R.; Couce-Rios, A.; Campbell, C. D.; Steven, A.; Paton, R. S.; Anderson, E. A. J. Am. Chem. Soc. 2017, 139, 10104–10114
kinetic isotope effect

#### Detailed Mechanistic Studies on Palladium-Catalyzed Selective C−H Olefination with Aliphatic Alkenes - A Significant Influence of Proton Shuttling

##### Deb, A.; Hazra, A.; Peng, Q.; Paton, R. S.; Maiti, D. J. Am. Chem. Soc. 2017, 139, 763–775
kinetic isotope effect


text:
# Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Construction 4.5.9.1 (Direct Images). Let $V: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{D}}$ and $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be morphisms of simplicial sets. For every integer $n \geq 0$, we let $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n}$ denote the collection of pairs $(\sigma , f)$, where $\sigma$ is an $n$-simplex of $\operatorname{\mathcal{C}}$ and $f: \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ is a morphism for which the composition

$\Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}\xrightarrow {V} \operatorname{\mathcal{D}}$

coincides with projection onto the second factor. Note that every nondecreasing function $\alpha : [m] \rightarrow [n]$ induces a map

$\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{n} \rightarrow \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{m} \quad \quad (\sigma ,f) \mapsto (\alpha ^{\ast }(\sigma ), f' ),$

where $f'$ denotes the composite map

$\Delta ^{m} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow { \alpha \times \operatorname{id}} \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}.$

This construction is compatible with composition, and therefore endows $\{ \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}})_{n} \} _{n \geq 0}$ with the structure of a simplicial set $\operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) = \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})_{\bullet }$ which we will refer to as the direct image of $\operatorname{\mathcal{E}}$ along $U$.

Note that the construction $(\sigma ,f) \mapsto \sigma$ determines a morphism of simplicial sets $\pi : \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{C}}$. Moreover, there is a tautological evaluation map $\operatorname{ev}: \operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}}) \rightarrow \operatorname{\mathcal{E}}$, which carries an $n$-simplex $( \widetilde{\sigma }, (\sigma ,f) )$ of the fiber product $\operatorname{\mathcal{D}}\times _{\operatorname{\mathcal{C}}} \operatorname{Res}_{\operatorname{\mathcal{D}}/\operatorname{\mathcal{C}}}(\operatorname{\mathcal{E}})$ to the $n$-simplex of $\operatorname{\mathcal{E}}$ given by the composite map $\Delta ^{n} \xrightarrow { \operatorname{id}\times \widetilde{\sigma } } \Delta ^{n} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{D}}\xrightarrow {f} \operatorname{\mathcal{E}}$.


text:
The most basic distributed radio network broadcasting primitive - called Decay - dates back to a PODC'87 result of Bar-Yehuda, Goldreich, and Itai. In any radio network with some informed source nodes, running Decay for O(d \log n + \log^2 n) rounds informs all nodes at most $d$ hops away from a source with high probability. Since 1987 this primitive has been the most important building block for implementing many other functionalities in radio networks. The only improvements to this decades-old algorithm are slight variations due to [Czumaj, Rytter; FOCS'03] and [Kowalski and Pelc, PODC'03] which achieve the same functionality in O(d \log \frac{n}{d} + \log^2 n) rounds. To obtain a speedup from this, $d$ and thus also the network diameter need to be near linear, i.e., larger than n^{1-\eps}.
Our new distributed primitive spreads messages for $d$ hops in O(d \frac{\log n \log \log n}{\log d} + \log^{O(1)} n) rounds. This improves over Decay for any super-polylogarithmic d = \log^{\omega(1)} n and achieves a near-optimal O(d \log \log n) running time for d = n^{\eps}. This also makes progress on an open question of Peleg.


text:
# 24  Optimization

To “optimize” means to make something as good as possible with the available resources. Optimization problems are common in science, logistics, industry, and any other area where one seeks the best solution to a problem. Some everyday examples:

• When to harvest trees being grown for lumber. Harvest too soon and you might be losing out on the prime growing years. On the other hand, wait too long and trees will show slow or zero growth.
• Walking up too steep a slope is tiring and slows you down; that is why hiking trails have switchbacks. However, when the switchbacks are too shallow, it takes a long time to cover the distance. What’s the most efficient angle to enable hikers to get up the hill in the shortest time.
• How much salt to add to a stew. Stews can be too salty, or they can be not salty enough. Somewhere in the middle is the optimum.

## 24.1 Structure of the problem

In an optimization problem, there is one or more input quantities whose value you have to choose. The amount of salt; the years to wait from planting to harvesting a tree; the angle of the trail with respect to the slope. We will call this the decision quantity.

Similarly, there is one or more output quantity that you value and want to make as good as possible. The taste of the stew; the amount of usable wood harvested; the time it takes to walk up the hill. The output quantities are called the objectives.

This chapter deals with optimization problems that involve only a single objective. Problems with multiple objectives are among the most interesting and important in real-world decision making. Single-objective optimization techniques are a component of the more complex decision making, but they are a good place to get started.

The model that relates inputs to the objective output is the objective function. Solving an optimization problem—once the modeling phase is complete—amounts to finding a value for the decision quantity (the input to the objective function) that produces the best output from the objective function.

Sometimes the objective is something that you want to minimize, make as small as possible. For instance, in the hiking trail problem, we seek to minimize the time it takes to walk up the trail. Sometimes you want to maximize the objective, as in the wood-harvest problem where the objective is to harvest the most wood per year.

Mathematically, maximization and minimization are the same thing. Every minimization problem can be turned into a maximization problem by putting a negative sign in front of the objective function. To simplify the discussion, in talking about finding the solution to an optimization problem we will imagine that the goal is to maximize. But keep in mind that many circumstances in the real world, “best” can mean minimization.

Recall from Section 6.7 that there are two components to the task of maximization or minimization. The argmax is the input to the objective function which produces the largest output. The maximum is the value of that output.1 Argmin and minimum are the words used in a situation where you seek the smallest value of the objective function.

Once you have found the argmax you can plug that value into the objective function to find the output value. That value is the maximum.

People often talk about “finding the maximum.” This is misleading. Instead, the idea is to find the input to the objective function—that is, the argmax—that produces the maximum output.

To illustrate the setup of an optimization problem, imagine yourself in the situation of a contest to see who can shoot a tennis ball the farthest into a field with a slingshot. During the contest, you will adjust the vertical angle of launch, place the ball into the slingshot’s cradle, pull back as far as possible, and let go. To win the contest, you need to optimize how you launch the ball.

The objective is to maximize the distance traveled by the ball. The objective function models the distance travelled as a function of the quantities you can control, for instance the vertical angle of launch or the amount by which you pull back the slingshot. For simplicity, we will imagine that the slingshot is pulled back by a standard amount, producing a velocity of the ball at release of $$v_0$$. You will win or lose based on the angle of launch you choose.

Before you head out into the field to experiment, let’s prepare by constructing the objective function. Using some principles of physics and mathematics (which you may not yet understand), we will model how far the ball will travel (horizontally) as a function of the angle of launch $$\theta$$ and the initial velocity $$v_0$$.

The mathematics of such problems involves an area called differential equations, an important part of calculus which we will come to later in the course. Since you don’t have the tools yet, we will just state a simple model of how long the ball stays in the air.

$\text{duration}(v_0, \theta) = 2 v_0 \sin(\theta)/g$ $$g$$ is the acceleration due to gravity, which is about $$9.8 \text{m}\text{s}^{-2}$$, assuming that the contest is being held on Earth.

