Read the equation
The first two terms are valid ring-Laplacian Fourier eigenvalues. Adding the position-varying diagonal cosine term does not in general produce eigenvalues of their sum. Independently reconstructing the default matrix and applying NumPy eigvalsh gives maximum sorted disagreement about 0.06062893; even the sum of the claimed values differs from the matrix trace.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| lambda_intra | Source’s claimed eigenvalue at index n |
| alpha | Ring coupling |
| kappa_intra | Modulation amplitude |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
int next = (i + 1) % NUM_STRANDS;
M_[i][prev] = params_.alpha;
M_[i][next] = params_.alpha;
}
}
// Analytical eigenvalues (Section 11.1):
// λ_intra(n) = -2α + 2α·cos(2πn/16) + κ_intra·cos(2πn/10.6)
double eigenvalue(int n) const {
double cos_16 = std::cos(2.0 * M_PI * n / NUM_STRANDS);
double cos_10_6 = std::cos(2.0 * M_PI * n / BASE_PAIRS_PER_TURN);
return -2.0 * params_.alpha + 2.0 * params_.alpha * cos_16 +
params_.kappa_intra * cos_10_6;
}
// Compute all eigenvalues
std::array<double, NUM_STRANDS> all_eigenvalues() const {
std::array<double, NUM_STRANDS> eigs;Exact original notation
λ_intra(n) = -2α + 2α·cos(2πn/16) + κ_intra·cos(2πn/10.6)