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**Our
Research**

Home KamLAND Experiment Our Research Results About Me

This research develops an empirical approximation of the appearance probability without assuming neutrino (or anti-neutrino) oscillations and takes into account most of the reactor distances.

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**Incorporating Multiple Reactor Distances**

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Number of Ep Counts equation: (For a derivation, look at the PowerPoint presentation attached in the Home page)

After a change of variables:

where, N =

Q =

P =

From this, we get:

After minimizing chi square, a way to invert the Q matrix is by:

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**Forming the Q matrix:**

• Test for l values ranging from 9-550 km per MeV

• If the Eν lies between 1.8 MeV-10 MeV, then the values are plugged into the Q equation

• If the Eν lies outside of that range, it does not contribute to the detector, so zero is inputted for that matrix element

• Obtain a different Q matrix for each reactor

• Superpose all the Q matrices

**Binning:**

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** **Why bin the Eps?

• The greater the counts per bin, the smaller the relative error

** **Why bin the l's?

• More functions than unknowns

• A higher sum in each l column will provide for a smaller error

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**Definitions:**

• Q0 = 76x1083 Q

• P0 = 1083x1 (no oscillations)

• N0 = 76x1

• Q’0 = 17x11 binned Q0

• N’0 = 17x1 binned N0

• P1 = 1083x1 (using oscillation formula)

• N1 = 76x1 (test N or real events data N)

• N’1 = 17x1 binned N1

• C is boxed

• Y is what is outside the box in the same equation as C, excluding N and P

• R is a matrix containing the orthonormal eigenvectors for each eigenvalue

• D is a diagonal matrix containing all the eigenvalues of C

• D tilde inverse has the smallest eigenvalue element equal to zero

**The Test**

**Why
omit the smallest eigenvalue?**

Hence, the smaller the eigenvalues get, the bigger the noise error gets

**Average
Estimated P(l) of the KamLAND Equation**

This program is funded by the National Science Foundation through grant number PHY-1157044. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.