The horizontal distance travelled by the tennis ball will be $\text{hdist}(v_0, \theta) = \cos(\theta) v_0\, \text{duration}(v_0, \theta) = 2 v_0^2 \cos(\theta)\sin(\theta) / g$ Our objective function is hdist(), and we seek to find the argmax. The input $$v_0$$ is (we have assumed) fixed, so the only decision quantity is the angle $$\theta$$.

The best choice of $$\theta$$ will make the quantity $$\cos(\theta)\sin(\theta)$$ as large as possible. So in finding the argmax, we don’t need to be concerned with $$v_0$$ or $$g$$.

Finding the argmax can be accomplished simply by plotting the function $$\cos(\theta)\sin(\theta)$$. We will implement the function so that the input is in units of degrees.

You can see that the maximum value is about 0.5 and that this occurs at an argmax $$\theta$$ that is a little bit less than 50$$^\circ$$.

Zooming in on the $$\theta$$ axis let’s you find the argmax with more precision:

From the graph, especially the zoomed-in version, you can read off the argmax as $$\theta = 45^\circ$$.

Finding the argmax solves the problem. You may also want to present your solution by reporting the value of hdist() when the argmax is given as input. You can read off the graph that the maximum of $$\cos(\theta)\sin(\theta)$$ is 0.5 at $$\theta = 45^\circ$$, so overall the distance will be $$v_0^2 / g$$

## 24.2 Interpreting the argmax

The graphical solution given to the slingshot problem is entirely satisfactory. Whether that solution will win the contest depends on whether the model we built for the objective function is correct. We have left out, for instance, air resistence, which is potentially important.

Solving the optimization problem has prepared us to test the result in the field. Perhaps we will find that the real-world optimum angle is somewhat steeper or shallower than $$\theta = 45^\circ$$.

Besides the argmax, another important quantity to read from the graph in Figure 24.1 is the precision of the argmax. In strict mathematical terms, the argmax for the tennis-ball problem is exactly 45 degrees at which point $$\cos(\theta)\sin(\theta) = 0.5$$. Suppose, however, that the ball were launched at only 40 degrees. Five degrees difference is apparent to the eye, but the result will be essentially the same as for 45 degrees: $$\cos(\theta)\sin(\theta) = 0.492$$. The same is true for a launch angle of 50 degrees. For both “sub-optimal” launch angles, the output is within 2 percent of the 45-degree result. It is easy to imagine that a factor outside the scope of the simple model—the wind, for instance—could change the result by as much or more than 2 percent, so a practical report of the argmax should reasonable be “40 to 50 degrees” rather than “exactly 45 degrees.”

Contests are won or lost by margins of less than 1%, so you should not casually deviate from the argmax. On the other hand, $$45^\circ$$ is the argmax of the model. Reality may deviate from the model. For instance, suppose that air resistance or wind might might affect distance by 1%. That is. the real-world result might deviate by as much as 1% of the model value. If so, we shouldn’t expect the real-world argmax to be any closer to 45$$^\circ$$ than $$\pm 5^\circ$$; anywhere in that domain interval generates an output that is within 1% of the maximum output for the model.

## 24.3 Derivatives and optimization

We are now going to reframe the search for the argmax and its interpretation in terms of derivatives of the objective function with respect to the decision quantity ($$\theta$$ in the slingshot problem). For a function with one input, this will not be an improvement from the look-at-the-graph technique to find the argmax. However, a genuine reason to use derivatives is to set us up in the future to solve problems with more than one input, where it is hard to draw or interpret a graph. Also, describing functions in the language of derivatives can help us think more clearly about aspects of the problem, such as the precision of the argmax.

With a graph such as Figure 24.1, it is easy to find the argmax; common sense carries the day. So it won’t be obvious at first why we will take the following approach:

Let’s denote an argmax of the objective function $$f(x)$$ by $$x^\star$$. Let’s look at the derivative $$\partial_x f(x)$$ in the neighborhood of $$x^\star$$. Referring to Figure 24.1, where $$x^\star = 45^\circ$$, you may be able to see that $$\partial_x f(x^\star)$$ is zero; the line tangent to the function’s graph at $$x^\star$$ is horizontal.

Seen another way, the slope of $$f(x)$$ to the left of $$x^\star$$ is positive. Moving a tiny bit to the right (that is, increasing $$x$$ by a very small amount) increases the output $$f(x)$$. On the other hand, just to the right of $$x^\star$$, the slope of $$f(x)$$ is negative; as you reach the top of a hill and continue on, you will be going downhill. So the derivative function is positive on one side of $$x^\star$$ and negative on the other, suggesting that it crosses zero at the argmax.

Common sense is correct: Walk uphill to get to the peak, walk downhill to move away from the peak. At the top of a smooth hill, the terrain is level. (Since our modeling functions are smooth, so must be the hills that we visualize the functions with.)

Inputs $$x^\star$$ such that $$\partial_x f(x^\star) = 0$$ are called critical points. Why not call them simply argmaxes? Because a the slope will also be zero at an argmin. And it is even possible to have the slope be zero at a point that is neither an argmin or an argmax.

At this point, we know that values $$x^\star$$ that give $$\partial_x f(x^\star) = 0$$ are “critical points,” but we haven’t said how to figure out whether a given critical point is an argmax, an argmin, or neither. This is where the behavior of $$\partial_x f(x)$$ near $$x=x^\star$$ is important. If $$x^\star$$ is an argmax, then $$\partial_x f(x)$$ will be positive to the left of $$x^\star$$ and negative to the right of $$x^\star$$; walk up the hill to get to $$x^\star$$, at the top the hill is flat, and just past the top the hill has a negative slope.

For an argmin, changing $$x$$ from less than $$x^\star$$ to greater than $$x^\star$$; you will be walking down into the valley, then level at the very bottom $$x=x^\star$$, then back up the other side of the valley after you pass $$x=x^\star$$. Figure 24.3 shows the situation.

The bottom row of graphs in Figure 24.3 shows the derivative of the objective function $$f(x)$$, that is, $$\partial_x f(x)$$. You can see that for the argmax of $$f(x)$$, the derivative $$\partial_x f(x)$$ is positive to the left and negative to the right. Similarly, near the argmin of $$f(x)$$, the derivative $$\partial_x f(x)$$ is negative to the left and positive to the right.

Stated another way, the derivative $$\partial_x f(x)$$ has a negative slope just to the left of an argmin and a positive slope to the left of an argmax.

The second derivative of the objective function $$f(x)$$ at a critical point $$x^\star$$ is what tells us whether the critical point is an argmax, an argmin, or neither.

Critical point $$x^\star$$ $$\partial_x f(x^\star)$$ $$\partial_{xx} f(x^\star)$$
argmax 0 negative
argmin 0 positive
neither 0 0

Throughout Block 2, we have translated features of functions that are evident on a graph into the language of derivatives:

1. The slope of a function $$f(x)$$ at any input $$x$$ is the value of the derivative function $$\partial_x f(x)$$ at that same $$x$$.
2. The concavity of a function $$f(x)$$ at any input is the slope of the derivative function, that is, $$\partial_{xx} f(x)$$.
3. Putting (i) and (ii) together, we get that the concavity of a function $$f(x)$$ at any input $$x$$ is the value of the second derivative function, that is, $$\partial_{xx} f(x)$$.
4. At an argmax $$x^\star$$ of $$f(x)$$, the value of the derivative function $$\partial_x f(x^\star)$$ is zero and the value of the second derivative function $$\partial_{xx} f(x^\star)$$ is negative. The situation at an argmin is along the same lines, the derivative of the objective function is zero and the second derivative is positive.

