High school mathematics at the research frontier
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High School Mathematics at the Research Frontier. Don Lincoln Fermilab. http://www-d0.fnal.gov/~lucifer/PowerPoint/HSMath.ppt. What is Particle Physics?. High Energy Particle Physics is a study of the smallest pieces of matter.

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High school mathematics at the research frontier

High School Mathematics at the Research Frontier

Don Lincoln

Fermilab

http://www-d0.fnal.gov/~lucifer/PowerPoint/HSMath.ppt


What is particle physics

What is Particle Physics?

High Energy Particle Physics is a study of the smallest pieces of matter.

It investigates (among other things) the nature of the universe immediately after the Big Bang.

It also explores physics at temperatures not common for the past 15 billion years (or so).

It’s a lot of fun.


High school mathematics at the research frontier

Fermilab

4x10-12

seconds

Stars form

(1 billion years)

Now

(15 billion years)

Atoms form

(300,000 years)

Nuclei form

(180 seconds)

Nucleons form

(10-10 seconds)

??? (Before that)


D detector run ii

DØ Detector: Run II

  • Weighs 5000 tons

  • Can inspect 3,000,000 collisions/second

  • Will record 50 collisions/second

  • Records approximately 10,000,000 bytes/second

  • Will record 1015(1,000,000,000,000,000) bytes in the next run (1 PetaByte).

30’

30’

50’


Remarkable photos

Remarkable Photos

In this collision, a top and

anti-top quark were created,

helping establish their existence

This collision is the most violent

ever recorded. It required that

particles hit within 10-19 m or

1/10,000 the size of a proton


How do you measure energy

How Do You Measure Energy?

  • Go to Walmart and buy an energy detector?

  • Ask the guy sitting the next seat over and hope the teacher doesn’t notice?

  • Ignore the problem and spend the day on the beach?

  • Design and build your equipment and calibrate it yourself.


Build an electronic scale

150 lbs

?? Volts

Volts are a unit of electricity

Car battery = 12 Volts

Walkman battery = 1.5 Volts

Build an Electronic Scale


Calibrating the scale

Make a line, solve slope and intercept

y = m x + b

Voltage = (0.05) weight + 3

Implies

Weight = 20 (Voltage – 3)

This implies that you can know

the voltage for any weight.

For instance, a weight of 60 lbs

will give a voltage of 6 V.

Now you have a calibrated scale.

(Or do you?)

Calibrating the Scale


Issues with calibrating

Issues with calibrating.

All four of these functions go through the two calibration points. Yet all give very different predictions for a weight of 60 lbs.

What can we do to resolve this?


Approach take more data

Easy

Hard

Approach: Take More Data


Solution pick two points

Solution: Pick Two Points

Dreadful representation of data


Solution pick two points1

Solution: Pick Two Points

Better, but still poor, representation of data


Why don t all the data lie on a line

Why don’t all the data lie on a line?

  • Error associated with each calibration point.

  • Must account for that in data analysis.

  • How do we determine errors?

  • What if some points have larger errors than others? How do we deal with this?


First retake calibration data

First Retake Calibration Data

  • Remeasure the 120 lb point

  • Note that the data doesn’t always repeat.

  • You get voltages near the 9 Volt ideal, but with substantial variation.

  • From this, estimate the error.


High school mathematics at the research frontier

Data

While the data clusters around 9 volts, it has a range. How we estimate the error is somewhat technical, but we can say

9  1 Volts


Redo for all calibration points

Redo for All Calibration Points


Redo for all calibration points1

Redo for All Calibration Points


High school mathematics at the research frontier

Both lines go through the data.

How to pick the best one?


State the problem

State the Problem

  • How to use mathematical techniques to determine which line is best?

  • How to estimate the amount of variability allowed in the found slope and intercept that will also allow for a reasonable fit?

  • Answer will be m Dm and b Db


The problem

Looks Intimidating!

The Problem

  • Given a set of five data points, denoted (xi,yi,si) [i.e. weight, voltage, uncertainty in voltage]

  • Also given a fit function f(xi) = m xi + b

  • Define


Forget the math what does it mean

Forget the math, what does it mean?

Each term in the sum is simply the separation between the data and fit in units of error bars. In this case, the separation is about 3.

f(xi)

yi - f(xi)

si

yi

xi


More translation

More Translation

So

Means

Since f(xi) = m xi + b, find m and b that minimizes the c2.


Approach

Calculus

Approach

Find m and b that minimizes c2

Back to algebra

Note the common term (-2). Factor it out.


