University Physics IV · topic UPIV-15.09
Reading a frontier claim, and what remains open
Connect the ideas, make your working visible, and test the result. Use the hints when you need a starting point; open the full solution after your attempt.
Further practice
Work through a fresh case before tackling the extension.
Reading a frontier claim: predict a finite change
A frontier claim is a statement about data together with a stated probability of having been fooled by background alone; until an independent experiment with different systematics reproduces it, that probability is the entire content of the claim. Use the supplied topic model Y_out = min(2500/max(s^2-25*f^2,2.5),999), where Y_out is EXPOSURE TO REACH 5σ (units). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; min(a,b) selects the smaller value; max(a,b) selects the larger value. s is Signal events per unit exposure, measured in events/unit. Keep f (Systematic on the background) = 3 %; t (Exposure collected) = 6 units fixed. For this exercise use only 15.15 events/unit ≤ s ≤ 36.2813 events/unit on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. Calculate Y_out at s = 19.7989 events/unit and 22.7572 events/unit. Find the signed output change ΔY_out and the average sensitivity ΔY_out/Δs across this interval. State whether the response increases or decreases; distinguish this interval average from a derivative at one point.
EXPOSURE TO REACH 5σ against s
Predictions from the supplied Reading a frontier claim model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): EXPOSURE TO REACH 5σ / units
Horizontal axis (x): s: Signal events per unit exposure / events/unit
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Signal events per unit exposure / events/unit | EXPOSURE TO REACH 5σ / units |
|---|---|
| 15.149999999999999 | 552.791597567723 |
| 20.432812499999997 | 12.987024681396521 |
| 25.715625 | 5.730089377531215 |
| 30.9984375 | 3.3971862691928205 |
| 36.28125 | 2.2907846921892294 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
Substitute each value of s separately while keeping every other input fixed, then subtract in the same order.
Reveal the worked solution
- At s = 19.7989 events/unit, substitute into Y_out = min(2500/max(s^2-25*f^2,2.5),999): Y₀ = 14.9705 units.
- At s = 22.7572 events/unit, the same fixed-parameter model gives Y₁ = 8.53556 units.
- Signed change: ΔY_out = Y₁ − Y₀ = -6.43491 units; the input change is Δs = 2.95838 events/unit.
- Average sensitivity is ΔY_out/Δs = -2.17515 units per events/unit. Its negative sign means that this response decreases over the specified interval.
- This is a secant slope between two settings. It equals a point derivative only for a locally linear relation or in a justified small-interval limit.
Answer Y₀ = 14.9705 units, Y₁ = 8.53556 units; ΔY_out = -6.43491 units; average sensitivity = -2.17515 units per events/unit.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.
Extra challenging questions
Work through the linked stages and explain what each result means.
Reading a frontier claim: infer an input with uncertainty
A frontier claim is a statement about data together with a stated probability of having been fooled by background alone; until an independent experiment with different systematics reproduces it, that probability is the entire content of the claim. Use the supplied topic model Y_out = min(2500/max(s^2-25*f^2,2.5),999), where Y_out is EXPOSURE TO REACH 5σ (units). The output symbol Y_out is distinct from all input symbols. ^ denotes a power; π = 3.141592653589793; min(a,b) selects the smaller value; max(a,b) selects the larger value. s is Signal events per unit exposure, measured in events/unit. Keep f (Systematic on the background) = 3 %; t (Exposure collected) = 6 units fixed. For this exercise use only 15.15 events/unit ≤ s ≤ 36.2813 events/unit on this continuous branch. A grid of test settings is an input restriction for this exercise, not by itself a claim of physical quantisation. This calculation retains the topic model's stated assumptions; it does not extend them to a different physical setup. A reading gives Y_out = 4.4543818 units, with a bounded reading uncertainty ±0.32454 units. Treat the model and all fixed inputs as exact for this exercise; the bound is not a standard deviation. First estimate the unknown s by straight-line interpolation between the two endpoint predictions. Then solve the actual model on the stated branch to an absolute input tolerance of 0.00211312 events/unit, using a numerical bracket if required. Find the full input interval consistent with the bounded reading uncertainty and compare it with a symmetric input-error assumption. Do not assume that the supplied calibration points prove monotonicity outside this branch.
EXPOSURE TO REACH 5σ against s
Predictions from the supplied Reading a frontier claim model, with all other inputs fixed as stated in the question. 5 calibration points support a trend comparison; they are not measured data or a straight-line law.
Vertical axis (y): EXPOSURE TO REACH 5σ / units
Horizontal axis (x): s: Signal events per unit exposure / events/unit
Axes show the supplied calibration interval; the origin may lie outside the plot.
Read the exact data as a table
| s: Signal events per unit exposure / events/unit | EXPOSURE TO REACH 5σ / units |
|---|---|
| 15.149999999999999 | 552.791597567723 |
| 20.432812499999997 | 12.987024681396521 |
| 25.715625 | 5.730089377531215 |
| 30.9984375 | 3.3971862691928205 |
| 36.28125 | 2.2907846921892294 |
These are the supplied values, not a worked answer. No fitted trend is drawn.
Show a hint
The model's endpoint predictions bracket the reading. Solve Y_out(s) = Y_measured and then solve again at Y_measured − u and Y_measured + u; a nonlinear inverse can give unequal input deviations.
Reveal the worked solution
- Endpoint predictions are Y_out(15.15) = 552.792 units and Y_out(36.2813) = 2.29078 units. The target lies between them. On the selected branch the response is strictly decreasing.
- Endpoint interpolation gives s_line = 15.15 + (4.45438 − 552.792)(36.2813 − 15.15)/(2.29078 − 552.792) = 36.1982 events/unit. This estimates the inverse of the endpoint chord, rather than assuming that chord is the physical model.
- Use the actual equation, not the chord, and bracket its root. Evaluating the midpoint and retaining the sign-changing half-interval gives s = 28.0401 events/unit; substituting back gives Y_out = 4.4543818 units.
- The interpolation error in the input is |s_line − s| = 8.15814 events/unit. This is larger than the requested numerical tolerance.
- Invert both limits Y_out = 4.1298418 units and 4.7789218 units. Sort the roots even if the response decreases: 27.352 events/unit ≤ s ≤ 28.8158 events/unit.
- The lower and upper input deviations are 0.688087 events/unit and 0.775733 events/unit. They differ because the inverse response is nonlinear; a symmetric error bar would only be a local approximation. This interval includes reading uncertainty only, not model error or uncertainty in fixed inputs.
Answer s ≈ 28.0401 events/unit; compatible interval [27.352, 28.8158] events/unit. Endpoint interpolation gives 36.1982 events/unit.
Compare the assumptions, method and units with your attempt. Explain any correction before trying another question.