What’s the critical point?

You’re familiar with the quadratic polynomial: $g(x) = a_0 + a_1 x + a_2 x^2$ The graph of a quadratic polynomial is a parabola, which might be concave up or concave down. A parabola has only one critical point, which might be an argmin or an argmax.

Let’s find the critical point. We know that the critical point is $$x^\star$$ such that $$\partial_x g(x^\star) = 0$$. Since we know how to differentiate a power law, we can see that $\partial_x g(x) = a_1 + 2 a_2 x$ and, more specifically, at the critical point $$x^\star$$ the derivative will be

$a_1 + 2 a_2 x^\star = 0$

The above is an equation, not a definition. It says that whatever $$x^\star$$ happens to be, the quantity $$a_1 + 2 a_2 x^\star$$ must be zero. Using plain old algebra, we can find the location of the critical point $x^\star = -\frac{a_1}{2 a_2}$

Math in the World: The demand “curve”

In economics, a monopoly or similar arrangement can set the price for a good or commodity. Monopolists can set the price at a level that generates the most income for themselves.

In 1836, early economist Antoine-Augustin Cournot published a theory of revenue versus demand based on his conception that demand will be a monotonically decreasing function of price. (That is, higher price means lower demand.) we will write as $$\text{Demand}(p)$$ demand as a function of price.

The revenue generated at price $$p$$ is $$R(p) \equiv p \text{Demand}(p)$$: price times demand.

To find the revenue-maximizing demand, differentiate $$R(p)$$ with respect to $$p$$ and find the argmax $$p^\star$$ at with $$\partial_p R(p^\star) = 0).$$ This can be done with the product rule.

$\partial_p R(p) = p \ \partial_p \text{Demand}(p) + \text{Demand}(p)$

At the argmax $$p^\star$$ we have: $p^\star \partial_p \text{Demand}(p^\star) + \text{Demand}(p^\star) = 0 \ \ \stackrel{\text{solving for}\ p^\star}{\Longrightarrow} \ \ p^\star = - \frac{\text{Demand}(p^\star)}{\partial_p \text{Demand}(p^\star)}$

If the monopolist knows the demand function $$D(p)$$, finding the revenue maximizing price is a simple matter. But in general, the monopolist does not know the demand function in advance. Instead, an informed guess is made to set the initial price $$p_0$$. Measuring sales $$D(p_0)$$ gives one point on the demand curve. Then, try another price $$p_1$$. This gives another point on the demand curve as well as an estimate

$\partial_p D(p_0) = \frac{D(p_1) - D(p_0)}{p_1 - p_0}$

Now the monopolist is set to model the demand curve as a straight-line function and easily to find $$p^\star$$ for the model. For instance, if the demand function is modeled as $$D_1 (p) = a + b p$$, the optimal price will be $$p^\star_1 = - \frac{a + b p^\star}{b}$$ which can be solved as $$p^\star_1 = - a/2b$$.

$$p^\star_1$$ is just an estimate of the optimum price. Still, the monopolist can try out that price, giving a third data point for the demand function. The new data can lead to a better model of the demand function. With the better estimate, find a new a argmax $$p^\star_2$$. This sort of iterative process for finding an argmax of a real-world function is very common in practice.

## 24.4 Be practical!

Decision making is about choosing among alternatives. In some engineering or policy contexts, this can mean finding a value for an input that will produce the “best” outcome. For those who have studied calculus, it is natural to believe that calculus-based techniques for optimization are the route to making the decision.

We emphasize that the optimization techniques covered in this chapter are only part of a broader set of techniques for real-world decision-making problems. In particular, most policy contexts involve multiple objectives. For example, in designing a car one goal is to make it cheap to manufacture, another to make it attractive, and still another to make it safe. These different objectives are often at odds with one another. In Block 4 of this text, we will discuss some calculus techniques that help policy-makers in multi-objective settings.

For now, sticking with the idealized (and often unrealistic) setting of maximizing a single objective, with one or more inputs. Recall the setting for calculus-type maximization. You have a function with one or more inputs, say, $$f(x)$$ or $$g(x,y)$$ or, often, $$h(x, y, z, \ldots)$$ where $$\ldots$$ might be standing for tens or hundreds or thousands of inputs or more.

If you can graph the function (feasible for one- or two-input functions), you can often easily scan the graph by eye to find the peak. The basis of the calculus techniques is the observation that, at the argmax of a smooth function, the derivative of the function is 0.

For example, consider a style problem that often appears in calculus textbooks. Suppose you have been tasked to design a container for a large volume V of liquid. The design specifications call for the weight of the container to be as little as possible. (This is a minimization problem, then.) In classical textbook fashion, the specifications might also stipulate that the container is to be a cylinder made out of a particular metal of a particular thickness.

The above is a lovely geometry/calculus problem. Whether it is relevant to any genuine, real-world problem is another question.

Using the notation in the diagram, the volume and surface area of the cylinder is $V(r, h) \equiv \pi r^2 h \ \ \ \text{and}\ \ \ A(r, h) \equiv 2 \pi r^2 + 2 \pi r h$

Minimizing the weight of the cylinder is our objective (according to the problem statement) and the weight is proportional to the surface area. Since the volume $$V$$ is given (according to the problem statement), we want to re-write the area function to use volume:

$h(r, V) \equiv V / \pi r^2 \ \ \ \implies\ \ \ A(r, V) = 2 \pi r^2 + 2 \pi r V/\pi r^2 = 2 \pi r^2 + 2 V / r$

Suppose $$V$$ is specified as 1000 liters. A good first step is to choose appropriate units for $$r$$ to make sure the formula for $$A(r, V)$$ is dimensionally consistent. Suppose we choose $$r$$ in cm. Then we want $$V$$ in cubic centimeters (cc). 1000 liters is 1,000,000 cc. Now we can plot a slice of the area function:

A <- makeFun(2*pi*r^2 + 2*V/r ~ r, V=1000000)
slice_plot(A(r) ~ r, bounds(r=c(10, 100))) %>%
gf_labs(x = "radius (cm)", y = "Surface area of container (square cm)")

As always, the function’s derivative is zero at the optimal $$r$$. In the graph, the argmin is near $$r=50$$ cm at which point the minimum is about 50,000 cm$$^2$$. Since $$h(r,V) = V/\pi r^2$$, the required height of cylinder will be near $$10^6 / \pi 50^2 = 127$$cm.

In calculus courses, the goal is often to find a formula for the optimal radius as a function of $$V$$. So we differentiate the objective function—that is, the area function for any $$V$$ and $$r$$ with respect to $$r$$,

$\partial_r A(r, V) = 4 \pi r - 2 V / r^2$

Setting this to zero (which will be true at the optimal $$r^\star$$) we can solve for $$r^\star$$ in terms of $$V$$:

$4 \pi r^\star - 2 \frac{V}{\left[r^\star\right]^2} = 0 \ \ \ \Longrightarrow\ \ \ 4\pi r^\star = 2\frac{V}{\left[r^\star\right]^2} \Longrightarrow\ \ \ \left[r^\star\right]^3 = \frac{1}{2\pi} V \ \ \ \Longrightarrow\ \ \ r^\star = \sqrt[3]{V/2\pi}$

For $$V = 1,000,000$$ cm$$^3$$, this gives $$r^\star = 54.1926$$ cm which in turn implies that the corresponding height $$h^\star = V/\pi (r^\star)^2 = 108.3852$$ cm.