Approach 2

Move terms to LHS

Factor out m and b terms

Rewrite as separate sums

Approach #2

Now distribute the terms


High school mathematics at the research frontier

Approach #3

Substitution

Note the common term in the denominator

Notice that this is simply two equations with two unknowns. Very similar to

You know how to solve this


High school mathematics at the research frontier

ohmigod….

yougottabekiddingme

So each number

isn’t bad


Approach 4

Approach #4

Inserting and evaluating, we get

m = 0.068781, b = 0.161967

What about significant figures?

2nd and 5th terms give biggest contribution to c2 = 2.587


Best fit

Best Fit


Best vs good

Best

Best vs. Good


Doesn t always mean good

Doesn’t always mean good


Goodness of fit

A new hypothetical set of data with the best line (as determined by the same c2 method) overlaid

Goodness of Fit

Our old buddy, in which the data and the fit seem to agree


New important concept

New Important Concept

  • If you have 2 data points and a polynomial of order 1 (line, parameters m & b), then your line will exactly go through your data

  • If you have 3 data points and a polynomial of order 2 (parabola, parameters A, B & C), then your curve will exactly go through your data

  • To actually test your fit, you need more data than the curve can naturally accommodate.

  • This is the so-called degrees of freedom.


Degrees of freedom dof

Degrees of Freedom (dof )

  • The dof of any problem is defined to be the number of data points minus the number of parameters.

  • In our case,

  • dof = 5 – 2 = 3

  • Need to define the c2/dof


Goodness of fit1

c2/dof = 22.52/(5-2) = 7.51

Goodness of Fit

c2/dof near 1 means the fit is good.

Too high  bad fit

Too small  errors were over estimated

Can calculate probability that data is represented by the given fit. In this case:

Top: < 0.1%

Bottom: 68%

In the interests of time, we will skip how to do this.

c2/dof = 2.587/(5-2) = 0.862


Uncertainty in m and b 1

Recall that we found

m = 0.068781, b = 0.161967

What about uncertainty and significant figures?

If we take the derived value for one variable (say m), we can derive the c2 function for the other variable (b).

The error in b is indicated by the spot at which the c2 is changed by 1.

So  0.35

Uncertainty in m and b #1


Uncertainty in m and b 2

Recall that we found

m = 0.068781, b = 0.161967

What about uncertainty and significant figures?

If we take the derived value for one variable (say b), we can derive the c2 function for the other variable (m).

Uncertainty in m and b #2

The error in m is indicated by the spot at which the c2 is changed by 1.

So  0.003


Uncertainty in m and b 3

So now we know a lot of the story

m = 0.068781  0.003

b = 0.161967  0.35

So we see that significant figures are an issue.

Finally we can see

Uncertainty in m and b #3

Voltage = (0.069  0.003) × Weight + (0.16  0.35)

Final complication: When we evaluated the error for m and b, we treated the other variable as constant. As we know, this wasn’t correct.


Error ellipse

c2min + 1

c2min + 2

c2min + 3

b

Best b & m

More complicated, but shows that uncertainty in one variable also affects the uncertainty seen in another variable.

m

Error Ellipse


High school mathematics at the research frontier

Increase intercept, keep slope the same

Increase intercept, keep slope the same

To remain ‘good’, if you increase the intercept, you must decrease the slope


High school mathematics at the research frontier

Decrease slope, keep intercept the same

Similarly, if you decrease the slope, you must increase the intercept


Error ellipse1

From both physical principles and strict mathematics, you can see that if you make a mistake estimating one parameter, the other must move to compensate. In this case, they areanti-correlated(i.e. if b, then m and if b, then m.)

b

m

Error Ellipse

Best b & m

new b

within errors

bbest

When one has an m below mbest, the range of preferred b’s tends to be above bbest.

mbest

new m

within errors


Back to physics

Data and error analysis is crucial, whether you work in a high school lab…

Back to Physics


High school mathematics at the research frontier

Or the Frontier!!!!


References

References

  • P. Bevington and D. Robinson, Data Reduction and Error Analysis for the Physical Sciences, 2nd Edition, McGraw-Hill, Inc. New York, 1992.

  • J. Taylor, An Introduction to Error Analysis, Oxford University Press, 1982.

  • Rotated ellipses

    • http://www.mecca.org/~halfacre/MATH/rotation.htm


High school mathematics at the research frontier

http://www-d0.fnal.gov/~lucifer/PowerPoint/HSMath.ppt


High school mathematics at the research frontier

http://worldscientific.com/books/physics/5430.html


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