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
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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).
• Its a lot of fun.

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• Now
• (15 billion years)

Stars form (1 billion years)
Atoms form (300,000 years)
Nuclei form (180 seconds)
Nucleons form (10-10 seconds)
??? (Before that)
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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).

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30
50
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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
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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 doesnt notice?
• Ignore the problem and spend the day on the
  beach?
• Design and build your equipment and calibrate it
  yourself.

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Build an Electronic Scale
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Calibrating the Scale
120 lb girl 9 V ? (120, 9)
180 lb guy 12 V ? (150, 12)
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Issues with calibrating.
Fit Value at 60 lb
Purple 6
Blue 10
Red -70
Green 11.5
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?
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Approach Take More Data
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Solution Pick Two Points
Dreadful representation of data
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Solution Pick Two Points
Better, but still poor, representation of data
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Why dont 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?

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First Retake Calibration Data

• Remeasure the 120 lb point
• Note that the data doesnt always repeat.
• You get voltages near the 9 Volt ideal, but
  with substantial variation.
• From this, estimate the error.

Attempt Voltage
1 9.26
2 9.35
3 9.08
4 8.72
5 8.58
6 9.02
7 9.25
8 8.86
9 8.94
10 9.12
11 8.72
12 9.33
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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
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Redo for All Calibration Points
Weight Voltage
60 4.2 ? 0.5
120 9.4 ? 1.0
150 10 ? 0.7
180 13.2 ? 1.2
300 13.2 ? 8.4
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Redo for All Calibration Points
Weight Voltage
60 4.2 ? 0.5
120 9.4 ? 1.0
150 10 ? 0.7
180 13.2 ? 1.2
300 13.2 ? 8.4
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Both lines go through the data. How to pick the
best one?
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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

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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

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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
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More Translation
So
Means
Since f(xi) m xi b, find m and b that
minimizes the c2.
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Approach
Find m and b that minimizes c2
Back to algebra Note the common term (-2).
Factor it out.
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Approach 2
Now distribute the terms
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Approach 3
Notice that this is simply two equations with two
unknowns. Very similar to
You know how to solve this
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ohmigod. yougottabekiddingme
So each number isnt bad
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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
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Best Fit
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Best vs. Good
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Doesnt always mean good
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Goodness of Fit
Our old buddy, in which the data and the fit seem
to agree
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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.

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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

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Goodness of Fit
c2/dof 22.52/(5-2) 7.51
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 lt 0.1 Bottom
68 In the interests of time, we will skip how
to do this.
c2/dof 2.587/(5-2) 0.862
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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).
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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).
The error in m is indicated by the spot at which
the c2 is changed by 1. So ? 0.003
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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
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 wasnt correct.
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Error Ellipse
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Error Ellipse
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 are
anti-correlated (i.e. if b?, then m? and if b?,
then m?.)
Best b m
new b within errors
bbest
When one has an m below mbest, the range of
preferred bs tends to be above bbest.
mbest
new m within errors
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Back to Physics
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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

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http//www-d0.fnal.gov/lucifer/PowerPoint/HSMath.
ppt
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http//worldscientific.com/books/physics/5430.html