We’ve presented the optimum $$r^\star$$ and $$h^\star$$ to the nearest micron. (Does that make sense to you? Think about it for a moment before reading on.)

In modeling, a good rule of thumb is this: “If you don’t know what a sensible precision is for reporting your result, you don’t yet have a complete grasp of the problem.” Here are two reasonable ways to sort out a suitable precision.

1. Solve a closely related problem that would have been equivalent for many practical purposes.
2. Calculate how much the input can deviate from the argmax while producing a trivial change in the output of the objective function.

Approach (2) is always at hand, since you already know the objective function. Let’s graph the objective function near $$r = 54.1926$$

Look carefully at the axes scales. Deviating from the mathematical optimum by about 5cm (that is, 50,000 microns) produces a change in the output of the objective function by about 400 units out of 55,000. In other words, about 0.7%.

It is true that $$r^\star = 54.1926$$ cm gives the “best” outcome. And sometimes such precision is warranted. For example, improving the speed of an elite marathon racer by even 0.1% would give her a 7 second advantage: often the difference between silver and gold!

What’s different is that you know exactly the ultimate objective of a marathon: finish faster. But you may not know the ultimate objective of the system your “optimal” tank will be a part of. For instance, your tank may be part of an external fuel pod on an aircraft. Certainly the aircraft designers want the tank to be as light as possible. But they also want to reduce drag as much as possible. A 54 cm diameter tube has about 17% more drag than a 50 cm tube. It is probably well worth increasing weight by 0.7% to save that much drag.

In reporting the results from an optimization problem, you ought to give the decision maker all relevant information. That might be as simple as including the above graph in your report.

We mentioned another technique for getting a handle on what precision is meaningful: (1) solve a closely related problem. It can requires some insight and creativity to frame the new problem. For instance, large capacity tanks often are shaped like a lozenge: a cylinder with hemi-spherical ends.

Using $$h$$ for the length of the cylindrical portion of the tank, and $$r$$ for the radius, the volume and surface area are:

$V(r, h) = \pi r^2 h + \frac{4}{3} \pi r^3 \ \ \ \text{and}\ \ \ A(r,h) = 2 \pi r h + 4 \pi r^2$

Again, $$V$$ is specified as 1000 liters. As detailed in Exercise 24.18, the surface area of this 1000-liter tank is about 48,400 cm$$^2$$. This is more than 10% less than for the cylindrical tank.

## 24.5 Exercises

#### Exercise 24.02

The simple model of the distance travelled by a tennis ball after launch from a slingshot is $\text{hdist}(v_0, \theta) = 2 v_0^2 \cos(\theta)\sin(\theta) / g$ where $$\theta$$ is the launch angle, measured from the horizontal, $$v_0$$ is the initial velocity, and $$g$$ is the acceleration due to gravity.

Part A What is the dimension of $$v_0$$?

L T L / T L / T$$^2$$ L$$^2$$ / T$$^2$$

Part B What is the dimension of $$g$$?

M L M L / T L / T$$^2$$ M L / T$$^2$$

Part C What is the dimension of $$\theta$$?

L L^2 T/L It is dimensionless

Part D What is the dimension of $$\sin(\theta$$)?

L L^2 T/L It is dimensionless

Part E What is the dimension of $$2 v_0^2 \cos(\theta)\sin(\theta) / g$$?

L L^2 L/T L/T$$^2$$

Part F Suppose the initial velocity of the ball is $$v_0 = 10$$ meters/second. Since we are on the surface of Earth, $$g=9.8$$ meters/second-squared. At the optimal launch angle $$\theta$$, how far does the model predict the ball will travel?

10 meters 25 meters 50 meters 75 meters

#### Exercise 24.04

Part A Consider the function $$f(x) \equiv x^3$$. Confirm that the value of the derivative $$\partial_x f(x = 0)$$ and so $$x^\star = 0$$ is a critical point. Which sort of critical point is $$x^\star=0$$? (Hint: Draw the graph of $$f(x)$$ near $$x=0$$ to see what’s going on.)

An argmax An argmin Neither

Part B Still working with the function $$f(x) \equiv x^3$$, find the value of the second-derivative $$\partial_{xx} f(x^\star)$$ evaluated at the critical point $$x = x^\star = 0$$. Which of these is $$\partial_{xx} f( x=0)$$?

Negative Positive Zero

#### Exercise 24.06

The graph shows three different functions labeled (A), (B), and (C).

Part A Function (A) is

concave down non-concave concave up

Part B Function (B) is

concave down non-concave concave up

Part C Function (C) is

concave down non-concave concave up

Part D The negative of function (A) is

concave down non-concave concave up

The graph shows a function $$\text{wave}(t) \equiv \sin(2 \pi t/4)$$ and labels four input values $$t$$.

Part E For what values of the input $$t$$ is the function concave up?

$$t = A$$ and $$t=D$$$$t = A$$ and $$t=C$$$$t = C$$ and $$t = D$$ none of the above

Part F For what values of the input $$t$$ is the function non-concave?

$$t = A$$$$t=B$$$$t=C$$ none of the above

Part G Where is the function steepest?

$$t = A$$$$t=B$$$$t=C$$$$t=D$$

#### Exercise 24.08

Here is a smooth function marked at a few points. Your task is, at each point, to estimate the value of the derivative, the sign of the second derivative, and the radius of the circle that would nicely match the function in a small region around each point. (Remember, we are asking for the radius of the circle, which is half the diameter.)

To simplify things, here is a table giving seven different combinations of the quantities you are to estimate. Some of them correctly match one of the labeled points, some do not. All you need to do is choose which is the correct set of quantities for each labeled point.

row value of 1st deriv sign of 2nd deriv radius
i -0.3 pos 0.25
ii 2.1 near 0 2000
iii -1.4 neg 12
iv 0.3 neg 0.3
v 2.1 pos 0.1
vi 1.3 neg 3
vii 0.5 pos 1

Part A Which row from the table best matches the function at point A?

i ii iii iv v vi vii

Part B Which row from the table best matches the function at point B?

i ii iii iv v vi vii

Part C Which row from the table best matches the function at point C?

i ii iii iv v vi vii

Part D Which row from the table best matches the function at point D?

i ii iii iv v vi vii

Part E Which row from the table best matches the function at point E?

i ii iii iv v vi vii

#### Exercise 24.10

You and your pet dog Swimmer often go to the beach and walk along the water’s edge. You throw a ball down the beach, but at an angle so it lands in the water. Swimmer goes to work. She runs down the beach (fast) and then plunges into the water, heading toward the ball. She can run fast on the beach: 400 m/minute. But she swims rather slower: 50 m/min.

Suppose you threw the ball to a point about 50 meters down the beach and 10 meters out in the water. The overall distance to the ball is therefore $$\sqrt{50^2 + 10^2} \approx 51$$ meters. If Swimmer entered the water immediately, she would take about a minute to reach the ball (51 m / 50 m/min). Swimmer can get to the ball faster by running down the beach a bit and then turning into the water. If Swimmer ran all 50 meters down the beach and then turned to swim the 10 meters, it would take her (50/400 + 10/50) minutes, about one-third of a minute.

Can Swimmer do better? You can set up the calculation like this. Imagine $$x$$ to be the distance down the beach that Swimmer runs. The time to run this distance will be $$x/400$$. The distance remaining to the ball can be found by the Pythagorean theorem. One leg of the triangle has length $$(50-x)$$, the other has length 10 m. So, the length of the third side is $$\sqrt{\strut (50-x)^2 + 10^2}$$. For instance, if $$x$$ were 45, the distance to swim in the water would be $$\sqrt{(50-45)^2 + 10^2} = 11.2$$ m. Divide this distance by 50 m/min to get the time spent in the water.

distance_in_water <- makeFun( __your_pythagorean_formula ~ x)
time_to_ball <- makeFun(x/400 + distance_in_water(x)/50 ~ x)

Time_to_ball() takes one argument, the distance $$x$$ Swimmer runs down the beach before turning into the water. Use a SANDBOX to find the distance that calculus-savvy Swimmer runs down the beach before turning into the water, if Swimmer’s goal is to get to the ball as fast as possible.

Part A What’s the optimal running distance for Swimmer?

46.75 47.5 48.75 49.75

Here’s a news story about a mathematician’s dog on the shore of Lake Michigan. It is not plausible that Swimmer has been trained in calculus. Perhaps the way Swimmer solves the running distance problem is simply to graph time_to_ball(x) ~ x over a suitable domain and find the argmax by eye!

#### Exercise 24.12

If you’re skeptical that a dog might do a calculus problem before running to fetch a ball, consider the path taken by a photon. “Fermat’s Principle” is that light takes the path of least time. To illustrate, consider the problem of a photon traveling from a point A to a point B, as in the diagram. The shortest path between the two points is a straight line. Along this straight-line path, the time taken by the photon will be the distance divided by the speed of light.

The diagram shows another path consisting of two segments, one of length $$l_1$$ and the other $$l_2$$. Obviously, the two-segment path is longer than the straight-line path. But according to Fermat’s principle, light “prefers” the longer path if the time taken to traverse it is shorter. This phenomenon is called refraction.

The reason the indirect path might be shorter is that the speed of light differs in different physical media. Light traveling in a vacuum famously has a speed of about 300,000 km per second. In air, the speed is smaller by a factor of 1/1.003. In water, the speed is smaller still: the factor is 1/1.3.

Imagine that the blue zone of the diagram is water and the clear zone air. The time for the photon to travel from point A to B is proportional to $$1.003\ l_1 + 1.3\ l_2$$.

To see the path taken by light, let’s imagine that point A is $$(x=0, y=10)$$ and point B is $$(x=20, y=-10)$$, and that the boundary between water and air is at $$y=0$$. We will place the point P at $$(x, 0)$$. The total time taken for light to traverse this path is 1.003 dist(A to P) + 1.3 dist(P to B).

Part A Which of these formulas gives the total time it takes for light to traverse the path from A to P at relative speed 1/1.003 and then the path from P to B at relative speed 1/1.3? A is located at $$(0, 10)$$, B is located at $$(20,-10)$$, and P is located at $$(x, 0)$$

1. $$1.003 \sqrt{(x-0)^2 +(0-10)^2}+ 1.3\sqrt{(20-x)^2 + (-10 - 0)^2}$$
2. $$\sqrt{(x-0)^2 +(0-10)^2}/1.003+ \sqrt{(20-x)^2 + (-10 - 0)^2}/1.3$$
3. $$1.003 \sqrt{(x-10)^2 +(0-0)^2}+ 1.3\sqrt{(-10 -x)^2 + (50 - 0)^2}$$

Implement the calculation of total_time() in R, then use a graph to find the argmin.

total_time <- makeFun( your_formula ~ x)
slice_plot(total_time(x) ~ x, bounds(x=0:20))

# For the next problem
dx_time  <- D(total_time(x) ~ x)
dxx_time <- D(total_time(x) ~ x & x)

Part B What value of $$x$$ (that is, the argmin) minimizes the travel time of light between points A and B? (Choose the best answer)

10.52 11.02 12.22 12.50 13.21 14.94

Part C Suppose that instead of being water, the blue area was glass. The speed of light in glass is roughly 1/1.5 times as big as in vacuum. What value of $$x$$ minimizes the travel time of light between points A and B? (Choose the best answer)

13.60 14.58 14.85 15.54

#### Exercise 24.18

Return to the problem of finding the optimal radius of a cylindrical tank with spherical ends. The point is to choose the sphere radius $$r$$ and the cylinder length $$h$$ that minimize the surface area of the tank while enclosing a specified volume. In terms of $$h$$ and $$r$$, the volume and surface area are:

$V(r, h) = \pi r^2 h + \frac{4}{3} \pi r^3 \ \ \ \text{and}\ \ \ A(r,h) = 2 \pi r h + 4 \pi r^2$

Part A Which of these is correct? (Hint: Only one of the answers is dimensionally consistent.)

1. $$h(r, V) = (V-4\pi r^3/3)/ \pi r^2$$
2. $$h(r, V) = (V-4\pi r^2/2)/ \pi r^2$$
3. $$h(r, V) = (V-4\pi r^3/3)/ \pi r^3$$
4. $$h(r, V) = \sqrt{(V-4\pi r^3/2)/\pi r^3}$$

Part B Which of these is the correct expression for $$A(r, V)$$

1. $$A(r, V) = 2 V/r + \frac{8 \pi}{3} r^2$$
2. $$A(r, V) = 2 V/r + \frac{4 \pi}{3} r^2$$
3. $$A(r, V) = V/r + \frac{20 \pi}{3} r^2$$
4. $$A(r, V) = V/r + 3 \pi r^2$$

Part C Find $$\partial_r A(r, V)$$ and set to zero. Solve for $$r^\star$$ in terms of $$V$$. Which of these is correct?

1. $$r^\star = \sqrt[3]{\frac{3}{4\pi} V}$$
2. $$r^\star = \sqrt[3]{\frac{4}{3\pi} V}$$
3. $$r^\star = \sqrt[3]{\frac{3\pi}{4} V}$$
4. $$r^\star = \sqrt[3]{\frac{3}{2\pi} V}$$

Find the optimum value of $$r$$ to minimize $$A(r,V)$$ when $$V = 1000$$ liters.

Part D What is the optimal value of $$r$$ in cm to a precision of one micron?

6.2035 46.0351 52.0351 62.0351

Use a sandbox to plot a graph of $$A(r, V)$$ versus r, setting $$V = 1000$$ liters.

Part E From the graph of $$A(r, V)$$ versus $$r$$ at $$V=1000$$ liters, read off a range of $$r$$ that produces $$A$$ no worse than 1% greater than the minimum. How wide is that range, approximately?

$$\pm 0.1$$cm $$\pm 1$$cm $$\pm 5$$cm $$\pm 10$$cm

1. Another word for an “input” is “argument.” Argmax is the contraction of argument producing the maximum output.↩


text:
Elastic Modulus

Written by Jerry Ratzlaff on . Posted in Classical Mechanics

Elastic modulus, abbreviated as $$\lambda$$, also known as modulus of elasticity, is the ratio of the stress applied to a body or substance to the resulting strain within the elastic limits.

Elastic Modulus formula

$$\large{ \lambda = \frac { \sigma } { \epsilon } }$$

Where:

$$\large{ \lambda }$$  (Greek symbol lambda) = elastic modulus

$$\large{ \epsilon }$$  (Greek symbol epsilon) = strain

$$\large{ \sigma }$$  (Greek symbol sigma) = stress

Solve for:

$$\large{ \epsilon = \frac { \sigma } { \lambda } }$$

$$\large{ \sigma = \lambda \; \epsilon }$$


text:
## Abstract

Electronic health records (EHRs), data generated and collected during normal clinical care, are increasingly being linked and used for translational cardiovascular disease research. Electronic health record data can be structured (e.g. coded diagnoses) or unstructured (e.g. clinical notes) and increasingly encapsulate medical imaging, genomic and patient-generated information. Large-scale EHR linkages enable researchers to conduct high-resolution observational and interventional clinical research at an unprecedented scale. A significant amount of preparatory work and research, however, is required to identify, obtain, and transform raw EHR data into research-ready variables that can be statistically analysed. This study critically reviews the opportunities and challenges that EHR data present in the field of cardiovascular disease clinical research and provides a series of recommendations for advancing and facilitating EHR research.

Over few past years, ‘big data’ has become a frequently used catchall phrase for research approaches involving the use of complex, large-scale datasets.1,2 There are many types of data that may fit this description, but within the sphere of clinically oriented research this term is often considered synonymous to electronic health record (EHR) data, or electronic medical record data. The powerful potential of these data for advancing biomedical research has been recognized by many.3–5 Funders in both the USA and the UK have recently made substantial investments in the area, such as the USD$215 million Precision Medicine Initiative6 announced by the US government and Genomics England, aiming to sequence 100 000 whole genomes during routine clinical care.7 Additionally, funding organizations are actively encouraging research utilizing large-scale biomedical data through specific initiatives. The Big Data To Knowledge programme8 was established by the National Institutes of Health (NIH) in 2012 to address the challenges and opportunities presented by big biomedical data through the provision of seed funding for biomedical data science-based research, methods, and training material development. In the UK, a consortium of 10 UK government and charity funders, led by the Medical Research Council have committed over £90 million across several initiatives that are aimed at supporting translational research using big data such as the national Farr Institute of Health Informatics Research,9 the UK Health Informatics Research Network, and the Medical Bioinformatics Initiative. The amount of EHR data being digitally generated and collected is vast and rapidly expanding, and presents multiple opportunities that have the potential to transform medical practice and cardiovascular research across all stages of translation. However, big data is not a panacea for all research problems, and for many researchers the path from big data to clinical impact for a specific research question is unclear. There are many factors that must be considered when planning to use EHR data for research related not just to ethical and policy issues raised by combining data sources10,11 but also the logistical and analytical decisions the process entails. One of the major impediments to the use of EHRs for research is that is the data they contain differs from data collected in a conventional cohort study or randomized controlled trial (RCT) in terms of both why and how it is recorded, and requires substantial processing before it can be statistically analysed. These data are generated and recorded throughout the patient pathway during interactions with primary, secondary, and tertiary healthcare providers. Data from specialized disease registries, which were originally set up for auditing clinical standards and benchmarking quality improvement initiatives, may also be incorporated. These different sources also record information in different ways. Electronic health record data can be structured [e.g. diagnosis recorded using medical classification systems such as the International Classification of Diseases—10th revision (ICD-10)12 or Systematic Nomenclature of Medicine – Clinical Terms (SNOMED-CT13)] or unstructured (e.g. textual narrative in clinical notes or coronary angiography reports in hospital information systems14). Electronic health records are also increasingly including cardiovascular imaging data from procedures such as echocardiography, angiography, magnetic resonance imaging, or computed tomography.15 For all sources of information, data collection will have been motivated by clinical care, administrative, or other reasons, and clinical information will be recorded using a variety of ways. The research-user is faced with substantial missing or incomplete information, data collected at irregular time-points, information that may be temporally inconsistent or conflicting, and potentially the task of integrating and harmonizing information contained in multiple sources. These challenges are not insurmountable and do not mean that EHR data cannot be widely used for research, but do require a clear identification of research areas that can best leverage EHR data, and the development of tools that smooth the path from research question to research result. ## Research opportunities well-placed to leverage electronic health record data ### High-resolution observational cohort studies Linkage of multiple EHR data sources permits the creation of large-scale cohorts of patients for whom extensive follow-up data are already available. This allows researchers to answer questions that reliance upon traditional investigator-led cohort studies would otherwise make impossible due to the scale, diagnostic resolution, timeframe, or cost. In addition, it allows researchers to define and examine the entire patient journey, from early presentations of non-acute manifestations through the various syndrome transitions to cardiac (or non-cardiac) death. This enables them to resolve the time sequence, examine and understand, the aetiological and prognostic differences between different coronary disease phenotypes.16 Chung et al.17 were able to take advantage of available EHR data in this way to conduct a comparative effectiveness study of acute coronary care on an international scale. Currently, Sweden and the UK are the only two countries in the world with ongoing, national registries for acute coronary syndrome events that cover all hospital care. Using these data, the authors showed that in the UK, compared with Sweden, 30-day mortality following acute myocardial infarction (AMI) was substantially higher, and that uptake of effective treatment was slower. The richness of the data meant that a substantial amount of clinical information could be incorporated into the casemix, including demography, risk factor comorbidity, and pre-hospital treatment. The researchers were also able to determine that diagnoses made in the two countries were comparable by examining troponin values and propensity to make a diagnosis. The results from this study are thus more robust than those based on a simple comparison of mortality rates, or focused on data from bespoke studies undertaken in hospitals that may not be representative of the broader healthcare system. Electronic health record cohorts can also be used to make timely contributions to debates of clinical importance, such as the controversy over the relationship between varenicline and adverse cardiovascular events. In 2011, a meta-analysis of 14 RCTs raised concerns that the use of varenicline for smoking cessation may increase risk for adverse cardiovascular events (ischaemia, arrhythmia, congestive heart failure, sudden death, or cardiovascular-related death).18 Three subsequent meta-analyses of RCTs did not find a significant association.19–21 However, the question remains controversial, partly due to disagreements over analytical methods used in these studies, but also because meta-analyses are limited to the analysis of existing studies.20,22,23 Svanström et al.24 were able to rapidly contribute new data to the debate by investigating the question in a cohort made up of the EHR data of over 35 000 Danish individuals who used either varenicline or bupropion for smoking cessation. In this observational study, published in 2012, there was no evidence for a higher number of adverse vascular events (acute coronary syndrome, ischaemic stroke, and cardiovascular death) in patients using varenicline. It would not have been feasible to take a comparable traditional cohort study from study design to publication within a similar timeframe, especially as very large number of patients would be required to ensure sufficient outcome numbers (only 117 were observed among the 35 000 patients in the EHR study). The capacity to investigate novel research questions has generally been limited by available data and funding for obtaining new data, but EHR data can potentially be used to address this problem. The relationship between auto-immune inflammatory conditions and atrial fibrillation (AF) is one example where EHR data have been able to fill a research niche. Although there is substantial research interest in this area,25 many of the large cardiovascular cohort studies (e.g. Framingham26) have limited data available on inflammatory conditions as this was not part of the original study design. However, researchers in the UK, USA, and Denmark have been able to use EHR resources to explore this research area using very large samples, finding associations with an increased risk of AF and a range of conditions including rheumatoid arthritis and psoriasis.27–30 Other researchers have taken an even broader, non-hypothesis-driven approach, using advanced computation techniques that consider any and all disease information available in EHR data to identify novel associations between diseases.31 The costs associated with using EHR data for these studies would have been much lower than comparable data collection, making them a cost-effective entry point into new areas of cardiovascular research. ### Enhanced clinical trials There is growing concern that the current model of discovering new interventions, evaluating them through RCTs, and implementing them in clinical care is significantly inefficient. The translation process itself is taking too long, with an average figure of 17 years reported in some cases.32 Additionally, the number of new drugs introduced to the market per year has been broadly flat since the 1950s yet the costs have steadily grown,33 and the cost of bringing a new licensed drug to the market has been estimated between USD$5 and 11 billion.34

In cardiovascular diseases (CVD), the problem is more acutely manifested through problems observed in the current clinical trials pipeline. There is a lack of contemporary and representative population data that can be utilized to draw accurate estimates of events and inform the selection of appropriate RCT primary and secondary endpoints. Clinical trials are often conducted in highly selected populations that are not necessarily representative of the populations presented in routine clinical care and as such, results obtained have limited generalizability and external validity.35 For example, the clinical characteristics, treatments, and inpatient outcomes of patients enrolled in a large trial of acute heart failure (Acute Study of Nesiritide in Decompensated Heart Failure) were found to be significantly different that those found in a contemporary disease registry.36 Furthermore, despite their growing importance in CVD research, non-drug interventions such as interventions based on clinical algorithms and decision support tools are not systematically evaluated through clinical trials since the process of randomization and outcome ascertainment is not seamlessly integrated into the clinical care pathway.

This has had a significantly negative impact on clinical trial conduct and findings. For example, recently there have been several late drug failures occurring within phase III clinical trials of therapeutic agents each costing several hundred million USD\$. HDL-cholesterol raising agents such as niacin, fibrates, and cholesteryl ester transfer protein failed to reduce all-cause mortality, coronary heart disease mortality, and myocardial infarction event rates in patients treated with statins.37 Likewise, heart rate lowering agents such as ivabradine when introduced to patients with stable coronary artery disease without clinical heart failure failed to improve cardiovascular mortality and non-fatal myocardial infarctions rates.38

There is growing optimism that EHR can enrich RCT design, delivery, and follow-up. Electronic health record can offer real-world phenotype-rich data that can directly inform trial design, enable the identification of optimal target populations, and offer accurate event rate estimates similar to those encountered in clinical care. The entire trial conduct pipeline, from recruitment at the point of care to randomization and adverse event capture can be integrated with routine clinical care enabling the cost-effective and efficient trialling of non-drug interventions. Additionally, EHR can provide richer contemporary data on trial participants at a fraction of the cost thus enabling the generalization of trial results to external populations.39

For example, the Thrombus Aspiration during ST-Segment Elevation Myocardial Infarction trial40 for assessing the clinical effect of routine intracoronary thrombus aspiration before primary percutaneous coronary intervention in patients with ST-segment elevation myocardial infarction recruited patients though the Swedish Coronary Angiography and Angioplasty Registry and utilizing national EHR and registry data for defining trial endpoints. Finally, EHR data provide valid, complete, long-term follow-up of phase III trials that would otherwise be too costly and complex to establish and too narrow in focus.41 While EHR offer a rich data-scaffolding for designing and implementing clinical trials, significant challenges still exist, mainly around information governance and recruitment of clinicians as outlined in the evaluation by van Staa et al.42,43

## Challenges in the pathway from electronic health record data to research results

Although the benefits of using EHR data for research are potentially large, the widespread use of EHR data is hampered by the fact that there are currently a number of additional steps, and many associated queries, in the pathway from research question to results and publication. As an example, consider a research project using existing data to investigate whether there is a relationship between gender and onset of AF. Most projects would involve applying standard analytical techniques to a bespoke investigator-led cohort of healthy individuals followed-up for cardiovascular conditions including AF (e.g. The Framingham Heart Study). For an existing dataset, only relatively minimal data preparation would be required before analyses could be conducted and data are often provided with detailed documentation. However, using EHR data to answer the same question would require a number of additional preparatory steps before statistical analyses could be conducted. Broadly speaking, these relate to: (i) identifying the EHR source(s) that contain the data needed for the research question; (ii) developing strategies for extracting the required information from the data source(s), and combining it where necessary; and (iii) creating a dataset that is ready for analysis using standard statistical techniques (see Figure 1).

Figure 1

Diagram of steps from research question to results and publication. The four central circles show the path from research question to results for a conventional study using existing data. Circles on outside of the spiral indicate the additional steps needed to conduct a research project using EHR data.

Figure 1

Diagram of steps from research question to results and publication. The four central circles show the path from research question to results for a conventional study using existing data. Circles on outside of the spiral indicate the additional steps needed to conduct a research project using EHR data.

### What electronic health record data sources are available?

The availability of diverse data sources, including EHR, is rapidly expanding, making the identification of relevant sources for a single project overwhelming. Selecting appropriate data sources for research is dependent upon knowledge of the patients included (e.g. inpatients, ambulatory care, and specialist treatment), the types of data recorded (e.g. diagnosis, prescriptions, test results, and procedures), and the format of those data (e.g. diagnostic codes, imaging, and free text), but often much of this can be difficult to determine in detail until data access has been granted. A recent Wellcome Trust report on the discoverability of EHR and other biomedical datasets for research44 found that for the vast majority of sources, no systematic method is used to capture, curate, and display information about the data contained in them, or to provide guidance on the information governance restrictions attached to them which determine how they can be accessed and used for research. The limited use of standardized methods (e.g. metadata) for describing such information hinders recognition of the limitations and opportunities these data sources present, and potentially results in under-utilization of data sources due to lack of knowledge about what they contain (Box 1).

Box 1
Definitions

Electronic health records. Electronic health records are data generated and recorded during routine clinical care. Electronic health records are diverse and encompass nationally and regionally available structured and unstructured data from primary care, hospitals, administrative data, and disease, procedure and death registries; increasingly including genomic, imaging, and patient sensor data.

Medical ontology. A structured controlled vocabulary of medical concepts and their semantic relations used to record, store, and transmit medical knowledge and patient-related clinical information efficiently.

Metadata. Metadata are data that describe aspects around a particular data element. For an EHR source, metadata can include information about the manner in which the data get generated and recorded, the medical ontologies used to record information, and the methods by which researchers can access the data for research.

Phenotyping. In the context of EHR, phenotyping is defined as the process of creating algorithms that define an observable trait (physical or biochemical) such as a clinical condition within EHR data.

However, overcoming this challenge is worth the additional effort, as combining data from multiple sources strengthen EHR-based cardiovascular research. For example, Herrett et al. explored the completeness of recording for AMI in four EHR sources: primary care (Clinical Practice Research Datalink; CPRD45), hospital admissions (Hospital Episode Statistics; HES46), a natonal MI disease registry (Myocardial Ischaemia National Audit Project47), and national mortality data (Office of National Statistics; ONS48). Compared with the disease registry, which was treated as the gold standard data source, none of the other data sources captured all MI events and consequently incidence rates based on data from a single source were underestimated by 25–50%.49 This finding is not limited to AMI; a similar investigation of AF diagnoses found that only ∼40% of the 72 793 AF patients identified had a diagnosis recorded in both primary and secondary care.50

Thus, for our example research question regarding gender and AF, we would likely decide to combine multiple EHR data sources, such as CPRD, HES, and ONS. This would enable us to use a sample of individuals broadly representative of the UK general population, and would include a more representative set of AF cases as diagnoses made in both primary and secondary care would be identified. However, individual access applications would need to be made for each EHR source prior to linkage of the different data sources, and information about what is contained within each would currently be limited to knowledge of the medical ontologies used to code clinical information.

### How can I define clinical conditions in electronic health record data?

Once the relevant data source(s) have been identified, researchers face another challenge: how to determine which patients have been diagnosed with a particular condition. Extracting phenotypic information (i.e. disease status), a process known as phenotyping, is a time-consuming and challenging task even in relation to a single data source, as multiple diagnosis codes may be used to describe similar or related conditions and their data. This challenge is amplified when data from multiple sources, recorded using different medical ontologies, are combined. Figure 2 illustrates this, using as an example data for one individual from the three EHR sources in our hypothetical research question. In this example, an AF diagnosis is recorded at three different time-points: as a secondary diagnosis during a hospital admission, in the primary care record after hospital admission information is transferred to their general practitioner, and as a primary diagnosis when the patient is admitted to hospital for an AF-related surgical procedure. This information needs to be reconciled in order to determine not only if, but also when, a diagnosis occurred.

Figure 2

Illustration of linked primary care data (Clinical Practice Research Datalink; CPRD), secondary care data (Hospital Episode Statistics; HES), and mortality data (Office of National Statistics; ONS) for a single patient. Circles on the top line show events recorded in one or more sources; red circles indicate a diagnosis. DVT, deep vein thrombosis; INR, international normalization ratio; AF, atrial fibrillation; HF, heart failure.

Figure 2

Illustration of linked primary care data (Clinical Practice Research Datalink; CPRD), secondary care data (Hospital Episode Statistics; HES), and mortality data (Office of National Statistics; ONS) for a single patient. Circles on the top line show events recorded in one or more sources; red circles indicate a diagnosis. DVT, deep vein thrombosis; INR, international normalization ratio; AF, atrial fibrillation; HF, heart failure.

Reconciling coded information from multiple sources is made more challenging by the different medical ontologies that are used by each source. For example, in the UK, primary care sources use Read codes, a subset of the SNOMED-CT clinical terminology, whereas secondary care and mortality sources use the ICD-10. Combining data recorded using these systems for a single condition, such as AF, is not straightforward as the clinical resolution they offer can vary substantially; there are 23 Read codes relating to AF, including disease subtype classification, but only 1 ICD-10 code. Data-driven computational methodologies, such as support vector machines (SVMs), can be applied on unstructured data (e.g. clinical text, electrocardiographic (ECG) monitoring data) to further enhance and fine-tune the accuracy of algorithms utilizing coded data.4,51,52 For example, Mohebbi and Ghassemian53 created an algorithm that consists of a linear discriminant analysis based feature reduction scheme and a SVM-based classifier and were able to accurately (sensitivity 99.07%, specificity 100%, and positive predictive value 100%) detect AF cases using RR intervals extracted from ECG signals.

No standardized methodologies and mechanisms exist to help research-users define, share, and evaluate EHR-derived phenotypes in a consistent way, or to apply algorithms for creating these phenotypes to their own data, although development of tools for this is very active.54–56 The USA-based Electronic Medical Records and Genomics (eMERGE) Consortium has developed an AF phenotype algorithm57 that focuses on clinical notes and electrocardiogram impression data. These data are not available in CPRD, HES, or ONS, although there is a UK-oriented EHR phenotype resource called CALIBER that does contain an AF phenotype based on coded data from primary and secondary care,50 which could be applied in this situation. However, if no phenotype algorithm existed, we would need to go through the process of developing a new phenotype algorithm for AF, and we would need to repeat this process for every other variable we wanted to include in our final dataset such as gender and any covariates such as other CVD, smoking status, or hypertension.

Validation, preferably against a gold standard, is a key step of defining disease phenotyping algorithms.58 The goal of the validation exercise is to evaluate the accuracy of the algorithm: it is the phenotyping algorithm including all patients that are eligible and excluding all patients that are ineligible, thus accurately allocating them in the case and control groups. Some phenotypes, such as type 2 diabetes,59 are inherently complex as they make use of multiple data elements (e.g. diagnostic codes, medication information, laboratory measurements, and clinical text) and should ideally be validated through manual review of case notes in primary or secondary healthcare providers in order to understand the information the physician had available at the time of diagnosis. Clinical notes, however, are not available at scale due to information governance restrictions and scaling this process for large cohorts of patients is challenging and time-consuming. An alternative approach is to validate the developed phenotyping algorithms by conducting epidemiological analyses of the association of known risk factors and the phenotype in question and compare with associations found in other studies. Other phenotypes, such as white blood cell count, the goal of the validation exercise is to ensure that the algorithm included all eligible patients and discarded outliers and incorrect values.

### How do I create a research-ready electronic health record dataset?

The process of applying phenotype algorithms to raw EHR data and creating a dataset that is ready to be statistically analysed requires several data transformations that are challenging due to data heterogeneity and complexity. Description of the process is rarely provided as part of academic outputs, and there is increasing recognition of the weaknesses that pervade the current landscape of EHR research in relation to sharing and standardization of data transformation methods.60 The prevalent scientific culture does not promote or reward sharing of standardized and re-usable data transformation libraries, which leads to substantial duplication of effort and increases the potential for a lack of reproducible results from EHR-based studies.

As for a conventional study, an EHR-based study requires a clear definition including the population from which individuals are sampled, inclusion and exclusion criteria, follow-up, and handling of missing data. For our example question, we may need to specify the age range of our patients, whether we are including individuals with prior cardiovascular conditions such as heart failure, and how missing data were handled, but there is additional information that should be reported for EHR data including: the data sources included, the end date of our follow-up data, whether there are exclusion/inclusion criteria based on data quality or other administrative information, details of new phenotype algorithms, and how data were multiply imputed if applicable. While this information can be described to some extent in the Methods section of a scientific paper, the associated computational manipulation and analyses are not standardized for EHR data, and there is currently no provision in scientific papers for detailed explanations of these methods or distribution of associated phenotype algorithms, computer software, or statistical/programming scripts.

## Recommendations for advancing electronic health record research

Many countries in Europe, and internationally, have EHR systems that could be utilized for research; national, centralized resources that facilitate the steps from research question to research dataset would substantially enhance the research potential of these data sources. Initiatives are already underway to achieve this in some countries, but few tackle all aspects of this process.

The UK-based CALIBER platform61 combines a repository of EHR phenotypes with curated record linkages combining primary care (Clinical Practice Research Datalink), hospital discharge (Hospital Episode Statistics), disease registry (Myocardial Ischaemia National Audit Project47), and death registry (Office of National Statistics) data in over 2 million adults with 10 million person years of follow-up. However, this resource does not provide any tools for bidirectional interactions with EHR data sources. In contrast, the Clinical Record Interactive Search system (based at the NIHR Mental Health Biomedical Research Centre and Dementia Unit at the South London and Maudsley NHS Foundation Trust) allows researchers to investigate anonymized secondary care data, including clinical notes and other text, via novel user-friendly tools that facilitate identification of patients meeting certain criteria and development of text-mining algorithms.62 Finally, the eMERGE Network,54 a US National Human Genome Research Institute-funded consortium, combines a phenotype repository with EHR data from multiple secondary healthcare providers, including imaging and text, linked to genotypic data for all participants.

National EHR portals could combine the strengths of all these projects by including: (i) a national catalogue of contemporary EHR sources curated using metadata standards; (ii) an interactive thesaurus of EHR-derived phenotype algorithms; (iii) standards-driven tools that will enable researchers to visually create observational and interventional research studies (population, inclusion/exclusion criteria, sources, phenotypes). The national catalogue should support the harvesting and integration of metadata from external sources, and manual curation by researchers within a standardized and reproducible framework, as well as providing guidance on data access and content. This will allow users to identify data sources that can provide information both within and across disease areas. The EHR phenotype algorithms and dataset creation tools need to be implemented in a fashion that supports reuse and modification by other users, as well as appropriate academic credit and/or citation. Creating this type of resource will help to foster an ‘open source’ approach to EHR research in which researchers can collaborate and learn from each other, and this will ultimately produce a greater advance in EHR research than could be achieved by any research group in isolation.

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